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adiabatic quantum computation | https://ncatlab.org/nlab/source/adiabatic+quantum+computation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By *adiabatic quantum computation* one means models of [[quantum computation]] on [[parameterized quantum systems]] where the [[quantum gates]] are [[unitary transformations]] on a [[topological phase of matter|gapped]] (and possibly [[topological order|topologically ordered]]) [[ground state]] which are induced, via the [[quantum adiabatic theorem]], by sufficiently slow movement of external [[dependent linear type theory|parameters]].
Often the term *adiabatic quantum computation* is used by default for [[optimization theory|optimization]] problems ("[[quantum annealing]]", see the references [below](#QuantumAnnealingReferences)).
On the other hand, the possibly most prominent example of adiabatic quantum computation is often not advertized as such (but see [CLBFN 2015](#CLBFN15)), namely [[topological quantum computation]] by adiabatic [[braid group|braiding]] of [defect anyons](braid+group+statistics#AsBraidingOfDefects) (whose positions is the external parameter, varying in a [[configuration space of points]]). This is made explicit in [Freedman, Kitaev, Larsen & Wang 2003, pp. 6](#FreedmanKitaevLarsenWang03); [Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6)](#NayakSimonSternFreedman08); and [Cheng, Galitski & Das Sarma 2011, p. 1](#ChengGalitskiDasSarma11); see also [Arovas, Schrieffer, Wilczek & Zee 1985, p. 1](#ArovasSchriefferWilczekZee85) and [Stanescu 2020, p. 321](#Stanescu20); [Barlas & Prodan 2020](#BarlasProdan20).
The following graphics shows this with labelling indicative of [momentum-space anyons](braid+group+statistics#ReferencesAnyonicBraidingInMomentumSpace):
\begin{imagefromfile}
"file_name": "AdiabaticBraidingOfBandNodes-220604.jpg",
"width": 600,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
> (graphics from [[schreiber:Topological Quantum Computation in TED-K|SS22]])
## Related concepts
* [[quantum adiabatic theorem]]
* [[Berry phase]]
* [[quantum computation]]
* [[measurement-based quantum computation]]
* [[topological quantum computation]]
## References
{#References}
### General
{#ReferencesGeneral}
* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Michael Sipser, *Quantum Computation by Adiabatic Evolution* [[arXiv:quant-ph/0001106](https://arxiv.org/abs/quant-ph/0001106)]
* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Joshua Lapan, Andrew Lundgren, Daniel Preda, *A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem*, Science **292** 5516 (2001) 472-475 [[doi:10.1126/science.1057726](https://doi.org/10.1126/science.1057726)]
* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]
Review:
* [[Andrew Childs]], *Overview of adiabatic quantum computation*, talk at *[CIFAR](https://cifar.ca/research-programs/quantum-information-science/) Workshop on Quantum Information Processing* (2013) [[pdf](https://www.cs.umd.edu/~amchilds/talks/cifar13-tutorial.pdf), [[Childs-AdiabaticQuantumComputation.pdf:file]]]
On robustness of adiabatic quantum computation (such as against [[decoherence]]):
* {#ChildsFarhiPreskill02} [[Andrew Childs]], [[Edward Farhi]], [[John Preskill]], *Robustness of adiabatic quantum computation*, Phys.Rev. A **65** (2002) 012322 [[arXiv:quant-ph/0108048](https://arxiv.org/abs/quant-ph/0108048), [doi:10.1103/PhysRevA.65.012322](https://doi.org/10.1103/PhysRevA.65.012322)]
### In optimization -- quantum annealing
{#QuantumAnnealingReferences}
Review with focus on optimization problems ([[quantum annealing]]):
* Catherine C. McGeoch, *Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice* Synthesis Lectures on Quantum Computing (2014) [[doi:10.2200/S00585ED1V01Y201407QMC008)](https://doi.org/10.2200/S00585ED1V01Y201407QMC008)]
* Tameem Albash, Daniel A. Lidar, *Adiabatic Quantum Computing*, Rev. Mod. Phys. **90** (2018) 015002 [[arXiv:1611.04471](https://arxiv.org/abs/1611.04471), [doi:10.1103/RevModPhys.90.015002](https://doi.org/10.1103/RevModPhys.90.015002)]
* Salvador E. Venegas-Andraca, William Cruz-Santos, Catherine McGeoch, Marco Lanzagorta, *A cross-disciplinary introduction to quantum annealing-based algorithms*, Contemporary Physics **59** 02 (2018) 174-196 [[arXiv:1803.03372](https://arxiv.org/abs/1803.03372), [doi:10.1080/00107514.2018.1450720](https://doi.org/10.1080/00107514.2018.1450720)]
* Erica K. Grant and Travis S. Humble, *Adiabatic Quantum Computing and Quantum Annealing*, Oxford research Encyclopedia (2020) [[doi:10.1093/acrefore/9780190871994.013.32](https://doi.org/10.1093/acrefore/9780190871994.013.32)]
* Atanu Rajak, Sei Suzuki, Amit Dutta, Bikas K. Chakrabarti, *Quantum Annealing: An Overview*, Philos Trans A Math Phys Eng Sci **381** 2241 (2023) 20210417 [[arXiv:2207.01827](https://arxiv.org/abs/2207.01827), [doi:10.1098/rsta.2021.0417](https://doi.org/10.1098/rsta.2021.0417)]
See also:
* Wikipedia, *[Adiabatic quantum computation](https://en.wikipedia.org/wiki/Adiabatic_quantum_computation)*
* Kristen L. Pudenz, Tameem Albash, Daniel A. Lidar: *Error-corrected quantum annealing with hundreds of qubits*, Nature Communications **5** 3243 (2014) [[doi:10.1038/ncomms4243](https://doi.org/10.1038/ncomms4243)]
> (with [[quantum error correction]])
* Minjae Jo, Michael Hanks, M. S. Kim, *Divide-and-conquer embedding for QUBO quantum annealing* [[arXiv:2211.02184](https://arxiv.org/abs/2211.02184)]
* Yusuke Kimura, Hidetoshi Nishimori, *Rigorous convergence condition for quantum annealing*, J. Phys. A: Math. Theor. **55** (2022) 435302 [[arXiv:2207.12096](https://arxiv.org/abs/2207.12096), [doi:10.1088/1751-8121/ac9dce](https://doi.org/10.1088/1751-8121/ac9dce)]
A more high-brow mathematical desription via "tangle machines":
* Avishy Y. Carmi, [[Daniel Moskovich]], §5 of: *Tangle Machines*, Proc. R. Soc. A **471** (2015) 20150111 [[arXiv:1404.2862](https://arxiv.org/abs/1404.2862), [doi:10.1098/rspa.2015.0111](https://doi.org/10.1098/rspa.2015.0111)]
On adiabatic quantum computation combined with [[parameterized quantum system|parameterized]] [[quantum circuits]]:
* Ioannis Kolotouros, Ioannis Petrongonas, Miloš Prokop, Petros Wallden, *Adiabatic quantum computing with parameterized quantum circuits* [[arXiv:2206.04373](https://arxiv.org/abs/2206.04373)]
### Geometric phase gates, holonomic quantum computation
References which consider quantum gates operating by (nonabelian) geometric Berry phases due to adiabatic parameter movement (*holonomic quantum computation*):
* [[Paolo Zanardi]], [[Mario Rasetti]], *Holonomic Quantum Computation*, Phys. Lett. A **264** (1999) 94-99 [<a href="https://doi.org/10.1016/S0375-9601(99)00803-8">doi:10.1016/S0375-9601(99)00803-8</a>, [arXiv:quant-ph/9904011](https://arxiv.org/abs/quant-ph/9904011)]
* [[Jiannis Pachos]], [[Paolo Zanardi]], [[Mario Rasetti]], *Non-Abelian Berry connections for quantum computation*, Phys. Rev. A **61** (2000) 010305 [[arXiv:quant-ph/9907103](https://arxiv.org/abs/quant-ph/9907103), [doi:10.1103/PhysRevA.61.010305](https://doi.org/10.1103/PhysRevA.61.010305)]
* Jonathan A. Jones, Vlatko Vedral, Artur Ekert, Giuseppe Castagnoli, *Geometric quantum computation using nuclear magnetic resonance*, Nature **403** (2000) 869–871 [[doi:10.1038/35002528](https://doi.org/10.1038/35002528)]
* Giuseppe Falci, Rosario Fazio, G. Massimo Palma, Jens Siewert & Vlatko Vedral, *Detection of geometric phases in superconducting nanocircuits*, Nature **407** 355–358 (2000) [[doi:10.1038/35030052](https://doi.org/10.1038/35030052)]
* [[Jiannis Pachos]], [[Paolo Zanardi]], *Quantum Holonomies for Quantum Computing*, Int. J. Mod. Phys. B **15** (2001) 1257-1286 [[arXiv:quant-ph/0007110](https://arxiv.org/abs/quant-ph/0007110), [doi:10.1142/S0217979201004836](https://doi.org/10.1142/S0217979201004836)]
* L. M. Duan, J. I. Cirac, P. Zoller, *Geometric Manipulation of Trapped Ions for Quantum Computation*, Science **292** (2001) 1695 [[arXiv:quant-ph/0111086](https://arxiv.org/abs/quant-ph/0111086), [doi:10.1126/science.1058835](https://doi.org/10.1126/science.1058835)]
* Jiang Zhang, Thi Ha Kyaw, Stefan Filipp, Leong-Chuan Kwek, Erik Sjöqvist, Dianmin Tong, *Geometric and holonomic quantum computation* [[arXiv:2110.03602](https://arxiv.org/abs/2110.03602)]
* Daniel Turyansky et al., *Inertial geometric quantum logic gates* [[arXiv:2303.13674](https://arxiv.org/abs/2303.13674)]
* Logan W. Cooke et al., *Demonstration of Floquet engineered non-Abelian geometric phase for holonomic quantum computing* [[arXiv:2307.12957](https://arxiv.org/abs/2307.12957)]
### In topological quantum computation
References which make explicit that [[topological quantum computation]] by [[braiding]] of [[anyon]] [[worldlines]] is a form of adiabatic quantum computation:
* {#ArovasSchriefferWilczekZee85} [[Daniel P. Arovas]], [[Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](anyon#Wilczek90)) [<a href="https://doi.org/10.1016/0550-3213(85)90252-4">doi:10.1016/0550-3213(85)90252-4</a>]
* [Childs, Farhi & Preskill (2002), p. 2](#ChildsFarhiPreskill02)
* {#FreedmanKitaevLarsenWang03} [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], pp. 6 of *Topological quantum computation*, Bull. Amer. Math. Soc. __40__ (2003) 31-38 [[arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf)]
* {#BonesteelHormoziZikosSimon05} [[Nicholas E. Bonesteel]], [[Layla Hormozi]], [[Georgios Zikos]], [[Steven H. Simon]], p. 1 of: *Braid Topologies for Quantum Computation*, Phys. Rev. Lett. **95** 140503 (2005) [[arXiv:quant-ph/0505065](https://arxiv.org/abs/quant-ph/0505065), [doi:10.1103/PhysRevLett.95.140503](https://doi.org/10.1103/PhysRevLett.95.140503)]
* {#NayakSimonSternFreedman08} [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], §II.A.2 (p. 6) of: _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. **80** 1083 (2008) [[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083)]
* {#ChengGalitskiDasSarma11} [[Meng Cheng]], [[Victor Galitski]], [[Sankar Das Sarma]], *Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors*, Phys. Rev. B **84** (2011) 104529 [[arXiv:1106.2549](https://arxiv.org/abs/1106.2549), [doi:10.1103/PhysRevB.84.104529](https://doi.org/10.1103/PhysRevB.84.104529)]
* [[Gustavo Rigolin]], [[Gerardo Ortiz]], p. 1 of: *The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy*, Phys. Rev. A **85** 062111 (2012) [[arXiv:1111.5333](https://arxiv.org/abs/1111.5333), [doi:10.1103/PhysRevA.85.062111](https://doi.org/10.1103/PhysRevA.85.062111)]
* [[Jiannis K. Pachos]], *Introduction to Topological Quantum Computation*, Cambridge University Press (2012) [[doi:10.1017/CBO9780511792908]( https://doi.org/10.1017/CBO9780511792908)]
> [p. 50]: "topological quantum computation resembles an adiabatic quantum computation with constant energy gap, where the quasiparticle coordinates provide the control parameters of the Hamiltonian."
> [p. 52]: "Holonomic quantum computation resembles the adiabatic scheme [...] topological quantum computation can be considered as holonomic computation where the employed adiabatic evolutions have topological characteristics."
* {#CLBFN15} Chris Cesare, Andrew J. Landahl, Dave Bacon, Steven T. Flammia, Alice Neels, *Adiabatic topological quantum computing*, Phys. Rev. A **92** (2015) 012336 [[arXiv:1406.2690](https://arxiv.org/abs/1406.2690), [doi:10.1103/PhysRevA.92.012336](https://doi.org/10.1103/PhysRevA.92.012336)]
* {#LahtinenPachos17} [[Ville Lahtinen]], [[Jiannis K. Pachos]], p. 1 of: _A Short Introduction to Topological Quantum Computation_, SciPost Phys. **3** 021 (2017) [[arXiv:1705.04103](https://arxiv.org/abs/1705.04103), [doi:10.21468/SciPostPhys.3.3.021](https://scipost.org/SciPostPhys.3.3.021)]
* E. Macaluso, T. Comparin, L. Mazza, and I. Carusotto, *Fusion Channels of Non-Abelian Anyons from Angular-Momentum and Density-Profile Measurements*, Phys. Rev. Lett. **123** 266801 (2019) [[arXiv:1903.03011](https://arxiv.org/abs/1903.03011), [doi:10.1103/PhysRevLett.123.266801](https://doi.org/10.1103/PhysRevLett.123.266801)]
* {#Stanescu20} [[Tudor D. Stanescu]], p. 321 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))
* {#BarlasProdan20} Yafis Barlas, Emil Prodan, *Topological braiding of non-Abelian mid-gap defects in classical meta-materials*, Phys. Rev. Lett. **124** (2020) 146801 [[arXiv:1903.00463](https://arxiv.org/abs/1903.00463), [doi:10.1103/PhysRevLett.124.146801](https://doi.org/10.1103/PhysRevLett.124.146801)]
[[!redirects quantum annealing]]
[[!redirects holonomic quantum computation]]
|
adiabatic switching | https://ncatlab.org/nlab/source/adiabatic+switching |
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###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### Adiabatic switching
In [[perturbative quantum field theory]], the term _adiabatic switching_ refers to considering a smooth transition between vanishing and non-vanishing [[interaction]] [[coupling constant|coupling]]: the interaction is slowly, hence (borrowing a term from [[thermodynamics]]) "[[adiabatic|adiabatically]]", switched on or off. This is mostly a mathematical device, not meant to directly reflect a physical situation of changing coupling, but it does serve to construct physical quantities. This is closely related to the role of [[operator-valued distributions]] which are quantities that give well defined [[linear operators]] (hence [[quantum observables]]) only when evaluated on any [[bump function]].
Originally adiabatic switching was considered ([Lippmann-Schwinger 50](#LippmannSchwinger50)) only in the [[time]]-direction (for a fixed choice of time on [[Minkowski spacetime]]) by multiplying the [[interaction]] term of the [[Lagrangian density]]/[[Hamiltonian]] by the [[exponential]] $\exp(- \epsilon {\Vert t \Vert})$ (for $\epsilon \in (0,\infty)$ a [[positive number|positive]] [[real number]] and for ${\Vert t\Vert}$ the [[absolute value]] of the time [[coordinate]]). Review is for instance in ([Strocchi 13, section 6.3](#Strocchi13)).
Using this, the _Gell-Mann and Low formula_ ([Gell-Mann & Low 51](#GellMannLow51), see [Molinari 06](#Molinari06)) expresses the [[eigenstates]] $\vert \psi \rangle$ of an interacting [[Hamiltonian]] $H = H_{free} + H_{int}$ in terms of the eigenstates $\vert \Psi_{free} \rangle$ of the free Hamiltonian by the "adiabatic limit"
$$
\vert \Psi^{\pm}_{int} \rangle
\;\propto\;
\underset{\epsilon \to 0}{\lim}
S_\epsilon(0, \pm \infty)
\vert \Psi_{free} \rangle
$$
(if the [[limit of a sequence|limit]] exists) where $S_\epsilon$ denotes the [[S-matrix]] of the adiabatically switched Hamiltonian $H_\epsilon \coloneqq H_{free} + e^{- \epsilon {\Vert t\Vert}}H_{int}$.
More generally, one may consider adiabatic switching taking place not just in time, but in all of [[spacetime]]. This the basis of [[causal perturbation theory]] and [[locally covariant perturbative quantum field theory]]:
In the construction of [[perturbative quantum field theory]] via the method of [[causal perturbation theory]] the [[interaction]] terms $L_{int}$ used in the mathematical construction of the [[S-matrix]] are multiplied with a "[[coupling constant]]" $g$ which is in fact taken to be a [[smooth function]] of [[compact support]] on [[spacetime]], hence a [[bump function]]:
$$
L_g = L_{free} + g L_{int}
\,.
$$
This means that the the [[interaction]] as modeled by the [[S-matrix]]
$$
S_g \coloneqq T \exp( \tfrac{i}{\hbar} \int_{X} g :L_{int}(x): )
$$
is non-trivial only on a [[compact topological space|compact]] [[subspace]] of [[spacetime]], towards its boundary it smoothly drops to zero. Hence outside this region the interaction is "switched off".
Since the actual interactions in [[physics]] are of course _not_ "switched off" anywhere, the use of an adiabatic switching is just an intermediate mathematical step.
Originally in ([Epstein-Glaser 73](#EpsteinGlaser73)) the idea was that after having constructed the [[S-matrix]] for any adiabatic switching $g$, the [[limit of a sequence|limit]] ("adiabatic limit") $g \to 1$ had to be taken to remove the switching in the end. Failure of this limit to exist is interpreted as "[[infrared divergency]]" of the [[perturbative quantum field theory]] (since the divergency comes from large scales, hence long [[wavelength]]).
But as observed in ([Il'in-Slavnov 78](#IlinSlavnov78)) and rediscovered in ([Brunetti-Fredenhagen 00](#BrunettiFredenhagen00)), an adiabatic switching map that is unity on a globally hyperbolic sub-spacetime $O \subset X$ is sufficient to compute the perturbative [[interacting field algebra]], hence the [[algebra of quantum observables]] $A(O)$ on that subspace, and the collection of all of these as $O$ ranges forms a [[causally local net of observables]] which fully captures the [[quantum field theory]] in the sense of the [[Haag-Kastler axioms]] ([this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet)). This perspective is now known as _[[locally covariant algebraic quantum field theory]]_.
### Adiabatic limit
The [[limit of a sequence|limit]] of the [[perturbative S-matrix]] as the [[adiabatic switching]] is removed (if it exists) is called the _adiabatic limit_ or _strong adiabatic limit_.
If one just asks that the corresponding limit exists for the [[n-point functions]] one speaks of a _weak adiabatic limit_.
Even with the adiabatically switched S-matrix elements (not taking a limit) the [[local net of quantum observables]] is well defined ([this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet)), this is hence a [[functor]]
$$
\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})
$$
that assigns [[algebras of observables]] to [[causally closed subsets]] of [[spacetime]]. The [[colimit]] algebra
$$
\mathcal{A} \coloneqq \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim} \mathcal{A}(\mathcal{O})
$$
over this [[functor]] (in the sense of [[category theory]]) always exists. This is also called the _algebraic adiabatic limit_.
(See around [Duch 17, section 4](#Duch17) for review of strong, weak and algebraic adiabaitc limit; and [Duch 17, chapter II](#Duch17) for results on the weak adiabatic limit)
Here
1. the _algebraic adiabatic_ limit defines the _[[quantum observables]]_ in the limit;
1. the weak adiabatic limit may serve to define also the _[[state on a star-algebra|states]]_, hence the [[interacting vacuum]] ([Duch 17, p. 113-114](#Duch17)).
## Related concepts
* [[adiabatic]]
## References
The concept of adiabatic switching in the time direction was introduced in
* {#LippmannSchwinger50} B. A. Lippmann, [[Julian Schwinger]], *Variational Principles for Scattering Processes. I*, Phys. Rev. 79, 469 (1950) ([doi:10.1103/PhysRev.79.469](https://doi.org/10.1103/PhysRev.79.469))
reviewed for instance in
* {#Strocchi13} [[Franco Strocchi]], section 6.3 of _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press, 2013
and the corresponding formula for the interacting eigenstates in terms of the free ones is due to
* {#GellMannLow51} [[Murray Gell-Mann]], F. Low, _Bound states in quantum field theory_ Phys. Rev. 84, 350 (1951)
* [[Murray Gell-Mann]], M. L. Goldberger, Phys. Rev. 91 398 (1953)
see
* {#Molinari06} Luca Guido Molinari, _Another proof of Gell-Mann and Low's theorem_, Journal of Mathematical Physics 48, 052113, 2007 ([arXiv:math-ph/0612030](https://arxiv.org/abs/math-ph/0612030))
See also:
* C. Lanczos, R. C. Clark, G. H. Derrick (eds.), Ch. 2, Sec 2 in: *Mathematical Methods in Solid State and Superfluid Theory*, Springer (1986) $[$[doi:10.1007/978-1-4899-6435-9](https://doi.org/10.1007/978-1-4899-6435-9)$]$
* Wikipedia, _[Gell-Mann and Low theorem](https://en.wikipedia.org/wiki/Gell-Mann_and_Low_theorem)_
The generalization to switching in all space-time directions was considered for the construction of [[causal perturbation theory]] in
* {#EpsteinGlaser73} [[Henri Epstein]] and [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211.
The observation that this in fact makes causal perturbation theory a tool for constructing [[local nets of observables]] for [[locally covariant perturbative quantum field theory]] is due to
* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32.
* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661,2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))
The term "algebraic adiabatic limit" for the resulting [[local net of observables]] (or its [[inductive limit]]) appears in
* {#FredenhagenLindner13} [[Klaus Fredenhagen]], [[Falk Lindner]], p. 7 of _Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics_, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014 ([arXiv:1306.6519](https://arxiv.org/abs/1306.6519))
The weak adiabatic limit in [[causal perturbation theory]] for massive fields was shown to exists in
* [Epstein-Glaser 73](#EpsteinGlaser73)
Extension of this result to [[quantum electrodynamics]] and [[phi^4 theory]] was given in
* {#BlanchardSeneor75} P. Blanchard and R. Seneor, _Green’s functions for theories with massless particles (in perturbation theory)_, Ann. Inst. H. Poincaré Sec. A
23 (2), 147–209 (1975) ([Numdam](http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0))
See also
* {#Scharf95} [[Günter Scharf]], section 3.11 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition
Further extension of the result is due to
* {#Duch17} [[Paweł Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))
[[!redirects adiabatic switchings]]
[[!redirects Gell-Mann and Low formula]]
[[!redirects adiabatic limit]]
[[!redirects adiabatic limits]]
[[!redirects algebraic adiabatic limit]]
[[!redirects algebraic adiabatic limits]]
[[!redirects strong adiabatic limit]]
[[!redirects strong adiabatic limits]]
[[!redirects weak adiabatic limit]]
[[!redirects weak adiabatic limits]]
|
adic noetherian ring | https://ncatlab.org/nlab/source/adic+noetherian+ring |
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### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[topological ring]] $R$ is an __adic noetherian ring__ if it is [[noetherian ring|noetherian]] as a [[ring]] and it has a [[topological basis]] consisting of all translations of the [[neighborhoods]] of zero of the form $I^n$ ($n\gt 0$) where $I\subset R$ is a fixed [[ideal]] of $R$, and $R$ is [[Hausdorff space|Hausdorff]] and [[complete space|complete]] in that topology. A choice of such an ideal is said to be the __defining ideal__ or (more French) the __ideal of definition__ of the topological ring $R$. If $R$ is an adic noetherian ring, an ideal $J\subset R$ is a defining ideal iff it is open and its powers tend to $\{0\}$.
The [[topology]] of an adic noetherian ring $R$ with the defining ideal $I$ is said to be the **$I$-adic topology** and the descending [[filtration]] of $R$ by the powers of $I$ to be the **$I$-adic filtration**.
For an adic noetherian ring $R$ there is a construction of a [[ringed space]], its [[formal spectrum]] $Spf(R)$, which does not depend on the choice of the ideal $I\subset R$ generating its (fixed in advance) topology. The underlying topological space of $Spf(R)$ is $Spec(R/I)$ which is (under the above assumptions on $R$ and $I$) a [[closed subspace]] of the [[spectrum (geometry)|spectrum]] $Spec(R)$ and it contains all closed points of $Spec(R)$.
## Related concepts
* [[completion of a ring]]
* [[linear topological ring]]
* [[pro-ring]]
## References
* PlanetMath _[I-adic topology](http://planetmath.org/iadictopology)_
* Wikipedia, _[Krull topology](https://en.wikipedia.org/wiki/Completion_%28algebra%29#Krull_topology)_
[[!redirects adic topology]]
[[!redirects adic topologies]]
[[!redirects adic filtration]]
[[!redirects adic ring]]
[[!redirects adic rings]]
|
adic residual | https://ncatlab.org/nlab/source/adic+residual |
#Contents#
* table of contents
{:toc}
## Idea
Forming the [[kernel]] of an [[adic completion]] map is sometimes called forming the _adic residual_, for instance the _$p$-residual_ for $p$-adic completion.
## Properties
### Explicit characterization
For $A$ a [[commutative ring]] and $\mathfrak{a} \subset A$ an ideal, and $N$ an $A$-module, then the $\mathfrak{a}$-adic residual of $N$ (hence the kernel of the map to the [[completion of a module]] $N \longrightarrow N^\wedge_{\mathfrak{a}}$) is equivalently the submodule of elements annihilated by $1 + \mathfrak{a}$.
E.g. theorem 4.3.2. here: [[Completion.pdf:file]]
### As a modality in arithmetic cohesion
For suitably well behaved ideals, forming the adic residual may be understood as the [[dR-flat modality]] in the [[cohesion]] of [[E-infinity arithmetic geometry]]:
[[!include arithmetic cohesion -- table]]
## References
|
adic ring > history | https://ncatlab.org/nlab/source/adic+ring+%3E+history | |
adic space | https://ncatlab.org/nlab/source/adic+space |
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###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
__Adic spaces__ are the basic objects in Huber's approach to [[non-archimedean analytic geometry]]. They are built by gluing [[valuation spectra]] of a certain class of [[topological rings]]. Unlike [[Berkovich analytic spectra]] the points of adic spaces correspond to valuations of arbitrary rank, not only rank one. If a [[Berkovich space]] is corresponding to a separated [[rigid analytic space]] then it can be obtained as the largest Hausdorff quotient of the corresponding adic space.
The framework of adic spaces are used to build [[perfectoid spaces]] out of perfectoid rings.
## Definitions
\begin{definition}
Let $(A,A^{+})$ be a Huber pair, i.e. $A$ is a Huber ring and $A^{+}\subseteq A$ is a ring of integral elements. The _adic spectrum_ $\mathrm{Spa}(A,A^{+})$ is the set of equivalence classes of continuous valuations $\vert\cdot\vert$ on $A$ such that $\vert A^{+}\vert\leq 1$.
If $x$ is a valuation, and $g\in A$, we also suggestively write $g\mapsto\vert g(x)\vert$ for the valuation $x$ applied to $g$. The topology on $\mathrm{Spa}(A,A^{+})$ is the one generated by open sets of the form
$$\lbrace x:\vert f(x)\vert\leq\vert g(x)\vert\neq 0\rbrace$$
where $f,g\in A$.
\end{definition}
## Examples
* The final object in the category of adic spaces is $\mathrm{Spa}(\mathbb{Z},\mathbb{Z})$.
* The adic closed disc over $\mathbb{Q}_{p}$ is given by $\mathrm{Spa}(A,A^{+})$ where $A=\mathbb{Q}_{p}\langle T\rangle$ and $A^{+}=\mathbb{Z}_{p}\langle T\rangle$.
* The adic open disc over $\mathbb{Q}_{p}$ is the generic fiber of $\mathrm{Spa}(A,A)\to\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$, where $A=\mathbb{Z}_{p}[[T]]$.
## Related concepts
* [[perfectoid space]]
* [[p-adic geometry]]
* [[analytic geometry]]
## References
* R. Huber, _Étale cohomology of rigid analytic varieties and adic spaces_, Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig, 1996. x+450 pp. ([MR2001c:14046](http://www.ams.org/mathscinet-getitem?mr=1734903))
* [[Sophie Morel]], _Adic spaces_ ([pdf](https://web.math.princeton.edu/~smorel/adic_notes.pdf))
* Torsten Wedhorn, _Adic spaces_ ([arXiv:1910.05934](https://arxiv.org/abs/1910.05934))
* [[Brian Conrad]], _A brief introduction to adic spaces_, [PDF](http://virtualmath1.stanford.edu/~conrad/papers/Adicnotes.pdf).
[[!redirects adic spaces]] |
adic topology > history | https://ncatlab.org/nlab/source/adic+topology+%3E+history | < [[adic topology]] |
adinkra | https://ncatlab.org/nlab/source/adinkra |
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###Context###
#### Superalgebra
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[super-algebra|super]]-[[representation theory]], what is called _adinkras_ ([Faux-Gates 04](#FauxGates04)) is a graphical tool for denoting those [[representations]] ([[super multiplets]]) of the $\mathcal{N}$-extended [[supersymmetry]] algebras in one dimension ([[supersymmetric quantum mechanics]] with [[number of supersymmetries|N supersymmetries]]) for which the [[supersymmetry]] generators act, up to [[derivatives]] and prefactors, by [[permutation]] of [[superfield]] components. These are called _adinkraic_ representations ([Zhang 13, p. 16](#Zhang13)).
While adinkraic representations are special among all representations of the 1-dimensional $\mathcal{N}$-extended supersymmetry algebra, the idea is that the [[dimensional reduction]] of representations ([[supermultiplets]]) of higher dimensional supersymmetry algebras down to 1d are of this form, at least for dimensional reduction from $d = 4$ and $N = \mathbf{4}$ ([GGMPPRW 09, section 3](#GGMPPRW09), [Gates-Hubsch-Stiffler 14](#GatesHubschStiffler14)).
The classification of adinkras, and hence of adinkraic representations, turns out to be controlled by [[linear codes]] ([Doran & Faux & Gates & HubschIgaLandweberMiller 11](#DoranFauxGatesHubschIgaLandweberMiller11)) and to be related to certain special [[super Riemann surfaces]] via [[dessins d'enfants]] ([Doran & Iga & Landweber & Mendez-Diez 13](#DoranIgaLandweberMendez-Diez13), [Doran & Iga & Kostiuk &Mendes-Diez 16](#DoranIgaKostiukMendes-Diez16)).
## Definition
For background, see at _[[geometry of physics -- supersymmetry]]_.
For $N \in \mathbb{N}$, write $\mathbb{R}^{1 \vert N}$
for the [[super Lie algebra]] over the [[real numbers]] that is spanned by a single generator $P$ in even (i.e., bosonic) degree and $N$ generators $Q_I$, $I \in \{1, 2, \cdots, N\}$ in odd (i.e., fermionic) degree, whose only non-trivial components of the super-[[Lie bracket]] are
$$
[Q_I, Q_J] = 2 \delta_{I J} P
=
\left\{
\array{
2 P & \vert & I = J
\\
0 & \vert & \text{otherwise}
}
\right.
$$
This is the 1-dimensional $N$-extended [[super translation super Lie algebra]]. We may think of this as the super-translational symmetry of 1-dimensional $N$-extended [[super Minkowski spacetime]].
Consider then super [[Lie algebra representations]] of $\mathbb{R}^{1 \vert N}$ on [[super vector spaces]] of smooth [[superfields]] on $\mathbb{R}^{1 \vert N}$ (regarded as a [[supermanifold]]) and such that the bosonic generator $P$ acts as the [[derivative]] operator on [[smooth functions]] on $\mathbb{R}^1$ in each component. If in addition the representation is such that in the canonical [[linear basis]] the odd generators $Q_I$ send even/odd basis elements $\phi_i$ to single odd/even basis elements $\psi_j$ (as opposed to [[linear combinations]] of them), hence if the $Q_I$ act apart from degree-shift and possibly [[differentiation]] by [[permutations]] on the components of the [[superfields]], then this representation of $\mathbb{R}^{1\vert N}$ is called _adinkraic_. ([Zhang 13, p. 16](#Zhang13)).
The corresponding _adinkra_ is the [[bipartite graph]] which expresses these permutations:
<img src="https://ncatlab.org/nlab/files/AdinkraRule.png" width="600">
> table grabbed from [Doran & Iga & Landweber & Mendez-Diez 13, p. 7](#DoranIgaLandweberMendez-Diez13)
<img src="https://ncatlab.org/nlab/files/1dAdinkraExample.png" width="600">
> graphics grabbed from [Iga-Zhang 15, p. 3](#IgaZhang15)
## Classification
The topology of an adinkra graph together with its edge coloring in $\{1,2, \cdots, N\}$ is called its _chromotopology_.
The set of adinkra chromotopologies is equivalent to the set of colored $N$-cubs modulo doubly even length-$N$ [[linear codes]] ([Doran-Faux-Gates-Hubsch-Iga-Landweber-Miller 11](#DoranFauxgatesHubschIgaLandweberMiller11))
(A [[linear code]] of length $N$ is a [[linear subspace]] of $(\mathbb{F}_2)^N$ for $\mathbb{F}_2$ the [[prime field]] with two elements and it is _doubly even_ if every element has weight a multiple of 4. )
see [Zhang 13, chapter 2](#Zhang13)
<img src="https://ncatlab.org/nlab/files/AdinkrasFromCode.png" width="600">
> graphics grabbed from [Iga-Zhang 15, p. 4](#IgaZhang15)
## From supermultiplets in higher dimensions
The [[dimensional reduction]] of the smallest [[supermultiplets]] of $d = 4, N = \mathbf{4}$ supersymmetry down to 1d yield adinkraic representations ([GGMPPRW 09, section 3](#GGMPPRW09), [Gates-Hubsch-Stiffler 14](#GatesHubschStiffler14)).
Corresponding adinkras for the chiral scalar supermultiplet (CM), the vector multiplet (VM) and the tensor multiplet (TM) look as follows:
<img src="https://ncatlab.org/nlab/files/AdinkrasFrom4d.png" width="450">
## References
For an introduction to adinkras, see the talk
* [[Lutian Zhao]], _What is an Adinkra?_, 2014 ([slides](http://math.sjtu.edu.cn/conference/Bannai/2014/data/20141213B/slides.pdf))
The concept of adinkras was introduced into [[supersymmetry]] [[representation theory]] in
* {#FauxGates04} [[Michael Faux]], [[Jim Gates]], _Adinkras: A Graphical Technology for Supersymmetric Representation Theory_, Phys.Rev. D71 (2005) 065002 ([hep-th/0408004](https://arxiv.org/abs/hep-th/0408004))
and further developed in the article
* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields_, Int. J. Mod. Phys. A22: 869-930, 2007 ([arXiv:math-ph/0512016](https://arxiv.org/abs/math-ph/0512016))
and many more ("DFGHIL collaboration"). For instance the relation to Clifford supermodules is discussed in
* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _Off-shell supersymmetry and filtered Clifford supermodules_ ([arXiv:math-ph/0603012](https://arxiv.org/abs/math-ph/0603012))
kinetic terms are discussed in
* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _Adinkras and the Dynamics of Superspace Prepotentials_ ([arXiv:hep-th/0605269](https://arxiv.org/abs/hep-th/0605269))
The classification of adinkras in terms of [[graphs]] and [[linear codes]] is due to
* {#DoranFauxGatesHubschIgaLandweberMiller11} [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], R. L. Miller, _Codes and Supersymmetry in One Dimension_, Adv. in Th. Math. Phys. 15 (2011) 1909-1970 ([arXiv:1108.4124](https://arxiv.org/abs/1108.4124))
and discussed in mathematical detail in
* {#Zhang11} [[Yan X Zhang]], _Adinkras for Mathematicians_ ([arXiv:1111.6055](https://arxiv.org/abs/1111.6055))
* {#Zhang13} [[Yan X Zhang]], _The combinatorics of Adinkras_, PhD thesis, MIT (2013) ([pdf](http://math.mit.edu/~yanzhang/math/thesis_adinkras.pdf))
The [[dimensional reduction]] of the standard [[supermultiplets]] of $D = 4, \mathcal{N} = 1$ supersymmetry to adinkraic representations of $D = 1, \mathcal{N}=4$ is due to
* {#GGMPPRW09} [[Jim Gates]], J. Gonzales, B. MacGregor, J. Parker, R. Polo-Sherk, V.G.J. Rodgers, L. Wassink, _$4D$, $N = 1$ Supersymmetry Genomics (I)_, JHEP 0912:008,2009 ([arXiv:0902.3830](https://arxiv.org/abs/0902.3830))
* {#GatesHubschStiffler14} [[Jim Gates]], [[Tristan Hübsch]], Kory Stiffler, _Adinkras and SUSY Holography_, Int. J. Mod. Phys. A29 no. 7, (2014) 1450041 ([arXiv:1208.5999](https://arxiv.org/abs/1208.5999))
See also
* {#IgaZhang15} [[Kevin Iga]], [[Yan Zhang]], _Structural Theory of 2-d Adinkras_ ([arXiv:1508.00491](https://arxiv.org/abs/1508.00491))
Discussion for $D=11$, $\mathcal{N}=1$:
* [[Jim Gates]], Yangrui Hu, S.-N. Hazel Mak, _Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, $\mathcal{N} = 1$ Superspace_ ([arXiv:2002.08502](https://arxiv.org/abs/2002.08502))
The relation of adinkras to special [[super Riemann surfaces]] via [[dessins d'enfants]] is due to
* {#DoranIgaLandweberMendez-Diez13} [[Charles Doran]], [[Kevin Iga]], [[Greg Landweber]], [[Stefan Méndez-Diez]], _Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras_ ([arXiv:1311.3736](https://arxiv.org/abs/1311.3736))
* {#DoranIgaKostiukMendes-Diez16} [[Charles Doran]], [[Kevin Iga]], Jordan Kostiuk, [[Stefan Méndez-Diez]], _Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II_ ([arXiv:1610.09983](https://arxiv.org/abs/1610.09983))
Further developments includes
* {#CalkinsGatesGatesStiffler15} Mathew Calkins, D. E. A. Gates, [[Jim Gates]] Jr., Kory Stiffler, _Adinkras, 0-branes, Holoraumy and the SUSY QFT/QM Correspondence_ ([arXiv:1501.00101](https://arxiv.org/abs/1501.00101))
Discussion in the context of [[spectral triples]] is in
* [[Matilde Marcolli]], Nick Zolman, _Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples_ ([arXiv:1606.04463](https://arxiv.org/abs/1606.04463))
See also
* Wes Caldwell, Alejandro Diaz, Isaac Friend, [[Jim Gates]], Jr., Siddhartha Harmalkar, Tamar Lambert-Brown, Daniel Lay, Karina Martirosova, Victor Meszaros, Mayowa Omokanwaye, Shaina Rudman, Daniel Shin, Anthony Vershov, _On the Four Dimensional Holoraumy of the $4D$, $\mathcal{N} = 1$ Complex Linear Supermultiplet_ ([arXiv:1702.05453](https://arxiv.org/abs/1702.05453))
* Kevin Iga, *Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and Codes* ([arXiv:2110.01665](https://arxiv.org/abs/2110.01665))
* Wikipedia, _<a href="https://en.wikipedia.org/wiki/Adinkra_symbols_(physics)">Adinkra symbols (physics)</a>_
[[!redirects adinkras]]
[[!redirects adinkraic representation]]
[[!redirects adinkraic representations]]
|
Adj | https://ncatlab.org/nlab/source/Adj |
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[2-category]] $Adj$ is the free-standing adjunction ([[walking]] adjunction).
A [[2-functor]] $Adj \to K$ is an [[adjunction]] in the 2-category $K$. These 2-functors form one version of the [[2-category of adjunctions]] of $K$.
## Definition
$Adj$ is the 2-category freely generated by
* two objects: $a$ and $b$,
* two morphisms: $L: a \to b$ and $R: b \to a$,
* and two 2-morphisms, called the "unit" and "counit": $i: 1_a \to L R$ and $e: R L \to 1_b$, satisfying two relations, called the "triangle equations".
The restrictions of the free-standing adjunction, $Adj$, to the sub-2-categories spanned by one endpoint, $a$, or the other, $b$, define the free-standing [[monad]] and the free-standing [[comonad]].
##Related entries
* [[2-category of adjunctions]]
* [[adjoint logic]]
* [[Hegelian taco]]
##References
* C. Auderset, _Adjonction et monade au niveau des 2-categories_, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. ([numdam](http://www.numdam.org/item/CTGDC_1974__15_1_3_0/))
* [[John Baez]], _This Week's Finds in Mathematical Physics (Week 174)_, ([TWF174](http://www.math.ucr.edu/home/baez/week174.html))
* Kevin Coulembier, [[Ross Street]], Michel van den Bergh, _Freely adjoining monoidal duals_, arXiv:2004.09697 (2020). ([abstract](https://arxiv.org/abs/2004.09697))
* Dieter Pumplün, _Eine Bemerkung über Monaden und adjungierte Funktoren_, Math. Annalen 185 (1970), 329-377.
* [[Stephen Schanuel]] and [[Ross Street]], The free adjunction, Cah. Top. Géom. Diff. 27 (1986), 81-83. ([numdam](http://archive.numdam.org/item/CTGDC_1986__27_1_81_0/))
[[!redirects walking adjunction]]
[[!redirects the walking adjunction]]
|
adjacency matrix | https://ncatlab.org/nlab/source/adjacency+matrix |
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### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--
\tableofcontents
## Definition
In [[graph theory]], an **adjacency matrix** for a [[finite set|finite]] [[multigraph]] or [[pseudograph]] with $n$ [[vertices]] is an $n$ by $n$ [[matrix]] of [[natural numbers]] which encodes the number of [[edges]] between each vertex: entry $a_{i, j}$ in the matrix is the number of edges between vertex $v_i$ and $v_j$.
## Properties
Let $G$ be a finite multigraph, and let $A$ be the associated adjacency matrix. Then the matrix power $A^n$ encodes the number of [[walks]] between each vertex: entry $b_{i, j}$ in $A^n$ is the number of walks between vertex $v_i \in G$ and $v_j \in G$.
## Related concepts
* [[matrix]]
## References
* Wikipedia, [Adjacency matrix](https://en.wikipedia.org/wiki/Adjacency_matrix) |
adjoint | https://ncatlab.org/nlab/source/adjoint |
## In linear algebra
The term _adjoint_ originates in [[linear algebra]], where it may refer, in increasing generality, to:
* [[adjoint matrix]]
* [[Hermitian adjoint]]
* [[adjoint operator]]
## In category theory
In [[category theory]], the term "adjoint" foremost refers to "[[adjoint functors]]", a terminology which is probably inspired by the similarity between the characteristic [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of such functors and the defining [[equality]] $\big\langle A v ,\, w \big\rangle
\,=\, \big\langle v ,\, A^\dagger w \big\rangle$
for [[Hermitian adjoint]] [[linear operators]].
Proceeding from here, there are various variants of adjointness in [[category theory]]:
* [[adjunction]], [[adjoint morphism]]
* [[adjoint functor]]
* [[strong adjoint functor]]
* [[adjoint triple]], [[adjoint quadruple]], [[recollement]]
* [[proadjoint]], [[Hopf adjunction]]
* [[2-adjunction]]
[[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]
* [[adjoint (infinity,1)-functor]]
* [[(∞,n)-category with adjoints]]
[[!redirects adjoints]]
|
adjoint (infinity,1)-functor | https://ncatlab.org/nlab/source/adjoint+%28infinity%2C1%29-functor | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of adjunction between two [[(∞,1)-functors]] generalizes the notion of [[adjoint functors]] from [[category theory]] to [[(infinity,1)-category|(∞,1)-category theory]].
There are many equivalent definitions of the ordinary notion of [[adjoint functor]]. Some of them have more evident generalizations to some parts of [[higher category theory]] than others.
* One definition of ordinary adjoint functors says that a pair of functors $C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ is an adjunction if there is a [[natural transformation|natural isomorphism]]
$$
Hom_C(L(-),-) \simeq Hom_D(-,R(-))
\,.
$$
The analog of this definition makes sense very generally in [[(∞,1)-category theory]], where $Hom_C(-,-) : C^{op} \times C \to \infty Grpd$ is the $(\infty,1)$-categorical hom-object.
* One other characterization of adjoint functors in terms of their [[cograph of a functor|cographs]]/[[heteromorphisms]]: the [[Cartesian fibrations]] to which the <a href="http://ncatlab.org/nlab/show/(infinity%2C1)-Grothendieck+construction#FibsOverInterval">functor is associated</a>. At [[cograph of a functor]] it is discussed how two functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if the cograph of $L$ coincides with the cograph of $R$ up to the obvious reversal of arrows
$$
(L \dashv R) \Leftrightarrow
(cograph(L) \simeq cograph(R^{op})^{op})
\,.
$$
Using the [[(∞,1)-Grothendieck construction]] the notion of cograph of a functor has an evident generalization to $(\infty,1)$-categories.
## Definition
### In terms of hom-equivalences
{#CharacterizationInTermsOfHomEquivalences}
+-- {: .num_defn #InTermsOfHomEquivalences}
###### Definition
**(in terms of hom equivalence induced by unit map)**
A pair of [[(∞,1)-functors]]
$$
C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
D
$$
is an adjunction, if there exists a _unit transformation_
$\epsilon : Id_D \to R \circ L$ -- a morphism in the [[(∞,1)-category of (∞,1)-functors]] $Func(D,D)$ -- such that for all $d \in D$ and $c \in C$ the induced morphism
$$
Hom_C(L(d),c)
\stackrel{R_{L(d), c}}{\to}
Hom_D(R(L(d)), R(c))
\stackrel{Hom_D(\epsilon, R(c))}{\to}
Hom_D(d,R(c))
$$
is an [[equivalence of ∞-groupoids]].
=--
In terms of the concrete incarnation of the notion of $(\infty,1)$-category by the notion of [[quasi-category]], we have that $Hom_C(L(d),c)$ and $Hom_D(d,R(c))$ are incarnated as [[hom-object in a quasi-category|hom-objects in quasi-categories]], which are [[Kan complexes]], and the above equivalence is a [[homotopy equivalence]] of Kan complexes.
In this form is due to [Lurie 09, Def. 5.2.2.7](#Lurie09).
{#RiehlVerityOnAdjunctionsViaHomEquivalences} Streamlined discussion is in
[Riehl & Verity 15, 4.4.2-4.4.4](#RiehlVerity15)
and [Riehl & Verity 20, 3.3.3-3.5.1](#RiehlVerity20)
and [Riehl & Verity "Elements", Prop. 4.1.1](#RVElements).
### In terms of cographs/heteromorphisms
{#InTermsOfCographsHeteromorphisms}
We discuss here the quasi-category theoretic analog of _[Adjoint functors in terms of cographs](cograph+of+a+functor#AdjointFunctorsInTermsOfCographs)_ ([[heteromorphisms]]).
We make use here of the explicit realization of the [[(∞,1)-Grothendieck construction]] in its incarnation for [[quasi-categories]]: here an [[(∞,1)-functors]] $L : D \to C$ may be regarded as a map $\Delta[1]^{op} \to $ [[(∞,1)Cat]], which corresponds under the Grothendieck construction to a [[Cartesian fibration]] of [[simplicial sets]] $coGraph(L) \to \Delta[1]$.
+-- {: .num_defn #InTermsOfCoCartesianFibrations}
###### Definition
**(in terms of Cartesian/coCartesian fibrations)**
Let $C$ and $D$ be [[quasi-categories]]. An **adjunction** between $C$ and $D$ is
* a morphism $K \to \Delta[1]$ of [[simplicial sets]], which is both a [[Cartesian fibration]] as well as a coCartesian fibration.
* together with [[equivalence of quasi-categories]] $C \stackrel{\simeq}{\to} K_{\{0\}}$ and $D \stackrel{\simeq}{\to} K_{\{1\}}$.
Two [[(∞,1)-functors]] $L : C \to D$ and $R : D \to C$ are called **adjoint** -- with $L$ _left adjoint_ to $R$ and $R$ _right adjoint_ to $L$ if
* there exists an adjunction $K \to I$ in the above sense
* and $L$ and $K$ are the <a href="http://ncatlab.org/nlab/show/(infinity%2C1)-Grothendieck+construction#FibsOverInterval">associated functors to</a> the Cartesian fibation $p \colon K \to \Delta[1]$ and the Cartesian fibration $p^{op} : K^{op} \to \Delta[1]^{op}$, respectively.
=--
### In the homotopy 2-category
{#InTheHomotopy2Category}
+-- {: .num_defn #InTermsOfTheHomotopy2Category}
###### Definition
**(in terms of the homotopy 2-category)**
Say that a 2-categorical pair of [[adjoint (∞,1)-functors]] is an [[adjunction]] in the [[homotopy 2-category of (∞,1)-categories]].
=--
This concept, in the spirit of [[formal (infinity,1)-category theory|formal $\infty$-category theory]], was mentioned, briefly, in
[Joyal 2008, p. 159 (11 of 348)](#Joyal08) and then expanded on in
[Riehl-Verity 15, Def. 4.0.1](#RiehlVerity15).
Such a 2-categorical adjunctions (Def. \ref{InTermsOfTheHomotopy2Category}) determines an adjoint pair of $\infty$-functors in the sense of [Lurie 2009](#Lurie) ([Riehl-Verity 15, Rem. 4.4.5](#RiehlVerity15)):
\begin{prop}\label{InfinityCollageTypeAdjunctionsAreAdjunctionsIn2Ho}
An anti-parallal pair of morphisms in [[(∞,1)Cat|$Cat_\infty$]] is a pair of adjoint $\infty$-functors in the sense of [Lurie 2009, Sec. 5.2](#Lurie) if and only its image in the [[homotopy 2-category]] [[homotopy 2-category of (infinity,1)-categories|$Ho_2\big(Cat_\infty\big)$]] forms an [[adjunction]] in the classical sense of [[2-category theory]] (Def. \ref{InTermsOfTheHomotopy2Category}).
\end{prop}
([Riehl & Verity 2022, Sec. F.5, Prop. F.5.6](#RVElements))
The conceptual content of Prop. \ref{InfinityCollageTypeAdjunctionsAreAdjunctionsIn2Ho} may be made manifest as follows:
\begin{proposition}
Every 2-categorical pair of adjoint $(\infty,1)$-functors in the sense of Def. \ref{InTermsOfTheHomotopy2Category} extends to a "homotopy coherent adjunction" in an essentially unique way.
\end{proposition}
([Riehl & Verity 2016, Thm. 4.3.11, 4.4.11](#RiehlVerity16))
## Properties
+-- {: .num_prop}
###### Proposition
For $C$ and $D$ [[quasi-categories]], the two definitions of adjunction,
1. in terms of Hom-equivalence induced by unit maps (Def. \ref{InTermsOfHomEquivalences})
1. in terms of Cartesian/coCartesian fibrations (Def. \ref{InTermsOfCoCartesianFibrations})
are equivalent.
=--
This is [[Higher Topos Theory|HTT, prop 5.2.2.8]].
+-- {: .proof}
###### Proof
First we discuss how to produce the unit for an adjunction from the data of a correspondence $K \to \Delta[1]$ that encodes an $\infty$-adjunction $(f \dashv g)$.
For that, define a morphism $F' : \Lambda[2]_2 \times C \to K$ as follows:
* on $\{0,2\}$ it is the morphism $F : C \times \Delta[1] \to K$ that exhibits $f$ as associated to $K$, being $Id_C$ on $C \times \{0\}$ and $f$ on $C \times \{2\}$;
* on $\{1,2\}$ it is the morphism $C \times \Delta[1] \stackrel{f \times Id}{\to} D \times \Delta[1] \stackrel{G}{\to} K$, where $G$ is the morphism that exhibits $g$ as associated to $K$;
Now observe that $F'$ in particular sends $\{1,2\}$ to [[Cartesian morphism]]s in $K$ (by definition of functor associated to $K$). By one of the equivalent characterizations of [[Cartesian morphism]]s, this means that the lift in the diagram
$$
\array{
\Lambda[2]_2 &\stackrel{F'}{\to}& K
\\
\downarrow &{}^{F''}\nearrow& \downarrow
\\
\Delta[2] \times C &\to & \Delta[1]
}
$$
exists. This defines a morphism $C \times \{0,1\} \to K$ whose components may be regarded as forming a [[natural transformation]] $u : d_C \to g \circ f$.
To show that this is indeed a unit transformation, we need to show that the maps of [[hom-object in a quasi-category]] for all $c \in C$ and $d \in D$
$$
Hom_D(f(f), d) \to Hom_C(g(f(c)), g(d)) \to Hom_C(c, g(d))
$$
is an equivalence, hence an isomorphism in the [[homotopy category]]. Once checks that this fits into a commuting diagram
$$
\array{
Hom_D(f(c), d) &\to& Hom_C(g(f(c)), g(d)) &\to& Hom_C(c, g(d))
\\
\downarrow &&&& \downarrow
\\
Hom_K(C,D) &&=&& Hom_K(C,D)
}
\,.
$$
For illustration, chasing a morphism $f(c) \to d$ through this diagram yields
$$
\array{
(f(c) \to d) &\mapsto& (g(f(c)) \to g(d)) &\mapsto&
(c \to g(f(c)) \to g(d))
\\
\downarrow && && \downarrow
\\
(c \to g(f(c)) \to f(c) \to d)
&&=&&
(c \to g(f(c)) \to g(d) \to d)
}
\,,
$$
where on the left we precomposed with the Cartesian morphism
$$
\array{
&& g(f(c))
\\
& \nearrow &\Downarrow^{\simeq}& \searrow
\\
c &&\to&& f(c)
}
$$
given by $F''|_{c} : \Delta[2] \to K$, by ...
=--
### Uniqueness of adjoints
{#UniquenessOfAdjoints}
The adjoint of a functor is, if it exists, essentially unique:
+-- {: .num_prop}
###### Proposition
If the $(\infty,1)$-functor between quasi-categories $L : D \to C$ admits a right adjoint $R : C \to D$, then this is unique up to homotopy.
Moreover, even the choice of homotopy is unique, up to ever higher homotopy, i.e. the collection of all right adjoints to $L$ forms a [[contractible]] [[∞-groupoid]], in the following sense:
Let $Func^L(C,D), Func^R(C,D) \subset Func(C,D)$ be the full sub-quasi-categories on the [[(∞,1)-category of (∞,1)-functors]] between $C$ and $D$ on those functors that are left adjoint and those that are right adjoints, respectively. Then there is a canonical [[equivalence of quasi-categories]]
$$
Func^L(C,D) \stackrel{\simeq}{\to}
Func^R(D,C)^{op}
$$
(to the [[opposite quasi-category]]), which takes every left adjoint functor to a corresponding right adjoint.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, prop 5.2.1.3]] (also remark 5.2.2.2), and [[Higher Topos Theory|HTT, prop. 5.2.6.2]].
The idea is to construct the category of right adjoints as an intersection of full subcategories
$$
\array{
Func^R(C,D) &\to& C^D
\\ \downarrow & & \downarrow
\\ (D^C)^{op} &\to& \infty Gpd^{C^{op} \times D}
}
$$
where the inclusions are given by the yoneda embedding. An element of $Func^R(C,D)$
corresponds to a functor $p : C^{op} \times D \to \infty Gpd$
for which there exists a pair of functors $g : D \to C$ and $f : C \to D$ such that $p \simeq D(f-,-) \simeq C(-,g-)$.
=--
### Uniqueness of unit and counit
Given functors $f : C \to D$ and $g : D \to C$, we can use the [[(∞,1)-end]] to determine compute a chain of equivalences
$$
\begin{aligned}
C^C(id, gf)
&\simeq \int_{c \in C} C(c, gf(c))
\\ &\simeq \int_{c \in C} \infty Gpd^D(D(f(c), -), C(c, g-))
\\ &\simeq Gpd^{C^{\op} \times D}(D(f-, -), C(-, g-))
\end{aligned}
$$
dually, we can identify the space of counits as
$$
D^D(fg, id) \simeq Gpd^{C^{\op} \times D}(C(-, g-), D(f-, -))
$$
So each half of the equivalence $D(f-,-) \simeq C(-,g-)$ corresponds essentially uniquely to a choice of unit and counit transformation.
### Preservation of limits and colimits {#PresOfLims}
Recall that for $(L \dashv R)$ an ordinary pair of [[adjoint functor]]s, the fact that $L$ preserves [[colimit]]s (and that $R$ preserves [[limit]]s) is a formal consequence of
1. the hom-isomorphism $Hom_C(L(-),-) \simeq Hom_D(-,R(-))$;
1. the fact that $Hom_C(-,-) : C^{op} \times C \to Set$ preserves all limits in both arguments;
1. the [[Yoneda lemma]], which says that two objects are isomorphic if all homs out of (into them) are.
Using this one computes for all $c \in C$ and diagram $d : I \to D$
$$
\begin{aligned}
Hom_C(L(\lim_{\to} d_i), c) & \simeq
Hom_D(\lim_\to d_i, R(c))
\\
& \simeq
\lim_{\leftarrow} Hom_D(d_i, R(c))
\\
& \simeq \lim_{\leftarrow} Hom_C(L(d_i), c)
\\
& \simeq Hom_C(\lim_{\to} L(d_i), c)
\,,
\end{aligned}
$$
which implies that $L(\lim_\to d_i) \simeq \lim_\to L(d_i)$.
Now to see this in $(\infty,1)$-category theory
(...) HTT Proposition 5.2.3.5
### Adjunctions on homotopy categories {#OnHomotopyCat}
+-- {: .num_prop}
###### Proposition
For $(L \dashv R) : C \stackrel{\leftarrow}{\to} D$ an $(\infty,1)$-adjunction, its image under decategorifying to [[homotopy category of an (infinity,1)-category|homotopy categories]] is a pair of ordinary [[adjoint functor]]s
$$
(Ho(L) \dashv Ho(R)) : Ho(C) \stackrel{\leftarrow}{\to} Ho(D)
\,.
$$
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, prop 5.2.2.9]].
This follows from that fact that for $\epsilon : Id_C \to R \circ L$ a unit of the $(\infty,1)$-adjunction, its image $Ho(\epsilon)$ is a unit for an ordinary adjunction.
=--
+-- {: .num_remark}
###### Remark
The converse statement is in general false. A near converse is given by [[Higher Topos Theory|HTT, prop 5.2.2.12]] if one instead considers $Ho$-enriched homotopy categories: if $Ho(L)$ has a right adjoint, then so does $L$.
It is important to consider the $Ho$-enriched homotopy category rather than the ordinary one. For a counterexample, when $Ho$ is considered as an ordinary category, $\pi_0 : Ho \to Set$ is both left and right adjoint to the inclusion $Set \subseteq Ho$. However, $\pi_0 : \infty Gpd \to Set$ does not have a left adjoint.
One way to find that an ordinary adjunction of homotopy categories lifts to an $(\infty,1)$-adjunction is to exhibit it as a [[Quillen adjunction]] between [[simplicial model category]]-structures. This is discussed in the Examples-section [Simplicial and derived adjunction](#SimplicialAndDerived) below.
=--
### Full and faithful adjoints
{#FullAndFaithfulAdjoints}
As for ordinary [[adjoint functors]] we have the following relations between full and faithful adjoints and idempotent monads.
+-- {: .num_prop}
###### Proposition
Given an $(\infty,1)$-adjunction $(L \dashv R) : C \to D$
* $R$ is a [[full and faithful (∞,1)-functor]] precisely is the counit $L R \stackrel{}{\to} Id$ is an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-functors]].
In this case $C$ is a [[reflective (∞,1)-subcategory]] of $D$.
* $L$ is a [[full and faithful (∞,1)-functor]] precisely is the unit $Id \to R L$ is an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-functors]].
=--
[Lurie, prop. 5.2.7.4](#Lurie), See also top of p. 308.
[[!include sliced adjoint functors -- section]]
### In terms of universal arrows {#UniversalArrows}
+-- {: .num_prop #UnivArr}
###### Proposition
An $(\infty,1)$-functor $G:D\to C$ admits a left adjoint if and only if for each $X\in C$, the [[comma (infinity,1)-category]] $(X \downarrow G)$ has an [[initial object in an (infinity,1)-category|initial object]], i.e. every object $X\in C$ admits a [[universal arrow]] $X\to G F X$ to $G$.
=--
This is stated explicitly as [Riehl-Verity, Corollary 16.2.7](#RVElements), and can be extracted with some work from [[Higher Topos Theory|HTT, Proposition 5.2.4.2]].
### Preservation by exponentiation
+-- {: .num_prop}
###### Proposition
Let $f : C \to D$ be left adjoint to $g : D \to C$.
Then for any $A$, $f^A$ is left adjoint to $g^A$ and
$A^g$ is left adjoint to $A^f$.
=--
+-- {: .proof}
###### Proof
Let $\eta : id_C \Rightarrow gf$ be a unit transformation. The property
of being a unit transformation can be detected at the level of enriched homotopy
categories, so $A^\eta: id_{A^C} \Rightarrow A^f A^g$ and $\eta^A : id_{C^A} \Rightarrow g^A f^A$ are also unit transformations.
=--
## Category of adjunctions
The functorality of adjunctions can be organized into the existence of two wide subcategories $LAdj \subseteq (\infty,1)Cat$ and $RAdj \subseteq (\infty,1)Cat$
whose functors are the left adjoints and the right adjoints respectively.
We can then define the functor categories
* $Func^L : LAdj^{op} \times LAdj \to (\infty,1)Cat$
is defined by taking $Func^L(C,D) \subseteq Func(C, D)$ to be the
full subcategory spanned by $LAdj(C, D)$.
* $Func^R : RAdj^{op} \times RAdj \to (\infty,1)Cat$
is defined by taking $Func^R(C,D) \subseteq Func(C, D)$ to be the
full subcategory spanned by $RAdj(C, D)$.
Lurie defines an adjunction to be a functor $X \to [1]$ that is both a cartesian and a cocartesian fibration. We can generalize this to
+-- {: .num_defn #AdjunctFibration}
###### Definition
A functor $p : X \to S$ is an _adjunct fibration_ iff it is both a cartesian fibration and a cocartesian fibration
=--
By the [[(∞,1)-Grothendieck construction]] construction, adjunct fibrations over $S$ correspond to category-valued functors on $S$ that send arrows of $S$ to adjoint pairs of categories.
+-- {: .num_lemma}
###### Lemma
For a functor $p : X \to S$ of (∞,1)-categories with small fibers.
* If $p$ is a cartesian fibration classified by $\chi : S^\op \to (\infty,1)Cat$,
$\chi$ factors through $RAdj$ iff $p$ is an adjunct fibration
* If $p$ is a cocartesian fibration classified by $\chi : S \to (\infty,1)Cat$,
$\chi$ factors through $LAdj$ iff $p$ is an adjunct fibration
=--
+-- {: .proof}
###### Proof
This is a restatement of [[Higher Topos Theory|HTT, corr. 5.2.2.5]].
=--
\begin{lemma}\label{LadjAndRadj}
There are anti-equivalences $ladj \,\colon\, RAdj^{op} \to LAdj$ and $radj \,\colon\, LAdj^{op} \to RAdj$ that are the identity on objects and the action on homspaces $LAdj(C, D) \simeq RAdj(D,C)$ is the equivalence sending a functor to its adjoint.
\end{lemma}
+-- {: .proof}
###### Proof
By the covariant Grothendieck construction, for any (∞,1)-category C, $Map(C, LAdj)$ can be identified with the ∞-groupoid of $(\infty,1)\widehat{Cat}_{/C}$ spanned by adjunct fibrations over $C$ with small fibers and all equivalences between them. The same is true of $Map(C^{\op}, RAdj)$.
Since the Grothendieck construction is natural in the base category, we obtain the asserted equivalence between $LAdj$ and $RAdj^{op}$. Taking $C = [1]$, this establishes the correspondence between an adjunction and its associated adjoint pair of functors.
=--
As discussed at [Uniqueness of Adjoints](#UniquenessOfAdjoints), this anti-equivalence extends to the (∞,2)-enrichment, in the sense they induce anti-equivalences $radj : Func^L(C, D)^{op} \to Func^R(D, C)$ and
$ladj : Func^R(C, D)^{op} \to Func^L(D, C)$.
The preservation of adjunctions by products and exponentials implies
+-- {: .num_lemma}
###### Lemma
The product and exponential on $(\infty,1)Cat$ restrict to functors
* $- \times - : LAdj \times LAdj \to LAdj$
and $- \times - : RAdj \times RAdj \to RAdj$
* $Func(-,-) : RAdj^{op} \times LAdj \to LAdj$
and $Func(-,-) : LAdj^{op} \times RAdj \to RAdj$
=--
## Examples
A large class of examples of $(\infty,1)$-adjunctions arises from [[Quillen adjunctions]] of [[model categories]], or adjunctions in [[sSet]]-[[enriched category theory]].
### Quillen adjunctions
Any [[Quillen adjunction]] induces an adjunction of [[(infinity,1)-categories]] on the [[simplicial localizations]]. See [Hinich 14](#Hinich14) or [Mazel-Gee 15](#MazelGee15).
### Simplicial and derived adjunctions {#SimplicialAndDerived}
We want to produce Cartesian/coCartesian fibration $K \to \Delta[1]$ from a given [[sSet]]-[[enriched category theory|enriched]] adjunction. For that first consider the following characterization
+-- {: .num_lemma}
###### Lemma
Let $K$ be a [[simplicially enriched category]] whose [[hom-objects]] are all [[Kan complexes]], regard the [[interval category]] $\Delta[1] := \{0 \to 1\}$ as an $sSet$-category in the obvious way using the embedding $const : Set \hookrightarrow sSet$ and consider an $sSet$-enriched functor $K \to \Delta[1]$. Let $C := K_0$ and $D := K_1$ be the $sSet$-enriched categories that are the fibers of this. Then under the [[homotopy coherent nerve]] $N : sSet Cat \to sSet$ the morphism
$$
N(p) : N(K) \to \Delta[1]
$$
is a [[Cartesian fibration]] precisely if for all objects $d \in D$ there exists a morphism $f : c \to d$ in $K$ such that postcomposition with this morphism
$$
C(c',f ) : C(c',c) = K(c',c) \to K(c',d)
$$
is a [[homotopy equivalence]] of [[Kan complex]]es for all objects $c' \in C'$.
=--
This appears as [[Higher Topos Theory|HTT, prop. 5.2.2.4]].
+-- {: .proof}
###### Proof
The statement follows from the characterization of [[Cartesian morphism]]s under homotopy coherent nerves ([[Higher Topos Theory|HTT, prop. 2.4.1.10]]), which says that for an $sSet$-enriched functor $p : C \to D$ between Kan-complex enriched categories that is [[hom-object]]-wise a [[Kan fibration]], a morphim $f : c' \to c''$ in $C$ is an $N(p)$-[[Cartesian morphism]] if for all objects $c \in C$ the diagram
$$
\array{
C(c,c') &\stackrel{C(c,f)}{\to}& C(c,c'')
\\
\downarrow^{\mathrlap{p_{c,c'}}} &&
\downarrow^{\mathrlap{p_{c,c''}}}
\\
D(p(c),p(c')) &\stackrel{D(p(c),p(f))}{\to}&
D(p(c), p(c''))
}
$$
is a [[homotopy pullback]] in the [[model structure on sSet-categories]].
For the case under consideration the functor in question is $p : K \to \Delta[1]$ and the above diagram becomes
$$
\array{
K(c,c') &\stackrel{K(c,f)}{\to}& K(c,c'')
\\
\downarrow
&&
\downarrow
\\
* &\to& *
}
\,.
$$
This is clearly a homotopy pullback precisely if the top morphism is an equivalence.
=--
Using this, we get the following.
+-- {: .num_prop}
###### Proposition
For $C$ and $D$ [[sSet]]-[[enriched categories]] whose hom-objects are all [[Kan complexes]], the image
$$
N(C)
\underoverset
{\underset{N(R)}{\longleftarrow}}
{\overset{N(L)}{\longrightarrow}}
{\bot}
N(D)
$$
under the [[homotopy coherent nerve]] of an [[sSet]]-enriched adjunction between $sSet$-[[enriched categories]]
$$
C \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}}
D
$$
is an adjunction of [[quasi-categories]].
Moreover, if $C$ and $D$ are equipped with the structure of a [[simplicial model category]] then the quasi-categorically [[derived functors]]
$$
N(C^\circ) \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}}
N(D^\circ)
$$
form an adjunction of quasi-categories.
=--
+-- {: .proof}
###### Proof
The first part is [[Higher Topos Theory|HTT, cor. 5.2.4.5]], the second
[[Higher Topos Theory|HTT, prop. 5.2.4.6]].
To get the first part, let $K$ be the $sSet$-category which is the join of $C$ and $D$: its set of objects is the disjoint union of the sets of objects of $C$ and $D$, and the [[hom-object]]s are
* for $c,c' \in C$: $K(c,c') := C(c,c')$;
* for $d,d' \in D$: $K(d,d') := D(d,d')$;
* for $c \in C$ and $d \in D$: $K(c,d) := C(L(c),d) = D(c,R(d))$;
and
$K(d,c) = \emptyset$
and equipped with the evident composition operation.
Then for every $d \in D$ there is the morphism $Id_{R(d)} \in K(R(d),d)$, composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence $N(K) \to \Delta[1]$ is a [[Cartesian fibration]].
By the analogous dual argument, we find that it is also a coCartesian fibration and hence an adjunction.
For the second statement, we need to refine the above argument just slightly to pass to the full $sSet$-subcategories on fibrant cofibrant objects:
let $K$ be as before and let $K^\circ$ be the full $sSet$-subcategory on objects that are fibrant-cofibrant (in $C$ or in $D$, respectively). Then for any fibrant cofibrant $d \in D$, we cannot just use the identity morphism $Id_{R(d)} \in K(R(d),d)$ since the right Quillen functor $R$ is only guaranteed to respect fibrations, not cofibrations, and so $R(d)$ might not be in $K^\circ$. But we can use the [[small object argument]] to obtain a functorial cofibrant replacement functor $Q : C \to C$, such that $Q(R(d))$ is cofibrant and there is an acyclic fibration $Q(R(d)) \to R(d)$. Take this to be the morphism in $K(Q(R(d)), d)$ that we pick for a given $d$. Then this does induce a homotopy equivalence
$$
C(c', Q(R(d))) \to C(c',R(d)) = K(c',d)
$$
because in an [[enriched model category]] the enriched hom out of a cofibrant object preserves weak equivalences between fibrant objects.
=--
### Localizations
A pair of adjoint $(\infty,1)$-functors $(L \dashv R) : C \stackrel{\leftarrow}{\hookrightarrow} D$ where $R$ is a [[full and faithful (∞,1)-functor]] exhibits $C$ as a [[reflective (∞,1)-subcategory]] of $D$. This subcategory and the composite $R \circ L : D \to D$ are a [[localization of an (∞,1)-category|localization]] of $D$.
## Related concepts
* [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]
* [[proadjoint]], [[Hopf adjunction]]
* [[2-adjunction]]
[[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]
* **adjoint $(\infty,1)$-functor
## References
The suggestion that a pair of adjoint $\infty$-functors should just be an [[adjunction]] in the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $\infty$-categories]] was originally stated, briefly, in:
* {#Joyal08} [[André Joyal]], p. 159 (11 of 348) in: _The theory of quasicategories and its applications_, lectures at: _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, Quadern **45** 2, Centre de Recerca Matemàtica, Barcelona 2008 ([[JoyalTheoryOfQuasiCategories.pdf:file]])
The definition as an isofibration of [[quasicategories]] over $\Delta[1]$ is due to:
* {#Lurie} [[Jacob Lurie]], Section 5.2 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))
The original suggestion of [Joyal 2008](#Joyal08) was then much expanded on (and generalized to [[∞-cosmoi]]), in the spirit of [[formal (infinity,1)-category theory|formal $\infty$-category theory]]:
* [[Emily Riehl]], §18.6 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]
* {#RiehlVerity15} [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 ([arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021)),
* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279), [doi:10.1016/j.aim.2015.09.011](https://doi.org/10.1016/j.aim.2015.09.011))
* {#RiehlVerity20} [[Emily Riehl]], [[Dominic Verity]], Def. 1.1.2 in: _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))
That the two definitions (of [Joyal 2008](#Joyal08) and [Lurie 2009](#Lurie)) are in fact equivalent is first indicated in [Riehl-Verity 15, Rem. 4.4.5](#RiehlVerity15) and then made fully explicit in:
* {#RVElements} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$
A proof that a [[Quillen adjunction]] of [[model categories]] induces an adjunction between [[(∞,1)-categories]] (in the sense of [Lurie 2009](#Lurie)) is recorded in:
* {#Hinich14} [[Vladimir Hinich]], *Dwyer-Kan Localization Revisited*, Homology, Homotopy and Applications Volume 18 (2016) Number 1 ([arXiv:1311.4128](https://arxiv.org/abs/1311.4128), [doi:10.4310/HHA.2016.v18.n1.a3](https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3))
and also in
* {#MazelGee15} [[Aaron Mazel-Gee]], _Quillen adjunctions induce adjunctions of quasicategories_, New York Journal of Mathematics Volume 22 (2016) 57-93 ([arXiv:1501.03146](https://arxiv.org/abs/1501.03146), [publisher](http://nyjm.albany.edu/j/2016/22-4.html))
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|
adjoint (infinity,1)-functor theorem | https://ncatlab.org/nlab/source/adjoint+%28infinity%2C1%29-functor+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The analogs in [[(∞,1)-category]] theory of the [[adjoint functor theorems]] in ordinary [[category theory]].
## Statement
+-- {: .num_defn}
###### Definition
* Let $G:\mathcal{D} \to \mathcal{C}$ be a functor between $(\infty, 1)$-categories. We say that $G$ satisfies the **solution set condition** if the $(\infty, 1)$-category $c/G$ admits a small weakly initial set for any $c$ in $\mathcal{C}$.
* $G$ satisfies the **$h$-initial object condition** if $c/G$ admits an [[h-initial object]] for any $c$ in $\mathcal{C}$.
* $\mathcal{C}$ is said to be **2-locally small** if for every pair of objects, $x$,$y$, of $\mathcal{C}$, the mapping space $map_{\mathcal{C}}(x,y)$ is [[locally small]].
=--
+-- {: .num_theorem #GeneralAdjointFunctorTheorem}
###### Theorem
* Let $G:\mathcal{D} \to \mathcal{C}$ be a [[continuous functor]]. Suppose that $\mathcal{D}$ is locally small and complete and $\mathcal{C}$ is 2-locally small. Then $G$ admits a left adjoint if and only if it satisfies the solution set condition.
* Let $G:\mathcal{D} \to \mathcal{C}$ be a [[finitely continuous functor]]. Suppose that $\mathcal{D}$ is finitely complete. Then $G$ admits a left adjoint if and only if it satisfies the $h$-initial object condition.
=--
+-- {: .proof}
###### Proof
See Section 3 of ([NRS18](#NRS18)).
=--
The following result is a consequence.
+-- {: .num_theorem}
###### Theorem
Let $F : C \to D$ be an [[(∞,1)-functor]] between [[locally presentable (∞,1)-categories]] then
1. it has a right [[adjoint (∞,1)-functor]] precisely if it preserves small [[limit in a quasi-category|colimits]];
1. it has a left [[adjoint (∞,1)-functor]] precisely if it is an [[accessible (∞,1)-functor]] and preserves small [[limit in a quasi-category|limits]].
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, cor. 5.5.2.9]].
=--
+-- {: .num_remark}
###### Remark
For the existence of right adjoints, we can weaken the hypotheses to merely requiring $D$ to be a [[locally small (infinity,1)-category]].
([[Higher Topos Theory|HTT, rem. 5.5.2.10]])
=--
## Related concepts
* [[adjoint functor theorem]]
## References
* {#NRS18} Hoang Kim Nguyen, [[George Raptis]], Christoph Schrade, _Adjoint functor theorems for ∞-categories_, Journal of the London Mathematical Society **101** 2 (2019) 659-681
([arXiv:1803.01664](https://arxiv.org/abs/1803.01664))
Section 5.5 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
[[!redirects adjoint (∞,1)-functor theorem]]
[[!redirects adjoint (∞,1)-functor theorems]] |
adjoint action | https://ncatlab.org/nlab/source/adjoint+action |
> This entry is about _conjugation_ in the sense of adjoint actions, as in forming [[conjugacy classes]]. For conjugation in the sense of [[anti-involutions]] on [[star algebras]] see at _[[complex conjugation]]_.
***
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _adjoint action_ is an [[action]] by _conjugation_ .
## Definition
### Of a group on itself
The _adjoint action_ of a [[group]] $G$ on itself is the [[action]] $Ad : G \times G \to G$ given by
$$
Ad : (g,h) \mapsto g^{-1} \cdot h \cdot g
\,.
$$
### Of a Lie group on its Lie algebra
The adjoint action $ad : G \times \mathfrak{g} \to \mathfrak{g}$ of a [[Lie group]] $G$ on its [[Lie algebra]] $\mathfrak{g}$ is for each $g \in G$ the [[derivative]] $d Ad(g) : T_e G \to T_e G$ of this action in the second argument at the neutral element of $G$
$$
ad : (g,x) \mapsto Ad(g)_*(x)
\,.
$$
This is often written as $ad(g)(x) = g^{-1} x g$ even though for a general Lie group the expression on the right is not the product of three factors in any way. But for a [[matrix Lie group]] $G$ it is: in this case both $g$ as well as $x$ are canonically identified with [[matrices]] and the expression on the right is the product of these matrices.
Since this is a linear action, it is called the _[[adjoint representation]]_ of a Lie group. The [[associated bundles]] with respect to this representation are called [[adjoint bundles]].
### Of a Lie algebra on itself
Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself
$$
ad : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}
$$
which is simply the Lie bracket
$$
ad_x : y \mapsto [x,y]
\,.
$$
### Of a Hopf algebra on itself
Let $k$ be a commutative unital ring and $H = (H,m,\eta,\Delta,\epsilon, S)$ be a [[Hopf algebra|Hopf]] $k$-algebra with multiplication $m$, unit map $\eta$, comultiplication $\Delta$, counit $\epsilon$ and the antipode map $S: H\to H^{op}$. We can use [[Sweedler notation]] $\Delta(h) = \sum h_{(1)}\otimes_k h_{(2)}$. The adjoint action of $H$ on $H$ is given by
$$
h\triangleright g = \sum h_{(1)} g S(h_{(2)})
$$
and it makes $H$ not only an $H$-module, but in fact a monoid in the monoidal category of $H$-modules (usually called $H$-[[module algebra]]).
### Of a simplicial group on itself
{#OfASimplicialGroupOnItself}
Let
* $\mathcal{G}$ be a [[simplicial group]],
and write
* $\mathcal{G}\Actions(sSet)$ for the [[category]] of $\mathcal{G}$-[[action objects]] [[internalization|internal to]] [[SimplicialSets]]l
* $W \mathcal{G} \in \mathcal{G}Actions(sSet)$ for its [[universal principal simplicial complex]];
* $\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet$ for the [[simplicial classifying space]];
* $\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet)$ for the [[adjoint action]] of $\mathcal{G}$ on itself:
\[
\label{ConjugationAction}
\array{
\mathcal{G}_{ad} \times \mathcal{G}
&\xrightarrow{\;\;\;}&
\mathcal{G}_{ad}
\\
(g_k,h_k) &\mapsto& h_k \cdot g_k \cdot h_k^{-1}
}
\]
which we may understand as the restriction along the [[diagonal morphism]] $\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G}$ of the following action of the [[direct product group]]:
$$
\array{
\mathcal{G}_{ad}
\times
(\mathcal{G} \times \mathcal{G})
&\xrightarrow{\;\;\;}&
\mathcal{G}_{ad}
\\
(g_k, (h'_k, h_k))
&\mapsto&
h'_k \cdot g_k \cdot h^{-1}_k
\mathrlap{\,.}
}
$$
\begin{proposition}\label{FreeLoopSpaceOfSimplicialClassifyingSpaceAsAdQuotient}
The [[free loop space object]] of the [[simplicial classifying space]] $\overline{W} \mathcal{G}$ is [[isomorphism|isomorphic]] in the [[classical homotopy category]] to the [[Borel construction]] of the [[adjoint action]] (eq:ConjugationAction):
$$
\mathcal{L}
\big(
\overline{W}\mathcal{G}
\big)
\;\;
\simeq
\;\;
\mathcal{G}_{ad} \sslash \mathcal{G}
\;\;\;\;\;\;
\in
\;\;
Ho\big(
sSet_{Qu}
\big)
$$
\end{proposition}
For proof and more background see at *[[free loop space of classifying space]]*.
## Related concepts
* [[coadjoint action]]
* [[adjoint representation]], [[adjoint bundle]]
* [[free loop space of classifying space]]
* [[conjugation action]]
* [[conjugacy class]]
* [[twisted ad-equivariant K-theory]]
## References
* Sigurdur Helgason, _Differential geometry, Lie groups, and symmetric spaces_
* [[Eckhard Meinrenken]], _Clifford algebras and Lie theory_, Springer
* [[eom]]: [adjoint representation of a Lie group](http://www.encyclopediaofmath.org/index.php/Adjoint_representation_of_a_Lie_group), [adjoint group](http://www.encyclopediaofmath.org/index.php/Adjoint_group)
[[!redirects adjoint actions]]
[[!redirects conjugation]]
[[!redirects conjugations]]
|
adjoint bundle | https://ncatlab.org/nlab/source/adjoint+bundle | [[!redirects adjoint bundles]]
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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $G$ a [[Lie group]] and $\mathfrak{g}$ its [[Lie algebra]], and for $P \to X$ a $G$-[[principal bundle]], the corresponding **adjoint bundle** is the [[associated bundle]] $P \times_G \mathfrak{g} \to X$ via the [[adjoint action]]/[[adjoint representation]] of $G$ on $\mathfrak{g}$.
## Related concepts
* [[Atiyah Lie algebroid]]
[[!redirects adjoint bundle]]
|
adjoint equivalence | https://ncatlab.org/nlab/source/adjoint+equivalence |
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###Context###
#### Category theory
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[[!include category theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--
# Adjoint equivalences
* table of contents
{: toc}
## Idea
In [[category theory]], the notion of *adjoint equivalence* is a more "coherent" or "structured" notion of [[equivalence of categories]], in which the [[2-morphism|2-]][[isomorphisms]] relating composites to identities are required to satisfy [[coherence laws]] (namely the [[zigzag identities]] for an [[adjunction]]).
## Definition
An **adjoint equivalence** between [[categories]] is an pair of [[adjoint functors]] $f\dashv g$ in which the [[unit of an adjunction|unit]] $\eta$ and [[unit of an adjunction|counit]] $\varepsilon$ are [[natural isomorphisms]]. It follows in particular that each of $f,g$ constitute an [[equivalence of categories]].
There is an identical definition internal to any [[2-category]], which reproduces the above notion when applied in [[Cat]].
## Properties
We work in any 2-category. First, we observe:
+-- {: .num_lemma}
###### Lemma
If $(f,g,\eta,\varepsilon)$ is an adjoint equivalence, then so is $(g,f,\varepsilon^{-1},\eta^{-1})$.
=--
Therefore, in an adjoint equivalence, each functor is both the [[left adjoint]] and the [[right adjoint]] of the other (i.e. it is an [[ambidextrous adjunction]]).
The definition as given above is also redundant:
+-- {: .num_lemma}
###### Lemma
If $(f,g,\eta,\varepsilon)$ is any equivalence, then it satisfies one [[zigzag identity]] iff it satisfies the other.
=--
+-- {: .proof}
###### Proof
Using [[string diagram]] notation, with strings progressing up the page and 1-morphisms progressing from left to right, we can draw the data of an equivalence (omitting labels for the regions denoting objects) as follows:
[[!include equivalence data - SVG]]
If we now suppose that one zigzag identity holds:
[[!include zigzag identity 1 - SVG]]
then we can verify the other as follows. (The first step uses the inverse of the first zigzag identity.)
[[!include zigzag identity 1 implies 2 - SVG]]
=--
Furthermore, although an adjoint equivalence is a "stronger" or "more structured" notion than a mere equivalence, the property of "being adjoint equivalent" is no stronger a condition than "being equivalent," since every equivalence may be refined to an adjoint equivalence by modifying one of the natural isomorphisms involved. More specifically:
+--{: .num_theorem}
###### Theorem
If $f\colon X\to Y$ is a morphism which is an equivalence, then given any morphism $g\colon Y\to X$ and any isomorphism $\eta\colon 1 \cong g f$, there exists a unique 2-isomorphism $\varepsilon\colon f g \cong 1$ such that $(f,g,\eta,\varepsilon)$ is an adjoint equivalence.
=--
+--{: .proof}
###### Proof
Since $f$ is an equivalence, there exists a $g'$ and isomorphisms $f g' \cong 1$ and $1\cong g' f$. However, we also have $g \cong g f g' \cong g'$, so the isomorphism $f g' \cong 1$ also induces an isomorphism $f g\cong 1$, which we denote $\xi$. Now $\eta$ and $\xi$ may not satisfy the zigzag identities, but if we define $\varepsilon$ as follows:
$$
f g
\xrightarrow{f g \xi^{-1}} f g f g
\xrightarrow{f \eta^{-1} g} f g
\xrightarrow{\xi} 1
$$
then we can verify, using string diagram notation as above, that $\varepsilon$ satisfies one zigzag identity, and hence (by the previous lemma) also the other:
[[!include adjointification zigzag identity - SVG]]
Finally, if $\varepsilon'\colon f g \to 1$ is any other isomorphism satisfying the zigzag identities with $\eta$, then we have
$$\varepsilon' = \varepsilon' . (\varepsilon f g) . (f \eta g) =
\varepsilon . (f g \varepsilon') . (f \eta g) = \varepsilon$$
using the [[interchange law]] and two zigzag identities. This shows uniqueness.
=--
In [[Categories Work]], IV.4, there is a different proof of the weaker fact that if a [[functor]] $f$ is part of an equivalence, then it is part of an adjoint equivalence. This proof is given in [[Cat]], but can be applied representably to any 2-category.
Since adjoints are unique up to unique isomorphism when they exist, it follows that any adjunction involving one functor which is an equivalence must be an adjoint equivalence. Therefore, for a fixed morphism $f$, the "category of adjoint equivalence data $(f,g,\eta,\varepsilon)$" is either empty (if $f$ is not an equivalence) or equivalent to the [[terminal category]] (if $f$ is an equivalence). In other words, it is a [[(-1)-category]].
Therefore, in any 2-category, the following data are all equivalent (i.e. form equivalent categories):
* A morphism $f\colon X\to Y$ with the [[property]] of being an equivalence.
* A morphism $f\colon X\to Y$ with the *structure* of a morphism $g\colon Y \to X$ and an isomorphism $\eta\colon 1 \cong g f$, together with the *property* that there exists an isomorphism $f g \cong 1$.
* A morphism $f$ together with the structure of adjoint equivalence data $(f,g,\eta,\varepsilon)$.
In other words, adjoint equivalences are the way to make the property of "being an equivalence" completely into "algebraic" structure. However, they are *not* equivalent to the category of the following data:
* A morphism $f$ together with the structure of a morphism $g\colon Y \to X$ and arbitrary isomorphisms $\eta\colon 1 \cong g f$ and $\varepsilon\colon f g \cong 1$.
## Applications
### Intervals in homotopy theory
One instance of the usefulness of adjoint equivalences is that the "[[walking structure|walking]] adjoint equivalence" 2-category is equivalent to the [[point]]. Thus, it can be used as an [[interval object]] in $2Cat$, and in fact it is one of the generating cofibrations for the [[canonical model structure|canonical (Lack) model structure]] on $2Cat$. This is not true of the "walking non-adjoint equivalence."
### Defining tricategories
The original definition of [[tricategory]] by Gordon-Power-Street involved coherence 2-morphisms with the property of being equivalences in the relevant hom-bicategories. This is fine for most purposes, but for others it is insufficient, such as the following.
* Since "being an equivalence" is not algebraic structure, the GPS definition of tricategory, taken literally, is not an algebraic structure. In particular, it is not [[monadic functor|monadic]] over 3-[[globular sets]], nor is it the algebras for a [[globular operad]]. Such monadicity is important if one wants to state [[coherence theorems]] as properties of [[free object|free]] structures.
* The definition of 3-functors and higher [[transfors]] between tricategories include data and axioms that involve composites incorporating not just the coherence equivalences, but their pseudo-inverses. Therefore, strictly speaking these definitions are not well-defined unless the definition of tricategory comes with chosen pseudo-inverses for these coherence equivalences---in which case one should certainly also choose full adjoint equivalence data in order that the space of choices be contractible.
These problems are, of course, easy to remedy by simply requiring adjoint equivalence data rather than merely single equivalence morphisms. This change was first written down by Gurski.
### Cartesian closed 2-categories
In a [[cartesian closed category]] with [[equalizers]], for any two objects $X$ and $Y$ one can construct the "object of isomorphisms from $X$ to $Y$" as the following equalizer:
$$ Iso(X,Y) \to X^Y \times Y^X \;\rightrightarrows\; X^X \times Y^Y $$
where the top arrow on the right side is (composition, reversed composition) and the bottom arrow factors through $(id,id)\colon 1 \to X^X \times Y^Y$. One can then prove that the maps $Iso(X,Y)\to X^Y$ and $Iso(X,Y)\to Y^X$ are monic, so that $Iso(X,Y)$ can be regarded either as "the object of maps $X\to Y$ which are isomorphisms" or "the object of maps $Y\to X$ which are isomorphisms" (or, as is most evident from its construction, "the object of pairs of maps $X\to Y$ and $Y\to X$ which are inverse isomorphisms").
In a [[cartesian closed 2-category]], however, the analogous "2-equalizer" $Eqv(X,Y)$, does not have similar properties: the projections $Eqv(X,Y)\to X^Y$ and $Eqv(X,Y)\to Y^X$ will not in general be [[fully faithful morphism|fully faithful]]. Thus, we can only regard $Eqv(X,Y)$ as "the object of not-necessarily-adjoint equivalence data $(f,g,\eta,\varepsilon)$." However, if we use a further [[equifier]] to construct its "subobject of adjoint equivalence data" $AdjEqv(X,Y)$, then the projections $AdjEqv(X,Y)\to X^Y$ and $AdjEqv(X,Y)\to Y^X$ will be fully faithful, so that $AdjEqv(X,Y)$ can also be regarded as "the object of maps $X\to Y$ which are equivalences" and dually.
## In higher category theory
In [[higher category theory]], one expects to have a similar "fully coherent" notion of "adjoint equivalence" in any [[n-category]] or [[infinity-category]], and one hopes to prove a similar theorem that any [[equivalence]] can be refined to an adjoint equivalence. This is known to be true at least in the following cases:
* For [[Gray-categories]], the statement and proof is in [[Steve Lack]]'s paper [1001.2366](http://arxiv.org/abs/1001.2366) on the [[model structure for Gray-categories]]. See [[adjoint 2-equivalence]].
* For [[tricategories]], the corresponding statement can be deduced from the Gray-categorical version using the [[coherence theorem for tricategories]]. A direct proof can also be found in [[Nick Gurski]]'s paper [Biequivalences in tricategories](http://www.tac.mta.ca/tac/volumes/26/14/26-14abs.html).
* For [[strict omega-categories]], more or less this fact can be found in the study of "generic squares" in the paper [0712.0617](http://arxiv.org/abs/0712.0617) on the [[model structure for strict omega-categories]].
* For [[quasicategories]], the theorem is true, where an "adjoint equivalence" means simply a map out of the [[nerve]] of the [[interval groupoid]]; see [[equivalence in a quasicategory]].
## Related concepts
* [[enriched adjoint equivalence]]
[[!redirects adjoint equivalences]]
[[!redirects adjoint equivalence of categories]]
[[!redirects adjoint equivalences of categories]]
|
adjoint functor | https://ncatlab.org/nlab/source/adjoint+functor | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
> (The special case in [[Cat]] of the general notion of *[[adjunction]]*.)
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
The concept of _adjoint functors_ [[Kan (1958)](#Kan58)] is a key concept in [[category theory]] --- if not _the_ key concept --- and it is in large part through the manifold identification of examples of adjoint functors appearing ubiquituously in the practice of [[mathematics]] that category theoretic tools are brought to use in general mathematics.
Abstractly, the notion of adjoint functors embodies the concept of *[[representable functors]]* and has as special cases the fundamental [[universal constructions]] of [[category theory]] such as notably *[[Kan extensions]]* and hence *([[colimit|co-]])[[limits]]* and *([[coend|co-]])[[ends]]*, while being itself the archetypical special case of a natural notion of *[[adjunction]]* which in [[2-category theory]] embodies a general [principle of duality](geometry+of+physics+--+categories+and+toposes#CategoryTheoryIsTheoryOfDuality).
Concretely, the concept of adjoint functors $L \dashv R \,\colon\, \mathcal{D} \leftrightarrows \mathcal{C}$ is immediately transparent and compelling in its incarnation as a [[natural isomorphism]] on [[hom-sets]] (see [below](#InTermsOfHomIsomorphism)) and more generally on [[hom-objects]] (see at *[[enriched adjoint functor]]*), where it just says that adjoint functors are those that may [[coherence condition|coherently]] be switched left$\leftrightarrow$right in a [[hom-set]]
$$
L \dashv R
\;\colon\;
\mathcal{D} \underoverset{R}{L}{\leftrightarrows} \mathcal{C}
\;\;\;\;\;\;\;\;\;\;\;\;\;
\text{means}
\;\;\;\;\;\;\;\;\;\;\;\;\;
\mathcal{D}\bigl(L(-),\,-\bigr)
\;\simeq\;
\mathcal{C}\bigl(-,\,R(-)\bigr)
\,.
$$
The striking analogy (in fact a kind of [[categorification]]) of this defining relation to the older notion of *[[adjoint operator|adjoint linear operators]]* between [[Hermitian vector spaces]]/[[Hilbert spaces]] $\mathscr{H}_i$ with [[inner products]] $\langle -,-\rangle_i$
$$
L
=
R^\dagger
\;\colon\;
\mathscr{H}_2 \leftrightarrows \mathscr{H}_1
\;\;\;\;\;\;\;\;\;\;\;
\text{means}
\;\;\;\;\;\;\;\;\;\;\;
\bigl\langle
L(-),\, -
\bigr\rangle_2
\;=\;
\bigl\langle
-,\, R(-)
\bigr\rangle_1
$$
is what gives the notion of adjoint functors its name (cf. further discussion at *[adjoint operator -- history](adjoint+operator#History)*).
> "the universality of the concept of adjointness, which was first isolated and named in the conceptual sphere of category theory" [[Lawvere (1969)](#Lawvere69)]
> "The multiple examples, here and elsewhere, of adjoint functors tend to show that adjoints occur almost everywhere in many branches of Mathematics. It is the thesis of [this book](#MacLane71) that a systematic use of all these adjunctions illuminates and clarifies these subjects." [[MacLane (1971), p. 103](#MacLane71)]
> "In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role." [first line from _[An interview with William Lawvere](https://ncatlab.org/nlab/show/William+Lawvere#Interview07)_, paraphrasing the first paragraph of _[Taking categories seriously](William+Lawvere#TakingCategoriesSeriously)_]
## Definition
There are various different but equivalent characterizations of adjoint functors, some of which are discussed below.
### In terms of Hom isomorphism
{#InTermsOfHomIsomorphism}
We discuss here the definition of adjointness of functors $L \dashv R$ in terms of a [[natural bijection]] between [[hom-sets]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} below):
$$
\{L(c) \to d\} \;\simeq\; \{ c \to R(d) \}
$$
We show that this is equivalent to the abstract definition, in terms of an [[adjunction]] in the [[2-category]] [[Cat]], in Prop. \ref{AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat} below.
$\,$
+-- {: .num_defn #AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}
###### Definition
**([[adjoint functors]] in terms of [[natural bijections]] of [[hom-sets]])**
Let $\mathcal{C}$ and $\mathcal{D}$ be two [[categories]], and let
$$
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{}
\mathcal{C}
$$
be a pair of [[functors]] between them, as shown. Then this is called a _pair of [[adjoint functors]]_ (or an _[[adjoint pair]] of [[functors]]_) with $L$ _[[left adjoint]]_ and $R$ _[[right adjoint]]_, denoted
\begin{center}
\begin{tikzcd}
\mathcal{D}
\arrow[r, shift right=6pt, "R"', "\bot"]
&
\mathcal{C}
\arrow[l, shift right=6pt, "L"']
\end{tikzcd}
\end{center}
if there exists a [[natural isomorphism]] between the [[hom-functors]] of the following form:
\[
\label{HomIsomorphismForAdjointFunctors}
Hom_{\mathcal{D}}\big(L(-),\,-\big)
\;\simeq\;
Hom_{\mathcal{C}}\big(-,\,R(-)\big)
\,.
\]
This means that for all [[objects]] $c \in \mathcal{C}$ and $d \in \mathcal{D}$ there is a [[bijection]] of [[hom-sets]]
$$
\array{
Hom_{\mathcal{D}}(L(c),d)
&\overset{\simeq}{\longrightarrow}&
Hom_{\mathcal{C}}(c,R(d))
\\
( L(c) \overset{f}{\to} d )
&\mapsto&
(c \overset{\widetilde f}{\to} R(d))
}
$$
which is [[natural bijection|natural]] in $c$ and $d$. This isomorphism is the **adjunction isomorphism** and the [[image]] $\widetilde f$ of a morphism $f$ under this bijections is called the _[[adjunct]]_ of $f$. Conversely, $f$ is called the adjunct of $\widetilde f$.
Naturality here means that for every [[morphism]] $g \colon c_2 \to c_1$ in $\mathcal{C}$ and for every [[morphism]] $h\colon d_1\to d_2$ in $\mathcal{D}$, the resulting square
\[
\label{NaturalitySquareForAdjointnessOfFunctors}
\array{
Hom_{\mathcal{D}}(L(c_1), d_1)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c_1, R(d_1))
\\
{}^{\mathllap{Hom_{\mathcal{D}}(L(g), h)}}\big\downarrow
&&
\big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(g, R(h))}}
\\
Hom_{\mathcal{D}}(L(c_2),d_2)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c_2,R(d_2))
}
\]
[[commuting square|commutes]] (see also at _[[hom-functor]]_ for the definition of the vertical maps here).
Explicitly, this commutativity, in turn, means that for every morphism $f \;\colon\; L(c_1) \to d_1$ with [[adjunct]] $\widetilde f \;\colon\; c_1 \to R(d_1)$, the adjunct of the [[composition]] is
$$
\array{
L(c_1) & \overset{f}{\longrightarrow} & d_1
\\
{}^{\mathllap{L(g)}}\big\uparrow && \big\downarrow^{\mathrlap{h}}
\\
L(c_2) && d_2
}
\;\;\;=\;\;\;
\array{
c_1 &\overset{\widetilde f}{\longrightarrow}& R(d_1)
\\
{}^{\mathllap{g}}\big\uparrow && \big\downarrow^{\mathrlap{R(h)}}
\\
c_2 && R(d_2)
}
$$
=--
+-- {: .num_defn #AdjunctionUnitFromHomIsomorphism}
###### Definition
**([[adjunction unit]] and [[adjunction counit|counit]] in terms of hom-isomorphism)
Given a pair of [[adjoint functors]]
$$
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot}
\mathcal{C}
$$
according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} one says that
1. for any $c \in \mathcal{C}$ the [[adjunct]] of the [[identity morphism]] on $L(c)$ is the _[[unit of an adjunction|unit morphism]]_ of the adjunction at that object, denoted
$$
\eta_c \coloneqq \widetilde{id_{L(c)}} \;\colon\; c \longrightarrow R(L(c))
$$
1. for any $d \in \mathcal{D}$ the [[adjunct]] of the [[identity morphism]] on $R(d)$ is the _[[counit of an adjunction|counit morphism]]_ of the adjunction at that object, denoted
$$
\epsilon_d \;\colon\; L(R(d)) \longrightarrow d
$$
=--
+-- {: .num_prop #GeneralAdjunctsInTermsOfAdjunctionUnitCounit}
###### Proposition
**(general [[adjuncts]] in terms of [[adjunction unit|unit/counit]])**
Consider a pair of [[adjoint functors]]
$$
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot}
\mathcal{C}
$$
according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}, with [[adjunction units]] $\eta_c$ and [[adjunction counits]] $\epsilon_d$ according to Def. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}.
Then
1. The [[adjunct]] $\widetilde f$ of any morphism $L(c) \overset{f}{\to} d$ is obtained from $R$ and $\eta_c$ as the [[composition|composite]]
\[
\label{AdjunctFormula}
\widetilde f
\;\colon\;
c
\overset{\eta_c}{\longrightarrow}
R(L(c))
\overset{R(f)}{\longrightarrow}
R(d)
\]
Conversely, the [[adjunct]] $f$ of any morphism $c \overset{\widetilde f}{\longrightarrow} R(d)$ is obtained from $L$ and $\epsilon_d$ as
\[
\label{ConverseAdjunctFormula}
f
\;\colon\;
L(c)
\overset{L(\widetilde f)}{\longrightarrow}
L(R(d))
\overset{\epsilon_d}{\longrightarrow}
d
\]
1. The [[adjunction units]] $\eta_c$ and [[adjunction counits]] $\epsilon_d$ are components of [[natural transformations]] of the form
$$
\eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L
$$
and
$$
\epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}
$$
1. The [[adjunction unit]] and [[adjunction counit]] satisfy the [[triangle identities]], saying that
$$
id_{L(c)}
\;\colon\;
L(c)
\overset{L(\eta_c)}{\longrightarrow}
L(R(L(c)))
\overset{\epsilon_{L(c)}}{\longrightarrow}
L(c)
$$
and
$$
id_{R(d)}
\;\colon\;
R(d)
\overset{\eta_{R(d)}}{\longrightarrow}
R(L(R(d)))
\overset{R(\epsilon_d)}{\longrightarrow}
R(d)
$$
=--
+-- {: .proof}
###### Proof
For the first statement, consider the [[naturality square]] (eq:NaturalitySquareForAdjointnessOfFunctors) in the form
$$
\array{
id_{L(c)}
\in
&
Hom_{\mathcal{D}}(L(c), L(c))
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c, R(L(c)))
\\
&
{}^{\mathllap{Hom_{\mathcal{D}}(L(id), f)}}\big\downarrow
&&
\big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(id, R(f))}}
\\
&
Hom_{\mathcal{D}}(L(c), d)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}( c, R(d) )
}
$$
and consider the element $id_{L(c_1)}$ in the top left entry. Its image under going down and then right in the diagram is $\widetilde f$, by Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}. On the other hand, its image under going right and then down is $ R(f)\circ \eta_{c}$, by Def. \ref{AdjunctionUnitFromHomIsomorphism}. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown, for the adjunct of $f$.
The converse formula follows analogously.
The third statement follows directly from this by applying these formulas for the [[adjuncts]] twice and using that the result must be the original morphism:
$$
\begin{aligned}
id_{L(c)}
& =
\widetilde \widetilde { id_{L(c)} }
\\
& = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) }
\\
& =
L(c)
\overset{L(\eta_c)}{\longrightarrow}
L(R(L(c)))
\overset{\epsilon_{L(c)}}{\longrightarrow}
L(c)
\end{aligned}
$$
For the second statement, we have to show that for every morphism $f \colon c_1 \to c_2$ the following [[commuting square|square commutes]]:
$$
\array{
c_1 &\overset{f}{\longrightarrow}& c_2
\\
{}^{\mathllap{\eta_{c_1}}}\big\downarrow
&&
\big\downarrow^{\mathrlap{\eta_{c_2}}}
\\
R(L(c_1))
&\underset{ R(L(f)) }{\longrightarrow}&
R(L(c_2))
}
$$
To see this, consider the [[naturality square]] (eq:NaturalitySquareForAdjointnessOfFunctors) in the form
$$
\array{
id_{L(c_2)}
\in
& Hom_{\mathcal{D}}(L(c_2), L(c_2))
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c_2, R(L(c_2)))
\\
&
{}^{\mathllap{Hom_{\mathcal{D}}(L(f),id_{L(c_2)})}}\big\downarrow
&&
\big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(f, R(id_{L(c_2)}))}}
\\
&
Hom_{\mathcal{D}}(L(c_1),L(c_2))
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c_1,R(L(c_2)))
}
$$
The image of the element $id_{L(c_2)}$ in the top left along the right and down is $ \eta_{c_2} \circ f$, by Def. \ref{AdjunctionUnitFromHomIsomorphism}, while its image down and then to the right is $\widetilde{L(f)} = R(L(f)) \circ \eta_{c_1}$, by the previous statement. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown.
The argument for the naturality of $\epsilon$ is directly analogous.
=--
+-- {: .num_prop #AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat}
###### Proposition
**(adjointness in terms of hom-isomorphism equivalent to adjunction in $Cat$)**
Two functors
$$
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{}
\mathcal{C}
$$
are an [[adjoint pair]] in the sense that there is a [[natural isomorphism]] (eq:HomIsomorphismForAdjointFunctors) according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}, precisely if they participate in an [[adjunction]] in the [[2-category]] [[Cat]], meaning that
1. there exist [[natural transformations]]
$$
\eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L
$$
and
$$
\epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}
$$
2. which satisfy the [[triangle identities]]
$$
id_{L(c)}
\;\colon\;
L(c)
\overset{L(\eta_c)}{\longrightarrow}
L(R(L(c)))
\overset{\epsilon_{L(c)}}{\longrightarrow}
L(c)
$$
and
$$
id_{R(d)}
\;\colon\;
R(d)
\overset{\eta_{R(d)}}{\longrightarrow}
R(L(R(d)))
\overset{R(\epsilon_d)}{\longrightarrow}
R(d)
$$
=--
+-- {: .proof}
###### Proof
That a hom-isomorphism (eq:HomIsomorphismForAdjointFunctors) implies units/counits satisfying the [[triangle identities]] is the statement of the second two items of Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}.
Hence it remains to show the converse. But the argument is along the same lines as the proof of Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}: We now _define_ forming of adjuncts by the formula (eq:AdjunctFormula). That the resulting assignment $f \mapsto \widetilde f$ is an [[isomorphism]] follows from the computation
$$
\begin{aligned}
\widetilde {\widetilde f}
& =
\widetilde{ c \overset{\eta_c}{\to} R(L(c)) \overset{R(f)}{\to} R(d) }
\\
& =
L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{L(R(f))}{\to} L(R(d)) \overset{\epsilon_d}{\to} d
\\
& =
L(c) \overset{L(\eta_c)}{\to} L(R(L(c)))
\overset{ \epsilon_{L(c)} }{\to} L(c)
\overset{f}{\longrightarrow} d
\\
& = L(c) \overset{f}{\longrightarrow} d
\end{aligned}
$$
where, after expanding out the definition, we used [[natural transformation|naturality]] of $\epsilon$ and then the [[triangle identity]].
Finally, that this construction satisfies the naturality condition (eq:NaturalitySquareForAdjointnessOfFunctors) follows from the functoriality of the functors involved, and the naturality of the unit/counit:
$$
\array{
c_2 &\overset{ \eta_{c_2} }{\longrightarrow}& R(L(c_2))
\\
{}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{R(L(g))}}
& \searrow^{\mathrlap{ R( L(g) \circ f ) }}
\\
c_1
&\overset{\eta_{c_1}}{\longrightarrow}&
R(L(c_1))
&\overset{R(f)}{\longrightarrow}&
R(d_1)
\\
&& & {}_{R( h\circ f)}\searrow & \downarrow^{\mathrlap{ R(h) }}
\\
&& && R(d_2)
}
$$
=--
### In terms of representable functors
{#InTermsOfRepresentableFunctors}
The condition (eq:HomIsomorphismForAdjointFunctors) on adjoint functors $L \dashv R$ in Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets} implies in particular that for every [[object]] $d \in \mathcal{D}$ the functor $Hom_{\mathcal{D}}(L(-),d)$ is a _[[representable functor]]_ with _[[representing object]]_ $R(d)$. The following Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} observes that the existence of such [[representing objects]] for all $d$ is, in fact, already sufficient to imply that there is a right adjoint functor.
This equivalent perspective on adjoint functors makes manifest that:
1. adjoint functors are, if they exist, unique up to natural isomorphism, this is Prop. \ref{UniquenessOfAdjoints} below;
1. the concept of adjoint functors makes sense also _[[relative adjoint functor|relative]]_ to a [[full subcategory]] on which representing objects exists, this is the content of Remark \ref{RelativeAdjointFunctors} below.
#### Global definition
+-- {: .num_prop #AdjointFunctorFromObjectwiseRepresentingObject}
###### Proposition
**(adjoint functor from objectwise [[representing object]])**
A [[functor]] $L \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ has a [[right adjoint]] $R \;\colon\; \mathcal{D} \to \mathcal{C}$, according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}, already if
for all [[objects]] $d \in \mathcal{D}$ there is an object $R(d) \in \mathcal{C}$ such that there is a [[natural isomorphism]]
$$
Hom_{\mathcal{D}}(L(-),d)
\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}
Hom_{\mathcal{C}}(-,R(d))
\,,
$$
hence for each [[object]] $c \in \mathcal{C}$ a [[bijection]]
$$
Hom_{\mathcal{D}}(L(c),d)
\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}
Hom_{\mathcal{C}}(c,R(d))
$$
such that for each [[morphism]] $g \;\colon\; c_2 \to c_1$, the following [[commuting diagram|diagram commutes]]
\[
\label{HalfNaturalitySquareForAdjointnessOfFunctors}
\array{
Hom_{\mathcal{D}}(L(c_1),d)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c_1,R(d))
\\
{}^{\mathllap{ Hom_{\mathcal{C}}(L(g),id_d) }}
\big\downarrow
&&
\big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, id_{R(d)} ) }}
\\
Hom_{\mathcal{D}}(L(c_2),d)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(c_2,R(d))
}
\]
(This is as in (eq:NaturalitySquareForAdjointnessOfFunctors), except that only naturality in the first variable is required.)
In this case there is a unique way to extend $R$ from a function on [[objects]] to a function on [[morphisms]] such as to make it a [[functor]] $R \colon \mathcal{D} \to \mathcal{C}$ which is [[right adjoint]] to $L$.
, and hence the statement is that with this, naturality in the second variable is already implied.
=--
+-- {: .proof}
###### Proof
Notice that
1. in the language of [[presheaves]] the assumption is that for each $d \in \mathcal{D}$ the presheaf
$$
Hom_{\mathcal{D}}(L(-),d)
\;\in\;
[\mathcal{C}^{op}, Set]
$$
is [[representable functor|represented]] by the object $R(d)$, and [[natural transformation|naturally]] so.
1. In terms of the [[Yoneda embedding]]
$$
y
\;\colon\;
\mathcal{C}
\hookrightarrow
[\mathcal{C}^{op}, Set]
$$
we have
\[
\label{YonedanotationForRepresentable}
Hom_{\mathcal{C}}(-,R(d))
=
y(R(d))
\]
The condition (eq:NaturalitySquareForAdjointnessOfFunctors) says equivalently that $R$ has to be such that for all [[morphisms]] $h \;\colon\; d_1 \to d_2 $ the following diagram in the [[category of presheaves]] $[\mathcal{C}^{op}, Set]$ [[commuting diagram|commutes]]
$$
\array{
Hom_{\mathcal{D}}(L(-),d_1)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(-,R(d_1))
\\
{}^{\mathllap{ Hom_{\mathcal{C}}( L(-) , h ) }}
\big\downarrow
&&
\big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( -, R(h) ) }}
\\
Hom_{\mathcal{D}}(L(-),d_2)
&\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}&
Hom_{\mathcal{C}}(-, R(d_2))
}
$$
This manifestly has a unique solution
$$
y(R(h))
\;=\;
Hom_{\mathcal{C}}(-,R(h))
$$
for every morphism $h \colon d_1 \to d_2$ under $y(R(-))$ (eq:YonedanotationForRepresentable). But the [[Yoneda embedding]] $y$ is a [[fully faithful functor]] ([this prop.](Yoneda+embedding#YonedaEmbeddingIsFullyFaithful)), which means that thereby also $R(h)$ is uniquely fixed.
=--
+-- {: .num_remark}
###### Remark
In more fancy language, the statement of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} is the following:
By precomposition $L$ defines a functor of [[presheaf categories]]
$$
L^* \;\colon\; [\mathcal{D}^{op}, Set] \to [\mathcal{C}^{op}, Set]
\,.
$$
By [[restriction]] along the [[Yoneda embedding]] $y \;\colon\; \mathcal{D} \to [\mathcal{D}^{op}, Set]$ this yields the functor
$$
\bar L
\;\colon\;
\array{
\mathcal{D}
&\overset{y}{\longrightarrow}&
[\mathcal{D}^{op}, Set]
&\overset{L^*}{\longrightarrow}&
[\mathcal{C}^{op}, Set]
\\
d
&\mapsto&
Hom_{\mathcal{D}}(-,d)
&\mapsto&
Hom_{\mathcal{D}}(L(-),d)
}
\,.
$$
The statement is that for all $d \in D$ this presheaf $\bar L(d)$ is [[representable functor|representable]], then it is functorially so in that there exists a functor $R \colon \mathcal{D} \to \mathcal{C}$ such that
$$
\bar L \;\simeq\; y \circ R
\,.
$$
=--
#### Local definition
{#LocalDefinition}
+-- {: .num_remark #RelativeAdjointFunctors}
###### Remark
**([[relative adjoint functors]])**
The perspective of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} has the advantage that it yields useful information even if the adjoint functor $R$ does not exist globally, i.e. as a functor on all of $\mathcal{D}$:
It may happen that
$$
\bar L(d) \coloneqq Hom_D(L(-),d) \in [C^{op}, Set]
$$
is [[representable functor|representable]] for _some_ [[object]] $d \in \mathcal{D}$ but not for all $d$. The representing object may still usefully be thought of as $R(d)$, and in fact it may be viewed as a right adjoint to $L$ _relative to_ the inclusion of the [[full subcategory]] determined by those $d$s for which $\bar L(d)$ is representable; see [[relative adjoint functor]] for more.
This _global_ versus _local_ evaluation of adjoint functors induces the global/local pictures of the definitions
* [[limit]] / [[homotopy limit]]
* [[Kan extension]]
as discussed there.
=--
### In terms of universal factorization through a (co)unit
{#UniversalArrows}
We have seen in Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit} that the [[unit of an adjunction]] and [[counit of an adjunction]] plays a special role. One may amplify this by characterizing these morphisms as _[[universal arrows]]_ in the sense of the following Def. \ref{UniversalArrow}. In fact the existence of these is already equivalent to the existence of an adjoint functor, this is the statement of Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor} below.
$\,$
+-- {: .num_defn #UniversalArrow}
###### Definition
**(universal arrow)**
Given a [[functor]] $R \;\colon\; \mathcal{D} \to \mathcal{C}$, and an object $c\in \mathcal{C}$, a _universal arrow_ from $c$ to $R$ is an [[initial object]] of the [[comma category]] $(c/R)$. This means that it consists of
1. an [[object]] $L(c)\in \mathcal{D}$
1. a [[morphism]] $\eta_c \;\colon\; c \to R\big(L(c)\big)$, to be called the _[[adjunction unit|unit]]_,
such that for any $d\in \mathcal{D}$, any morphism $f \colon c\to R(d)$ factors through this unit $\eta_c$ as
\[
\label{UniversalArrowFactorization}
\array{
&& c
\\
&
{}^{\mathllap{\eta_c}}\swarrow
&&
\searrow^{\mathrlap{f}}
\\
R\big(L(c)\big)
&&
\underset
{R (\widetilde f)}
{\longrightarrow}
&&
R(d)
\\
\\
L(c)
&&
\underset{ \widetilde f}{\longrightarrow}
&&
d
}
\]
for a unique $\widetilde f \;\colon\; L(c) \longrightarrow d$, to be called the [[adjunct]] of $f$.
=--
(e.g. [Borceux, Vol. 1, Definition 3.1.1](#Borceux94))
+-- {: .num_prop #UniversalMorphismsAreInitialObjectsInCommaCategory}
###### Proposition
**([[universal morphisms]] are [[initial objects]] in the [[comma category]])**
Let $R: \mathcal{D} \to \mathcal{C}$ be a [[functor]] and $c \in \mathcal{C}$ an [[object]]. Then the following are equivalent:
1. $c \overset{\eta_c}{\to} R(L(c))$ is a [[universal morphism]] into $R(L(c))$ (Def. \ref{UniversalArrow});
1. $(c, \eta_c)$ is the [[initial object]] in the [[comma category]] $c/R$.
=--
+-- {: .num_prop #CollectionOfUniversalArrowsEquivalentToAdjointFunctor}
###### Proposition
**(collection of [[universal arrows]] equivalent to [[adjoint functor]])**
Let $R \;\colon\; \mathcal{D} \to \mathcal{C}$ be a [[functor]]. Then the following are equivalent:
1. $R$ has a [[left adjoint]] functor $L \colon \mathcal{C} \to \mathcal{D}$ according to Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets},
1. for every [[object]] $c \in \mathcal{C}$ there is a universal arrow $c \overset{\eta_c}{\longrightarrow} R(L(c))$, according to Def. \ref{UniversalArrow}.
=--
+-- {: .proof}
###### Proof
In one direction, assume a [[left adjoint]] $L$ is given. Define the would-be universal arrow at $c \in \mathcal{C}$ to be the [[unit of an adjunction|unit of the adjunction]] $\eta_c$ via Def. \ref{AdjunctionUnitFromHomIsomorphism}. Then the statement that this really is a universal arrow is implied by Prop. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit}.
In the other direction, assume that universal arrows $\eta_c$ are given. The uniqueness clause in Def. \ref{UniversalArrow} immediately implies [[bijections]]
$$
\array{
Hom_{\mathcal{D}}(L(c),d)
&\overset{\simeq}{\longrightarrow}&
Hom_{\mathcal{C}}(c,R(d))
\\
\left(
L(c) \overset{\widetilde f}{\to} d
\right)
&\mapsto&
\left(
c \overset{\eta_c}{\to} R(L(c)) \overset{ R(\widetilde f) }{\to} R(d)
\right)
}
$$
Hence to satisfy (eq:HomIsomorphismForAdjointFunctors) it remains to show that these are [[natural transformation|natural]] in both variables. In fact, by Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} it is sufficient to show naturality in the variable $d$. But this is immediate from the functoriality of $R$ applied in (eq:UniversalArrowFactorization): For $h \colon d_1 \to d_2$ any [[morphism]], we have
$$
\array{
&& c
\\
& {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}}
\\
R (L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d_1)
\\
&& {}_{\mathllap{ R( h\circ \widetilde f ) }}\searrow && \downarrow^{\mathrlap{R(h)}}
\\
&& && R(d_2)
}
$$
=--
+-- {: .num_example}
###### Example
**([[localization]] via [[universal arrows]])**
The characterization of adjoint functors in terms of universal factorizations through the unit and counit (Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor}) is of particular interest in the case that $R$ is a [[full and faithful functor]]
$$
R \;\colon\;
\mathcal{D} \hookrightarrow \mathcal{C}
$$
exhibiting $\mathcal{D}$ as a [[reflective subcategory]] of $\mathcal{C}$. In this case we may think of $L$ as a [[localization]] and of objects in the [[essential image]] of $L$ as **local objects**. Then the above says that:
* every morphism $c \to R d$ from $c$ into a local object factors throught the localization of $c$.
=--
### In terms of comma categories
{#InTermsOfCommaCategories}
A [[functor]] $L \colon C \to D$ is [[left adjoint]] to a functor $R \colon D \to C$ if and only if there is an isomorphism (not [[equivalence]]) of [[comma categories]] $L \downarrow D \cong C \downarrow R$ and this isomorphism commutes with the [[forgetful functors]] to the [[product category]] $C \times D$. See §B.I.2 of *[[Functorial Semantics of Algebraic Theories]]*.
This characterisation generalises (in the unenriched setting) to [[relative adjunctions]] by replacing $C \downarrow R$ by $J \downarrow R$.
### In terms of cographs/correspondences/heteromorphisms
{#InTermsOfCographsHeteromorphisms}
Every [[profunctor]]
$$
k : C^{op} \times D \to S
$$
defines a category $C *^k D$ with $Obj(C *^k D) = Obj(C) \sqcup Obj(D)$ and with [[hom set]] given by
$$
Hom_{C^{op} \times D}(X,Y)
=
\left\{
\array{
Hom_C(X,Y) & if X, Y \in C
\\
Hom_{D}(X,Y) & if X,Y \in D
\\
k(X,Y) & if X \in C and Y \in D
\\
\emptyset & otherwise
}
\right.
$$
($k(X,Y)$ is also called the _[[heteromorphisms]]_).
This category naturally comes with a functor to the [[interval]] category
$$
C *^k D \to \Delta^1
\,.
$$
Now, every functor $L : C \to D$ induces a [[profunctor]]
$$
k_L(X,Y) = Hom_D(L(X), Y)
$$
and every functor $R : D \to C$ induces a [[profunctor]]
$$
k_R(X,Y) = Hom_C(X, R(Y))
\,.
$$
The functors $L$ and $R$ are adjoint precisely if the [[profunctors]] that they define in the above way are equivalent. This in turn is the case if
$C \star^L D \simeq (D^{op} \star^{R^{op}} C^{op})^{op}$.
We say that $C \star^k D$ is the [[cograph of a functor|cograph of the functor]] $k$. See there for more on this.
### In terms of graphs/2-sided discrete fibrations
Functors $L \colon C \to D$ and $R \colon D \to C$ are adjoint precisely if we have a [[commutative diagram]] of the form
$$
\array{
(L \downarrow Id_D)
&&\stackrel{\cong}{\to}&&
(Id_C \downarrow R)
\\
& \searrow && \swarrow
\\
&&
C \times D
\mathrlap{\,,}
}
$$
where the downwards arrows denote the [[maps]] induced by the canonical projections out of the [[comma categories]]. This definition of adjoint functors was introduced by [[Lawvere]] in *[[Functorial Semantics of Algebraic Theories]]*, and was the original motivation for [[comma categories]].
The above [[diagram]] may be recovered directly from the image under the equivalence $[C^{op} \times D, Set] \stackrel{\simeq}{\to} DFib(D,C) $ described at [[2-sided fibration]] of the isomorphism of induced [[profunctors]] $C^{op} \times D \to Set$ (see above at "In terms of Hom isomorphism"). Its relation to the hom-set definition of adjoint functors can thus be understood within the general paradigm of [[Grothendieck construction]]-like correspondences. Consequently, this description is not viable for [[enriched category|enriched adjunctions]].
This description generalises to [[relative adjunctions]] by replacing $Id_C$ with $J$.
### In terms of Kan extensions/liftings
Given $L \colon C \to D$, we have that it has a [[right adjoint]] $R\colon D \to C$ precisely if the [[left Kan extension]] $Lan_L 1_C$ of the [[identity]] along $L$ exists and is [absolute](/nlab/show/Kan+extension#AbsoluteKanExtension), in which case
$$
R \simeq \mathop{Lan}_L 1_C
\,.
$$
In this case, the universal 2-cell $1_C \to R L$ corresponds to the
[[unit of an adjunction|unit of the adjunction]]; the counit and the verification of the triangular identities can all be obtained through properties of Kan extensions and absoluteness.
It is also possible to express this in terms of Kan liftings: $L$ has a right adjoint $R$ if and only if:
- $R \simeq \mathop{Rift}_L 1_D$ and this [[Kan lift]] is [[Kan lift|absolute]]
In this case, we get the counit as given by the universal cell $L R \to 1_D$, while the rest of the data and properties can be derived from it through the absolute Kan lifting assumption.
Dually, we have that for $R\colon D \to C$, it has a left adjoint $L \colon C \to D$ precisely if
- $L \simeq \mathop{Ran}_R 1_D$, and this Kan extension is [[Kan extension|absolute]]
or, in terms of left Kan liftings:
- $L \simeq \mathop{Lift}_R 1_C$, and this Kan lifting is [[Kan lift|absolute]]
This follows from the fact that the adjunction $L \dashv R$ [induces](#PrePostcompositeAdjunctions) adjunctions $- \circ R \dashv - \circ L$ and $L \circ - \dashv R \circ -$.
The formulations in terms of liftings generalize to (unenriched) [[relative adjoint|relative adjoints]] by allowing an arbitrary functor $J$ in place of the identity; see there for more.
### Transformation of adjoints
{#TransformationOfAdjoints}
There are several layers of generality at which one may consider a notion of [[homomorphism]] *between* adjoint functors.
Here is a basic but important notion:
\begin{definition}\label{ConjugateTransformationOfAdjoints}
**([[conjugate transformations of adjoints]])**
\linebreak
Given a [[pair]] of pairs of adjoint functors between the same [[categories]]
$$
\array{
\mathcal{C}
\underoverset
{\underset{R_1}{\longleftarrow}}
{\overset{L_1}{\longrightarrow}}
{\;\;\; \bot \;\;\;}
\mathcal{D}
\\
\mathcal{C}
\underoverset
{\underset{R_2}{\longleftarrow}}
{\overset{L_2}{\longrightarrow}}
{\;\;\; \bot \;\;\;}
\mathcal{D}
}
$$
then a [[pair]] of [[natural transformations]] between the adjoints of the same chirality, of this form
$$
\lambda \,\colon\, L_1 \to L_2
\;\;\;\;\;\;
\rho \,\colon\, R_2 \to R_1
$$
is called *conjugate for* [[MacLane (1971), §IV.7 (5)](#MacLane71)] or a *pseudo-transformation of* [[Harpaz & Prasma (2015), Sec. 2.2](#HarpazPrasma15)] the given adjunctions if they make the following [[diagram]] of [[natural transformations]] between [[hom-sets]] [[commuting diagram|commute]]:
$$
\array{
\mathcal{C}\big(
L_2(-)
,\,
(-)
\big)
&\overset{\sim}{\longrightarrow}&
\mathcal{D}\big(
(-)
,\,
R_2(-)
\big)
\\
\mathllap{{}^{
\mathcal{C}\big(\lambda_{(-)},\,-\big)
}}
\Big\downarrow
&&
\Big\downarrow
\mathrlap{{}^{
\mathcal{C}\big(-,\,\rho_{(-)}\big)
}}
\\
\mathcal{C}\big(
L_1(-)
,\,
(-)
\big)
&\overset{\sim}{\longrightarrow}&
\mathcal{D}\big(
(-)
,\,
R_1(-)
\big)
\mathrlap{\,,}
}
$$
where the horizontal maps are the given hom-isomorphisms (eq:HomIsomorphismForAdjointFunctors).
\end{definition}
This condition is compatible with [[horizontal composition|horizontal]] and [[vertical composition]] of [[natural transformations]] as [[2-morphisms]] in [[Cat]] and hence yields:
\begin{definition}\label{CatAdj}
The ([[very large category|very large]]) [[wide subcategory|wide]] and [[locally full sub-2-category]] [[CatAdj|$Cat_{Adj}$]] of [[Cat]]
\[
\label{CatAdjInsideCat}
Cat_{Adj} \longrightarrow Cat
\]
whose
* [[objects]] are categories,
* [[1-morphisms]] are [[left adjoint]] [[functors]]
* [[2-morphisms]] are [[natural transformations]] which are conjugate in the sense of Def. \ref{ConjugateTransformationOfAdjoints}.
\end{definition}
\begin{proposition}
Under the [[Grothendieck construction]], the [[Grothendieck fibrations]] which arise from [[pseudofunctors]] $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{Adj}$ (eq:CatAdjInsideCat) are equivalently the [[bifibrations]].
\end{proposition}
This may be [[category theory]] [[folklore]]; a proof has been spelled out in [Harpaz & Prasma (2015), Prop. 2.2.1](#HarpazPrasma15).
## Properties
{#Properties}
### Basic properties
+-- {: .num_prop #UniquenessOfAdjoints}
###### Proposition
**([[adjoint functors]] are unique up to [[natural isomorphism]])**
The [[left adjoint]] or [[right adjoint]] to a [[functor]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}), if it exists, is unique up to [[natural isomorphism]].
=--
+-- {: .proof}
###### Proof
Suppose the functor $L \colon \mathcal{D} \to \mathcal{C}$ is given, and we are asking for uniqueness of its right adjoint, if it exists. The other case is directly analogous.
Suppose that $R_1, R_2 \;\colon\; \mathcal{C} \to \mathcal{D}$ are two [[functors]] which are [[right adjoint]] to $L$. Then for each $d \in \mathcal{D}$ the corresponding two hom-isomorphisms (eq:HomIsomorphismForAdjointFunctors) combine to say that there is a [[natural isomorphism]]
$$
\Phi_d
\;\colon\;
Hom_{\mathcal{C}}(-,R_1(d))
\;\simeq\;
Hom_{\mathcal{C}}(-,R_2(d))
$$
As in the proof of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject}, the [[Yoneda lemma]] implies that
$$
\Phi_d \;=\; y( \phi_d )
$$
for some [[isomorphism]]
$$
\phi_d \;\colon\; R_1(d) \overset{\simeq}{\to} R_2(d)
\,.
$$
But then the uniqueness statement of Prop. \ref{AdjointFunctorFromObjectwiseRepresentingObject} implies that the collection of these isomorphisms for each object constitues a [[natural isomorphism]] between the functors.
=--
+-- {: .num_prop #AdjointsPreserveCoLimits}
###### Proposition
**([[left adjoints preserve colimits and right adjoints preserve limits]])**
Let $(L \dashv R) \colon \mathcal{D} \to \mathcal{C}$ be a pair of [[adjoint functors]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}). Then
* $L$ [[preserved limit|preserves]] all [[colimits]] that exist in $\mathcal{C}$,
* $R$ preserves all [[limits]] in $\mathcal{D}$.
=--
+-- {: .proof}
###### Proof
Let $y : I \to \mathcal{D}$ be a [[diagram]] whose [[limit]] $\lim_{\leftarrow_i} y_i$ exists. Then we have a sequence of [[natural isomorphism]]s, natural in $x \in C$
$$
\begin{aligned}
Hom_{\mathcal{C}}(x, R {\lim_\leftarrow}_i y_i)
& \simeq
Hom_{\mathcal{D}}(L x, {\lim_\leftarrow}_i y_i)
\\
& \simeq
{\lim_\leftarrow}_i Hom_{\mathcal{D}}(L x, y_i)
\\
& \simeq
{\lim_\leftarrow}_i Hom_{\mathcal{C}}( x, R y_i)
\\
& \simeq
Hom_{\mathcal{C}}( x, {\lim_\leftarrow}_i R y_i)
\,,
\end{aligned}
$$
where we used the hom-isomorphism (eq:HomIsomorphismForAdjointFunctors) and the fact that any [[hom-functor]] preserves limits (see there). Because this is natural in $x$ the [[Yoneda lemma]] implies that we have an [[isomorphism]]
$$
R {\lim_\leftarrow}_i y_i
\simeq
{\lim_\leftarrow}_i R y_i
\,.
$$
The argument that shows the preservation of colimits by $L$ is analogous.
=--
+-- {: .num_remark #AdjointFunctorTheorem}
###### Remark
A partial converse to Prop. \ref{AdjointsPreserveCoLimits} is provided by the [[adjoint functor theorem]]. See also _[Pointwise Expression](#PointwiseExpression)_ below.
=--
+-- {: .num_prop #FullyFaithfulAndInvertibleAdjoints}
###### Proposition
Let $L \dashv R$ be a pair of adjoint functors (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}). Then the following holds:
* $R$ is [[faithful functor|faithful]] precisely if the component of the [[unit of an adjunction|counit]] over every object $x$ is an [[epimorphism]] $L R x \stackrel{}{\to} x $;
* $R$ is [[full functor|full]] precisely if the component of the [[unit of an adjunction|counit]] over every object $x$ is a [[split monomorphism]] $L R x \stackrel{}{\to} x $;
* $L$ is [[faithful functor|faithful]] precisely if the component of the [[unit of an adjunction|unit]] over every object $x$ is a [[monomorphism]] $x \hookrightarrow R L x $;
* $L$ is [[full functor|full]] precisely if the component of the [[unit of an adjunction|unit]] over every object $x$ is a [[split epimorphism]] $x \to R L x $;
* $R$ is [[full and faithful functor|full and faithful]]
(exhibits a [[reflective subcategory]])
precisely if
the [[unit of an adjunction|counit]] is a [[natural isomorphism]]
$\epsilon : L \circ R \stackrel{\simeq}{\to} Id_D$
* $L$ is [[full and faithful functor|full and faithful]]
(exhibits a [[coreflective subcategory]]) precisely if
the [[unit of an adjunction|unit]]
is a natural isomorphism
$\eta : Id_C \stackrel{\simeq}{\to} R \circ L$.
* The following are equivalent:
* $L$ and $R$ are both [[full and faithful functor|full and faithful]];
* $L$ is an [[equivalence of categories|equivalence]];
* $R$ is an [[equivalence of categories|equivalence]].
=--
{#FurtherProperties} (Further properties are listed in [this MathOverflow discussion](https://mathoverflow.net/questions/100808/properties-of-functors-and-their-adjoints).)
| | | |
|---|----|----|
| $\phantom{A}$**[[adjunction]]**$\phantom{A}$ | | $\phantom{A}$[[adjunction unit|unit]] is [[isomorphism|iso]]:$\phantom{A}$ |
| | | $\phantom{A}$[[coreflective subcategory|coreflection]]$\phantom{A}$ |
| $\phantom{A}$[[adjunction counit|counit]] is [[isomorphism|iso]]:$\phantom{A}$ | $\phantom{A}$[[reflective subcategory|reflection]]$\phantom{A}$ | $\phantom{A}$[[adjoint equivalence of categories|adjoint equivalence]]$\phantom{A}$ |
{: style='margin:auto'}
+-- {: .proof}
###### Proof
For the characterization of faithful $R$ by epi counit components, notice (as discussed at _[[epimorphism]]_ ) that $L R x \to x$ being an epimorphism is equivalent to the induced [[function]]
$$
Hom(x, a) \to Hom(L R x, a)
$$
being an [[injection]] for all objects $a$. Then use that, by adjointness, we have an isomorphism
$$
Hom(L R x , a ) \stackrel{\simeq}{\to} Hom(R x, R a)
$$
and that, by the formula for [[adjuncts]] and the [[zig-zag identity]], this is such that the composite
$$
R_{x,a} : Hom(x,a) \to Hom(L R x, a) \stackrel{\simeq}{\to}
Hom(R x, R a)
$$
is the component map of the functor $R$ ([this Prop.](geometry+of+physics+–+basic+notions+of+category+theory#ReExpressingMiddleFunctorInAdjointTriple)):
$$
\begin{aligned}
(x \stackrel{f}{\to} a) & \mapsto
(L R x \to x \stackrel{f}{\to} a)
\\
& \mapsto
(R L R x \to R x \stackrel{R f}{\to} R a)
\\
& \mapsto
(R x \to R L R x \to R x \stackrel{R f}{\to} R a)
\\
& = (R x \stackrel{R f}{\to} R a)
\end{aligned}
\,.
$$
Therefore $R_{x,a}$ is injective for all $x,a$, hence $R$ is faithful, precisely if $L R x \to x$ is an epimorphism for all $x$. The characterization of $R$ full is just the same reasoning applied to the fact that $\epsilon_x \colon L R x \to x$ is a [[split monomorphism]] iff for all objects $a$ the induced function
\[
Hom(x, a) \to Hom(L R x, a)
\]
is a surjection.
For the characterization of faithful $L$ by monic units notice that analogously (as discussed at [[monomorphism]]) $x \to R L x$ is a monomorphism if for all objects $a$ the function
$$
Hom(a,x ) \to Hom(a, R L x)
$$
is an injection. Analogously to the previous argument we find that this is equivalent to
$$
L_{a,x} : Hom(a,x ) \to Hom(a, R L x) \stackrel{\simeq}{\to} Hom(L a, L x)
$$
being an injection. So $L$ is faithful precisely if all $x \to R L x$ are monos. For $L$ full, it's just the same applied to $x \to R L x$ [[split epimorphism]] iff the induced function
$$
Hom(a,x ) \to Hom(a, R L x)
$$
is a surjection, for all objects $a$.
The proof of the other statements proceeds analogously.
=--
Parts of this statement can be strengthened:
+-- {: .num_prop #ReflectionRecognizedByIdempotency}
###### Proposition
Let $(L \dashv R) : D \to C$ be a pair of [[adjoint functors]] such that there is _any_ [[natural isomorphism]]
$$
L R \simeq Id
\,,
$$
then also the [[unit of an adjunction|counit]] $\epsilon : L R \to Id$ is an [[isomorphism]].
=--
This appears as ([Johnstone, lemma 1.1.1](#Johnstone)).
+-- {: .proof}
###### Proof
Using the given [[isomorphism]], we may transfer the [[comonad]] structure on $L R$ to a comonad structure on $Id_D$. By the [[Eckmann-Hilton argument]] the [[endomorphism monoid]] of $Id_D$ is commutative. Therefore, since the coproduct on the comonad $Id_D$ is a [[left inverse]] to the counit (by the co-[[unitality]] property applied to this degenerate situation), it is in fact a two-sided [[inverse]] and hence the $Id_D$-counit is an [[isomorphism]]. Transferring this back one finds that also the counit of the comand $L R$, hence of the adjunction $(L \dashv R)$ is an isomorphism.
=--
### Pointwise expression
{#PointwiseExpression}
+-- {: .num_prop #PointwiseExpressionOfLeftAdjoints}
###### Proposition
**(pointwise expression of [[left adjoints]] in terms of [[limits]] over [[comma categories]])**
A [[functor]] $R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ has a [[left adjoint]] $L \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ precisely if
1. $R$ [[preserved limit|preserves]] all [[limits]] that exist in $\mathcal{C}$;
1. for each [[object]] $d \in \mathcal{D}$, the [[limit]] of the canonical functor out of the [[comma category]] of $R$ under $d$
$$
d/R \longrightarrow \mathcal{C}
$$
exists.
In this case the value of the [[left adjoint]] $L$ on $d$ is given by that limit:
\[
\label{FormulaForLeftAdjointByPointwiseLimit}
L(d)
\;\simeq\;
\underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c
\]
=--
(e.g. [[Categories Work|MacLane, chapter X, theorem 2]])
+-- {: .proof}
###### Proof
First assume that the left adjoint exist. Then
1. $R$ is a [[right adjoint]] and hence preserves limits since all [[right adjoints preserve limits]];
1. by Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor} the [[adjunction unit]] provides a [[universal morphism]] $\eta_d$ into $L(d)$, and hence, by Prop. \ref{UniversalMorphismsAreInitialObjectsInCommaCategory}, exhibits $(L(d), \eta_d)$ as the [[initial object]] of the [[comma category]] $d/R$. The limit over any category with an initial object exists, as it is given by that initial object.
Conversely, assume that the two conditions are satisfied and let $L(d)$ be given by (eq:FormulaForLeftAdjointByPointwiseLimit). We need to show that this yields a left adjoint.
By the assumption that $R$ preserves all limits that exist, we have
\[
\label{RAppliedtoFormulaForLeftAdjointByPointwiseLimit}
\array{
R(L(d))
& =
R\left(
\underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c
\right)
\\
& \simeq
\underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} R(c)
}
\]
Since the $d \overset{f}{\to} R(d)$ constitute a [[cone]] over the [[diagram]] of the $R(d)$, there is universal morphism
$$
d \overset{\phantom{AA} \eta_d \phantom{AA}}{\longrightarrow} R(L(d))
\,.
$$
By Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor} it is now sufficient to show that $\eta_d$ is a [[universal morphism]] into $L(d)$, hence that for all $c \in \mathcal{C}$ and $d \overset{g}{\longrightarrow} R(c)$ there is a unique morphism $L(d) \overset{\widetilde f}{\longrightarrow} c$ such that
$$
\array{
&& d
\\
& {}^{\mathllap{ \eta_d }}\swarrow && \searrow^{\mathrlap{f}}
\\
R(L(d)) && \underset{\phantom{AA}R(\widetilde f)\phantom{AA}}{\longrightarrow} && R(c)
\\
L(d)
&&\underset{\phantom{AA}\widetilde f\phantom{AA}}{\longrightarrow}&&
c
}
$$
By Prop. \ref{UniversalMorphismsAreInitialObjectsInCommaCategory}, this is equivalent to $(L(d), \eta_d)$ being the [[initial object]] in the [[comma category]] $c/R$, which in turn is equivalent to it being the [[limit]] of the [[identity functor]] on $c/R$ ([this prop.](initial+object#LimitOverIdentityFunctorIsInitialObject)). But this follows directly from the limit formulas (eq:FormulaForLeftAdjointByPointwiseLimit) and (eq:RAppliedtoFormulaForLeftAdjointByPointwiseLimit).
=--
See at _[[adjoint functor theorem]]_ for more.
[[!include relation between adjunctions and monads -- section]]
### Opposite adjoint functors
Given a pair of [[adjoint functors]]
$$
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}
{\overset{L}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
\mathcal{C}
$$
there is an induced *[[opposite adjunction]]* of [[opposite functors]] between their [[opposite categories]] of the form
$$
\mathcal{D}^{op}
\underoverset
{\underset{L^{op}}{\longleftarrow}}
{\overset{R^{op}}{\longrightarrow}}
{\;\;\;\;\bot\;\;\;\;}
\mathcal{C}^{op}
\,.
$$
Hence where $L$ was the [[left adjoint]], its [[opposite functor|opposite]] becomes the [[right adjoint]], and dually for $R$.
This is immediate from the definition of [[opposite categories]] and the characterization of adjoint functors via the corresponding [hom-isomorphism](#InTermsOfHomIsomorphism).
The [[adjunction unit]] of the opposite adjunction has as components the components of the original [[adjunction counit]], regarded in the opposite category, and dually:
$$
\epsilon^{R^{op} L^{op}}_{c}
\;\colon\;
R^{op}\circ L^{op}(c)
\xrightarrow{\;\; \big( \eta^{R L}_c \big) ^{op} \;\;}
c
\,,
{\phantom{AAAAAA}}
\eta^{L^{op} R^{op}}_{d}
\;\colon\;
d
\xrightarrow{\;\; \big( \epsilon^{L R}_d \big) ^{op} \;\;}
L^{op} \circ R^{op}(d)
\,.
$$
### Composing adjunctions
Given two pairs of [[adjoint functors]]
$$
\mathcal{E}
\underoverset
{\underset{R'}{\longrightarrow}}
{\overset{L'}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}
{\overset{L}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
\mathcal{C}
$$
there is an induced *composite adjunction* $L' \circ L \dashv R \circ R'$ between $\mathcal{C}$ and $\mathcal{E}$.
This is immediate from the characterization of adjoint functors in terms of [hom-isomorphisms](#InTermsOfHomIsomorphism):
$$
Hom_{\mathcal{E}}(L'(L(-)),-) \;\simeq\; Hom_{\mathcal{D}}(L(-),R'(-)) \;\simeq\; Hom_{\mathcal{C}}(-,R(R'(-)))
$$
### Pre- and postcomposite adjoint functors
{#PrePostcompositeAdjunctions}
Given a pair of [[adjoint functors]]
$$
\mathcal{D}
\underoverset
{\underset{R}{\longrightarrow}}
{\overset{L}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
\mathcal{C}
$$
and a [[category]] $\mathcal{E}$, there is an induced adjunction of precomposition functors between the [[functor categories]] $[\mathcal{C}, \mathcal{E}]$ and $[\mathcal{D}, \mathcal{E}]$ of the form
$$
[\mathcal{D}, \mathcal{E}]
\underoverset
{\underset{- \circ L}{\longrightarrow}}
{\overset{- \circ R}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
[\mathcal{C}, \mathcal{E}]
\,.
$$
Hence where $L$ was the [[left adjoint]], its precomposition functor $- \circ L$ becomes the [[right adjoint]], and dually for $R$.
The components $\eta_F : F \Rightarrow F \circ R \circ L$ of the [[unit of an adjunction|unit]] and $\epsilon_F : F \circ L \circ R \Rightarrow F$ of the [[counit of an adjunction|counit]] are given by [[whiskering]] the original unit and counit with $F$ on the left.
By uniqueness of adjoints, this implies that [[Kan extension|left Kan extensions]] along $L$ are given by precomposition with $R$, which is another way of saying that $R$ is the [absolute left Kan extension](#in_terms_of_kan_extensionsliftings) of the [[identity functor]] along $L$. Dually, [[Kan extension|right Kan extensions]] along $R$ are given by precomposition with $L$.
There is also an induced adjunction of postcomposition functors between $[\mathcal{E}, \mathcal{C}]$ and $[\mathcal{E}, \mathcal{D}]$ of the form
$$
[\mathcal{E}, \mathcal{D}]
\underoverset
{\underset{R \circ -}{\longrightarrow}}
{\overset{L \circ -}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
[\mathcal{E}, \mathcal{C}]
\,.
$$
The components $\eta_F : F \Rightarrow R \circ L \circ F$ of the [[unit of an adjunction|unit]] and $\epsilon_F : L \circ R \circ F \Rightarrow F$ of the [[counit of an adjunction|counit]] are given by [[whiskering]] the original unit and counit with $F$ on the right.
By uniqueness of adjoints, this implies that [[Kan lift|left Kan lifts]] along $R$ are given by postcomposition with $L$, which is another way of saying that $L$ is the [absolute left Kan lift](#in_terms_of_kan_extensionsliftings) of the [[identity functor]] along $R$. Dually, [[Kan lift|right Kan lifts]] along $L$ are given by postcomposition with $R$.
[[!include sliced adjoint functors -- section]]
## Examples
{#Examples}
The central point about examples of adjoint functors is:
_Adjoint functors are ubiquitous_ .
To a fair extent, [[category theory]] is all about adjoint functors and the other [[universal construction]]s: [[Kan extension]]s, [[limit]]s, [[representable functor]]s, which are all special cases of adjoint functors -- and adjoint functors are special cases of these.
Listing examples of adjoint functors is much like listing examples of [[integral]]s in [[analysis]]: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see [[coend]] for more).
Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.
### General
* A pair of adjoint functors between [[posets]] is a _[[Galois correspondence]]_.
* A pair of adjoint functors $(L \dashv R)$ where $R$ is a [[full and faithful functor]] exhibits a [[reflective subcategory]].
In this case $L$ may be regarded as a [[localization]]. The fact that the adjunction provides universal factorization through unit and counit in this case means that every morphism $f : c \to R d$ into a local object factors through the localization of $c$.
* A pair of adjoint functors that is also an [[equivalence of categories]] is called an [[adjoint equivalence]].
* A pair of adjoint functors where $C$ and $D$ have finite [[limit]]s and $L$ preserves these finite limits is a [[geometric morphism]]. These are one kind of morphisms between [[topos]]es. If in addition $R$ is full and faithful, then this is a [[geometric embedding]].
* The left and right adjoint functors $p_!$ and $p_*$ (if they exist) to a functor $p^* : [K',C] \to [K,C]$ between [[functor categories]] obtained by precomposition with a functor $p : K \to K'$ of [[diagram]] categories are called the left and right [[Kan extension]] functors along $p$
$$
(Lan_p \dashv p^* \dashv Ran_p)
:=
(p_! \dashv p^* \dashv p_*) :
[K,C]
\stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}}}
[K',C]
\,.
$$
If $K' = {*}$ is the [[terminal category]] then this are the [[limit]] and [[colimit]] functors on $[K,C]$.
If $C = $ [[Set]] then this is the [[direct image]] and [[inverse image]] operation on [[presheaves]].
* if $R$ is regarded as a [[forgetful functor]] then its left adjoint $L$ is a regarded as a [[free functor]].
* If $C$ is a category with small [[colimit]]s and $K$ is a [[small category]] (a [[diagram]] category) and $Q : K \to C$ is any functor, then this induces a [[nerve and realization]] pair of adjoint functors
$$
(|-|_Q \dashv N_Q) : C \stackrel{\overset{|-|_Q}{\leftarrow}}{\underset{N_Q}{\to}}
[K^{op}, Set]
$$
between $C$ and the [[category of presheaves]] on $K$, where
* the [[nerve]] functor is given by
$$
N_Q(c) := Hom_C(Q(-),c) : k \mapsto Hom_C(Q(k),c)
$$
* and the realization functor is given by the [[coend]]
$$
|F|_Q := \int^{k \in K} Q(k)\cdot F(k)
\,,
$$
where in the integrand we have the canonical [[copower|tensoring]] of
$C$ over [[Set]] ($Q(k) \cdot F(k) = \coprod_{s \in F(k)} Q(k)$).
A famous examples of this is obtained for $C = $ [[Top]], $K = \Delta$
the [[simplex category]] and $Q : \Delta \to Top$ the functor that sends
$[n]$ to the standard topological $n$-[[simplex]]. In this case
the nerve functor is the
[[fundamental infinity-groupoid|singular simplicial complex]] functor
and the realization is ordinary [[geometric realization]].
## Related concepts
{#RelatedConcepts}
* **adjoint functor**, [[adjunction]]
* [[strong adjoint functor]]
* {#RelatedSelfAdjointFunctor} [[self-adjoint functor]]
* [[adjoint triple]], [[adjoint quadruple]], [[recollement]]
* [[dual adjunction]]
* [[adjunction of two variables]]
* [[ambidextrous adjunction]]
* [[adjoint monad]]
* [[fixed point of an adjunction]]
* [[enriched adjunction]]
* [[transformation of adjoints]]
[[CatAdj|$Cat_{Adj}$]]
* [[proadjoint]], [[Hopf adjunction]]
* [[2-adjunction]]
[[adjoint 2-functor]], [[strict adjoint 2-functor]]
[[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]
* [[adjoint (∞,1)-functor]]
* [[(∞,n)-category with adjoints]]
* [[multi-adjoint]], [[weak adjoint]], [[solution set condition]]
## References
{#References}
For the basics, see any text on [[category theory]] (and see the references at _[[adjunction]]_), for instance:
* {#MacLane71} [[Saunders MacLane]], Chapter IV of: _[[Categories Work|Categories for the working mathematician]]_, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
* {#Borceux94} [[Francis Borceux]], Section 3 in Volume 1 and Section 4.2 in Volume 2 of _[[Handbook of Categorical Algebra]]_ (1994)
* {#Johnstone} [[Peter Johnstone]], first pages of _[[Sketches of an Elephant]]_ (2022)
* _[[geometry of physics -- categories and toposes]] -- [Adjunctions](https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes#Adjunctions)_
Though the definition of an [[adjoint equivalence]] appears in [[Grothendieck|Grothendieck's]] [[Tohoku]] paper, the idea of adjoint functors in general goes back to
* {#Kan58} [[Daniel Kan]], *Adjoint functors*, Transactions of the American Mathematical Society **87** 2 (1958) 294-329 [[jstor:1993102](http://www.jstor.org/stable/1993102)]
and in [relation](monad#RelationToAdjunctionsAndMonadicity) to [[comonad|co]]/[[monads]] to
* [[Peter J. Huber]], §4 in: *Homotopy theory in general categories*, Mathematische Annalen **144** (1961) 361–385 [[doi:10.1007/BF01396534](https://doi.org/10.1007/BF01396534)]
and its fundamental relevance for [[category theory]] was highlighted in
* {#Freyd64} [[Peter Freyd]], _Abelian categories -- An introduction to the theory of functors_, Harper's Series in Modern Mathematics, Harper & Row, New York (1964) [[pdf](http://www.maths.ed.ac.uk/~aar/papers/freydab.pdf)]
* {#Lawvere69} [[William Lawvere]], _Adjointness in Foundations_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296
The history of the idea that adjoint functors formalize aspects of [[dialectics]] is recounted in
* {#Lambek82} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) ([doi:10.1007/978-94-009-8462-2_6](https://doi.org/10.1007/978-94-009-8462-2_6))
(more along these lines at _[[adjoint modality]]_)
See also:
* Wikipedia, [Adjoint Functors](http://en.wikipedia.org/wiki/Adjoint_functors)
* [[The Catsters]] ([list](http://www.youtube.com/view_play_list?p=54B49729E5102248))
More on the notion of [[transformation of adjoints]]:
* {#HarpazPrasma15} [[Yonatan Harpaz]], [[Matan Prasma]], Section 2.2. of: _The Grothendieck construction for model categories_, Advances in Mathematics **281** (2015) 1306-1363 [[arXiv:1404.1852](https://arxiv.org/abs/1404.1852), [10.1016/j.aim.2015.03.031](https://doi.org/10.1016/j.aim.2015.03.031)]
> (in the context of the [[model structure on a Grothendieck construction]])
[[!redirects adjoint functors]]
[[!redirects adjoint pair of functors]]
[[!redirects adjoint pairs of functors]]
[[!redirects universal arrow]]
[[!redirects universal arrows]]
[[!redirects universal morphism]]
[[!redirects universal morphisms]]
|
adjoint functor theorem | https://ncatlab.org/nlab/source/adjoint+functor+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
_Adjoint functor theorems_ are theorems stating that under certain conditions a [[functor]] that preserves [[limit]]s is a [[right adjoint]], and that a functor that preserves [[colimit]]s is a [[left adjoint]].
A basic result of [[category theory]] is that right [[adjoint functors]] [[preserved limit|preserve]] all [[limits]] that exist in their [[domain]], and, dually, left adjoints preserve all [[colimits]]. An _adjoint functor theorem_ is a statement that (under certain conditions) the converse holds: a functor which preserves limits is a right adjoint.
The basic idea of an adjoint functor theorem is that _if_ we could assume that a [[large category]] $D$ had all [[limits]] over [[small category|small]] and [[large category|large]] [[diagrams]], then for $R : D \to C$ a [[functor]] that [[preserved limit|preserves]] all these limits we might define its would-be left adjoint $L$ by taking $L c$ to be the limit
$$
L c \coloneqq \lim_{c\to R d} d
$$
This notation stands for a limit over the [[comma category]] $c/R$ (whose [[objects]] are [[pairs]] $(d,f:c\to R d)$ and whose [[morphisms]] are arrows $d\to d'$ in $D$ making the obvious triangle [[commuting diagram|commute]] in $C$) of the projection functor $\pi: c/R \to D$ that forgets the morphism $f$:
$$
L c = \lim_{c\to R d} \pi
\,.
$$
Because with this definition there would be, for every $d$, an obvious morphism
$$
L R d \stackrel{=}{\to} \lim_{R d \to R d'} d' \to d
$$
(the component map over $d$ of the limiting [[cone]]). Moreover, because $R$ preserves limits, we would have an [[isomorphism]]
$$
R L c \simeq \lim_{c\to R d} R d
$$
(the limit of the functor $R \pi$), and hence an obvious morphism of [[cone]] tips
$$
c \to R L c
\,.
$$
It is easy to check that these would be the [[unit of an adjunction|unit]] and [[counit of an adjunction|counit]] of an [[adjunction]] $L\dashv R$. See _[[adjoint functor]]_ for more.
The problem with this would-be argument is that in general the comma category $(c/G)$ may not be [[small category]]. But one can generally not expect a large category to have all large limits: even if we pass to a [[universe]] in which $(c/G)$ is considered small, a [classical theorem of Freyd](complete+small+category#CompleteSmallCategoriesArePosets) says that any [[complete small category]] is a [[preorder]] (see [[complete small category]] for the proof, which is valid in [[classical logic]] and also holds classically in any [[Grothendieck topos]]). Thus, the argument we gave above is necessarily only an **adjoint functor theorem for preorders**:
+-- {: .num_theorem #ForPosets}
###### Theorem
If $G:D\to C$ is any functor between (small) preorders such that $D$ has, and $G$ preserves, all small [[meets]], then $G$ has a left adjoint.
=--
(This theorem holds in [[constructive mathematics]], although not in [[predicative mathematics]]; the classical reasoning before this explains why the theorem is not more general, but the proof itself is already constructive.)
To obtain adjoint functor theorems for categories that are not preorders, one must therefore impose various additional "size conditions" on the category $D$ and/or the functor $G$.
## Statement
+-- {: .num_theorem #StandardAdjointFunctorTheorem}
###### Theorem
Sufficient conditions for a limit-preserving functor $R : C \to D$ to be a [[right adjoint]] include:
* $C$ is [[complete category|complete]] and [[locally small category|locally small]], and $R$ satisfies the [[solution set condition]].
This is Freyd's original version, sometimes called the "**General Adjoint Functor Theorem**".
* $C$ is complete, locally small, [[well-powered category|well-powered]], and has a [[small set|small]] [[cogenerating set]], and $D$ is [[locally small category|locally small]].
This is sometimes called the "**Special Adjoint Functor Theorem**", and abbreviated to SAFT.
* $C$ is locally small and [[total category|cototal]], and $D$ is locally small.
=--
In the first two cases, which work by replacing large limits by small ones, it suffices to assume that $R$ preserves small limits (that it preserves all limits will follow). The third case works by assuming that $C$ has, while not all large limits, enough so that the theorem goes through; thus is this case $R$ must be already known to preserve [[large category|large]] limits as well.
+-- {: .proof}
###### Proof
Here is a proof of the General Adjoint Functor Theorem: that a functor $R : C \to D$ out of a [[locally small category]] $C$ with all small limits has a left adjoint if it preserves these [[limit]]s and satisfies the [[solution set condition]].
From the discussion at <a href="http://ncatlab.org/nlab/show/adjoint%20functor#UniversalArrows">adjoint functors -- In terms of universal arrows</a> we have that the existence of the adjoint is equivalent to the existence for each $d \in D$ of an [[initial object]] $i_d : d \to R L d$ in the [[comma category]] $(d \downarrow R)$: an object such that for each $f : d \to R d'$ there is a unique $\tilde f$ such that
$$
\array{
&& d
\\
& {}^{\mathllap{i_d}}\swarrow && \searrow^{\mathrlap{f}}
\\
R L d &&\underset{R \tilde f}{\to}&& R d'
\\
\\
L d &&\underset{\tilde f}{\to}&& d'
}
$$
commutes. Now an initial object is the limit of the identity functor, but this is generally a large limit; we replace this with some small limit conditions that guarantee existence of an initial object.
1. Let $Y$ be a category. Call a small family of objects $F$ _weakly initial_ if for every object $y$ of $Y$ there exists $x \in F$ and a morphism $f: x \to y$.
1. Suppose $Y$ has small products. If $F$ is a [[weak multilimit|weakly initial family]], then $\prod_{x \in F} x$ is a [[weakly initial object]].
1. **Claim:** Suppose $Y$ is locally small and has joint equalizers of small families. If $x$ is a weakly initial object, then the domain $e$ of the joint equalizer $i: e \to x$ of all arrows $x \to x$ is an initial object. **Proof:** clearly $e$ is weakly initial. Suppose given an object $y$ and arrows $f, g: e \to y$; we must show $f = g$. Let $j: d \to e$ be the equalizer of $f$ and $g$. There exists an arrow $k: x \to d$. The arrow $i: e \to x$ equalizes $1_x$ and $i j k: x \to x$, so $i j k i = i$. Since $i$ is monic, $j (k i) = 1_e$. Thus $j$ is an epi, and $f = g$ follows.
If $C$ is locally small and small-complete and $R: C \to D$ preserves limits, then $d \downarrow R$ is locally small and small-complete for every object $d$ of $D$.
If in addition each $d \downarrow R$ has a weakly initial family (solution set condition), then by 2. and 3. each $d \downarrow R$ has an initial object. This restates the condition that $R$ has a left adjoint.
=--
In fact, it suffices for $C$ to be [[Cauchy complete]] and for $R$ to be $\alpha$-[[flat functor|flat]] for every $\alpha$ (i.e. for its [[Yoneda extension]] to be [[continuous functor|continuous]]). See [Borceux](#Bouceux).
+-- {: .proof}
###### Proof of SAFT
As before, the proof proceeds by constructing initial objects of comma categories. We assume that $C$ is locally small, small-complete, well-powered, has a cogenerating set $\{c_\alpha: \alpha \in A\}$, and that $R: C \to D$ is a small-[[continuous functor]] into a [[locally small category]] $D$.
As before, for each object $d$ of $D$, the comma category $d \downarrow R$ is locally small and small-complete. Moreover, it is easy to check that it is well-powered, and that the set of all objects of the form $d \to R c_\alpha$ is a cogenerating set for $d \downarrow R$.
It then remains to prove that any locally small, small-complete, well-powered category $X$ with a cogenerating set $\{k_s: s \in S\}$ has an initial object. The initial object $0$ is constructed as the intersection = pullback of all subobjects of $\prod_s k_s$, i.e., the minimal subobject. Then, given $f, g: 0 \to x$, the equalizer $Eq(f, g)$ is isomorphic to $0$ because $0$ is minimal, and so $f = g$: there is at most one arrow $0 \to x$ for each $x$.
On the other hand, for each $x$ the canonical map
$$i: x \to \prod_{s \in S} k_{s}^{\hom(x, k_s)}$$
is monic since the $k_s$ cogenerate. The following pullback of $i$,
$$\array{
k & \to & x \\
\downarrow & & \downarrow \mathrlap{i} \\
\prod_s k_{s}^1 & \stackrel{\prod_s k_{s}^!}{\to} & \prod_s k_{s}^{\hom(x, k_s)}
},$$
gives a subobject $k$ of $\prod_s k_s$ that maps to $x$, and into which $0$ embeds. Thus there exists a map $0 \to x$, and we conclude $0$ is initial.
=--
### In locally presentable categories
In practice an important special case is that of functors between
[[locally presentable categories]]. For these there is the following version of an adjoint functor theorem.
+-- {: .num_theorem #AdjFuncTheoremForLocallyPresentableCats}
###### Theorem
Let $F \colon C \to D$ be a functor between [[locally presentable categories]]. Then
* $F$ has a [[right adjoint]] if and only if it preserves all small [[colimits]].
* $F$ has a [[left adjoint]] if and only if it is an [[accessible functor]] and preserves all small [[limits]].
=--
The second statement, characterizing when $F$ has a left adjoint, is ([AdamekRosicky, theorem 1.66](#AdamekRosicky)). In the "if" direction, this is an application of the general adjoint functor theorem: any accessible functor satisfies the solution set condition. The "only if", particularly that having a left adjoint forces accessibility, takes a little work. But in any case there are easy examples that show that continuity alone is insufficient, i.e., examples of [[continuous functors]] between [[locally presentable categories]] that do not have [[left adjoints]]. See below in the section [In locally presentable categories](#InLocallyPresentableCategories).
The first statement, characterizing when $F$ has a right adjoint, can be proven using the special adjoint functor theorem: by a non-trivial theorem ([AdamekRosicky, theorem 1.58](#AdamekRosicky)), any locally presentable category is co-wellpowered.
Thus, the first statement can be strengthened by removing the assumption that $D$ is locally presentable: it is enough that $D$ be locally small. For example, if $C$ is locally presentable, then every [[continuous functor]]
$$C^{op} \to Set$$
has a left adjoint (is representable), because its opposite $C \to Set^{op}$ is cocontinuous and therefore has a right adjoint, even though $Set^{op}$ is not locally presentable.
A [[right adjoint]] to any [[cocontinuous functor]] $F \colon C \to D$ between [[locally presentable categories]] can also be constructed directly. If $C$ is locally $\lambda$-presentable and $P_\lambda$ is the subcategory of $\lambda$-small objects, then $C$ is equivalent to the full subcategory of $[P_\lambda^{op},Set]$ of presheaves that preserve $\lambda$-small limits ([AdamekRosicky, theorem 1.46](#AdamekRosicky)). The presheaves in the image of the functor $D \to [P_\lambda^{op},Set]$ defined by $d \mapsto \hom(F-,d)$ preserve $\lambda$-small limits because $F$ is cocontinuous. So this functor factors through the subcategory $C$. The functor $D \to C$ so-constructed is a right adjoint to $F$.
## Examples {#Examples}
### In locally presentable categories
{#InLocallyPresentableCategories}
The following is a counter-example, indicating the need for something more than just continuity to force a functor between locally presentable categories to be a right adjoint; as stated in theorem \ref{AdjFuncTheoremForLocallyPresentableCats}, the missing extra condition is precisely accessibility.
+-- {: .num_example}
###### Example
For every infinite [[cardinal number]] $\kappa$, let
$G_\kappa$ be a [[simple group]] of [[cardinality]] $\kappa$.
Define the functor $ML:$ [[Group]] $\to$ [[Set]] to be the
[[product]] of all the [[representable functors]]
$Hom(G_\kappa,-)$.
Since no group can admit a nontrivial
[[homomorphism]] from proper-[[class]]-many of the $G_\kappa$,
this functor does indeed land (or can be redefined to land) in [[Set]].
Since it is a product of representables, it is
[[continuous functor|continuous]]
(and of course [[Group]] and [[Set]] are
[[locally presentable categories]]),
but it is not itself representable (hence has no [[left adjoint]]).
=--
[[André Joyal]] has been attributing this example to [[Saunders MacLane]]; it appears in print for instance right at the beginning of ([AdámekKoubekTrnková01](#AdamekKoubekTrnkova)).
### In cocomplete categories
Suppose $C$ and $D$ are categories that admit [[small colimits]]
(i.e., are [[cocomplete]])
and $F\colon C\to D$ is a functor that preserves [[small colimits]]
(i.e., is cocontinuous).
The $F$ has a [[right adjoint functor|right adjoint]]
if and only if for any object $d\in D$
the functor
$$C^{op} \to Set$$
that sends
$$c\mapsto D(F(c),d)$$
is a [[small presheaf]] on $C$.
See the [MathOverflow answer](https://mathoverflow.net/questions/346153/surmounting-set-theoretical-difficulties-in-algebraic-geometry/346162#346162) by [[Ivan Di Liberti]].
### In toposes
+-- {: .num_prop}
###### Proposition
Every [[sheaf topos]] is a [[total category]] and a [[cototal category]].
=--
See the discussion at _[[Grothendieck topos]]_.
It follows that
+-- {: .num_cor}
###### Corollary
Let $F : C \to D$ be a [[functor]] between [[sheaf toposes]]. Then
* $F$ has a [[right adjoint]] precisely if it preserves all small [[colimits]];
* $F$ has a [[left adjoint]] precisely if it preserves all small [[limits]].
=--
### In presheaf categories
{#InPresheafCategories}
It is instructive to spell out the construction of the [[right adjoint]] from a colimit preserving functor $L$ in the simple case where all categories are [[categories of presheaves]]. This is a particularly simple case, but is useful in itself and serves as a template for the general case.
So let now $C$ and $D$ be [[small categories]] and $L$ a [[colimit]]-[[preserved colimit|preserving]] functor between their [[categories of presheaves]] (which we abbreviate $\widehat C \;\coloneqq\; [C^{op}, Set]$, etc):
$$
L \;\colon\; \widehat{C} \longrightarrow \widehat{D}
$$
Then its [[right adjoint]] $ R \;\colon\; \widehat{D} \longrightarrow \widehat{C}$ is given (with $y_c \coloneqq C(-, c)$ denoting the [[Yoneda embedding]]) by
\[
\label{RightAdjointOfColimitPreservingFunctorOfPresheaves}
R
\;\colon\;
A
\;\mapsto\;
R(A)
\;\coloneqq\;
\widehat{D}
\left(
L(y_{(-)}),
A
\right)
\,,
\]
as we shall check in a moment. But first notice that using the [[co-Yoneda lemma]] this may be rewritten as
$$
\cdots
\;\simeq\;
\int^{c \in C}
\widehat{D}
\left(
L(y_c), A
\right)
\cdot y_c
\,,
$$
where the [[coend]] is equivalently given by the [[colimit]]
$$
\cdots \;=\; \lim_{\underset{L c \to A}{\to}} c
\,.
$$
This is the formula for the would-be right adjoint from the general discussion above, only that here the colimit is only over the representables, hence over a small category.
Now we check that the functor $R$ thus obtained is indeed right adjoint to $L$, by explicitly checking the hom-isomorphism ([here](adjoint+functor#InTermsOfHomIsomorphism)) of the pair of [[adjoint functors]]:
We compute $\widehat{D}(L(X),A)$. In the first step
$$
\widehat{D}(L(X), A)
\simeq
\widehat{D}
\left(
L
\left(
\int^{c \in C} X(c) \cdot y_c
\right),
A
\right)
$$
we use the [[co-Yoneda lemma]] for $X$. Then, because $L$ [[preserved limit|preserves]] colimits, this is
$$
\cdots
\simeq
\widehat{D}
\left(
\int^c X(c) \cdot L(y_c), A
\right)
\,.
$$
Since the [[hom-functor]] preserves limits in both arguments, we can take the [[coend]] out to get an [[end]]
$$
\cdots
\simeq
\int_{c \in C}
\widehat{D}(X(c) \cdot L(y_c), A)
\,.
$$
Then we use the standard [[copower|tensoring]] of our categories over [[Set]] to get
$$
\cdots
\simeq
\int_{c \in C}
Set\left(
X(c),
\widehat{D}(L(y_c),A)
\right)
\,.
$$
And finally this is recognized as the formula for the [[hom-set]] of presheaves (see the discussion at _[[functor category]]_):
$$
\cdots
\simeq
\widehat{C}
\left(
X,
\widehat{D}(L(y_{(-)}),A)
\right)
\;=\;
\widehat{C}(X, R A)
\,,
$$
where on the right we identified (eq:RightAdjointOfColimitPreservingFunctorOfPresheaves).
The [[composition]] of this sequence of [[natural isomorphisms]] is hence the desired hom-isomorphism:
$$
\widehat{D}(L(X), A) \simeq \widehat{C}(X, R(A))
\,.
$$
## Related concepts
* ([[cocontinuous functor|co]])[[continuous functor]]
* [[indexed adjoint functor theorem]]
* [[adjoint (∞,1)-functor theorem]]
* [[adjoint triangle theorem]]
* [[adjoint lifting theorem]]
## References
The classical adjoint functor theorems originate in the exercise section of ch.3 (pp.84ff) in
* [[Peter Freyd]], *Abelian Categories*, Harper & Row New York 1964. (Reprinted with author's comment as [TAC reprints no. 3 (2003)](http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html))
A more recent exposition is in
* [[Peter Freyd]], Andre Scedrov, *Categories, Allegories*, North-Holland, 1990. (Also Dover reprint New York 2014, pp.144-148)
where the [[solution set condition]] is called "pre-adjointness".
Careful discussions can be found in
* {#Bouceux} [[Francis Borceux]], *[[Handbook of Categorical Algebra]] I*, Cambridge UP, 1994. (sections 3.3, 6.6)
* [[Saunders MacLane]], §V.6, V.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
A brief introductory discussion is around [theorem 5.4](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf#page=54) of
* [[Jaap van Oosten]], *Basic category theory*, ms. ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf))
A detailed expository survey is
* Oliver Kullmann, *The adjoint functor theorem*, ms. ([pdf slides](http://www.cs.swan.ac.uk/~csoliver/Hauptseminar/Quellen/200702Swansea_b.pdf))
The adjoint functor theorem in context with [[Yoneda embedding]] is discussed in
* Friedrich Ulmer, *The adjoint functor theorem and the Yoneda embedding*, Illinois J. Math. **15** no.3 (1971), pp. 355-361. ([euclid](http://projecteuclid.org/euclid.ijm/1256052605))
The connection between the solution set condition and the Čech homology construction is discussed in
* Renato Betti, _Čech methods and the adjoint functor theorem_ , Cah. Top. Géom. Diff. Cat. **XXVI** no.3 (1985) pp.245-257. ([numdam](http://www.numdam.org/item/?id=CTGDC_1985__26_3_245_0))
An enriched adjoint functor theorem is given in:
* [[Francis Borceux]]. *Limites enrichies et existence de $ V $-foncteur adjoint*. Cahiers de topologie et géométrie différentielle catégoriques 16.4 (1975): 395-408. ([pdf](http://www.numdam.org/item/CTGDC_1975__16_4_395_0.pdf))
The parallels between the adjoint functor theorem for categories and the computation of colimits from limits in a lattice as well as similar parallels between co/completeness for Boolean algebras and Paré's theorem for finite completeness of toposes is studied from a type-theoretical perspective in
* [[Dusko Pavlovic|Duško Pavlović]], _On completeness and cocompleteness in and around small categories_ , APAL **74** (1995) pp.121-152.
The case for [[locally presentable categories]] is discussed in
* [[Jiří Adámek]], [[Jiri Rosicky]], *[[Locally presentable and accessible categories]]*, Cambridge UP, 1994.
{#AdamekRosicky}
* [[Jiří Adámek]], V. Koubek and V. Trnková, *How large are left exact functors?*, Theory and Applications of Categories **8** (2001), pp. 377-390. ([abstract](http://www.emis.de/journals/TAC/volumes/8/n13/8-13abs.html))
{#AdamekKoubekTrnkova}
A specialization to the locally $\alpha$-presentable case is given in Theorem 2.11 of
* [[Giacomo Tendas]], *On continuity of accessible functors*, Applied Categorical Structures, 2022, [doi](https://doi.org/10.1007/s10485-022-09677-x).
A relative version of Freyd's classical results is in
* [[Brian Day]], *An adjoint-functor theorem over topoi*, Bull. Austral. Math. Soc. **15** (1976), pp. 381-394.
[[indexed adjoint functor theorem|Adjoint functor theorems for indexed categories]] are discussed in
* [[Robert Paré]], [[Dietmar Schumacher]], _Abstract Families and the Adjoint Functor Theorems_ , pp.1-125 in LNM **661** Springer Heidelberg 1978.
A history of the general adjoint functor theorem is discussed in:
* [[Hans-E. Porst]], _The history of the General Adjoint Functor Theorem_ [[arxiv:2310.19528](https://arxiv.org/abs/2310.19528)]
A stronger version for finitary functors between locally finitely presentable categories whose domain is ranked, requiring only the preservation of countable limits for the existence of a left adjoint, is discussed in
* [[Jiří Adámek]], Lurdes Sousa, _A Finitary Adjoint Functor Theorem_, [arXiv](https://arxiv.org/abs/2311.14965).
[[!redirects adjoint functor theorems]] |
adjoint lifting theorem | https://ncatlab.org/nlab/source/adjoint+lifting+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Statement ##
+-- {: .un_theorem}
###### Theorem
**(The adjoint lifting theorem)**. Consider the following [[commutative diagram|commutative square]] of [[functor]]s:
$$
\begin{array}{cccc}\mathcal{A} & \overset{Q}{\to} & \mathcal{B} \\
^{U}\downarrow & & \downarrow^{V} \\
\mathcal{C} & \underset{R}{\to} & \mathcal{D}
\end{array}
$$
and suppose that
* $U$ and $V$ are [[monadic functor|monadic]], and
* $\mathcal{A}$ has [[coequalizer]]s of reflexive pairs.
Then, if $R$ has a [[left adjoint]], then $Q$ also has a [[left adjoint]].
=--
A detailed proof may be found in Sec. 4.5 of Vol. 2 of [Borceux](#Borceux)
(see especially Theorem 4.5.6 on p. 226 and Ex. 4.8.6 on p. 252). Also ([Johnstone, prop. 1.1.3](#Johnstone)) For a sketch of proof, see ahead.
+-- {: .un_cor}
###### Corollary
If the bottom functor $R$ of the above square is the
identity arrow (so that $U=V\circ Q$), if $U$ and $V$ are monadic, and
if $\mathcal{A}$ has coequalizers of reflexive pairs, then $Q$ is monadic.
=--
+-- {: .proof}
###### Proof
The adjoint lifting theorem implies the existence of a left
adjoint, and the rest is a straightforward application of the
[[monadicity theorem]].
=--
## Sketch of proof ##
We may assume the situation of the following diagram (with $V Q = R U$):
$$
\begin{array}{cccc}\mathcal{C}^\mathbb{T} & \underoverset{Q}{K(?)}{\leftrightarrows} & \mathcal{D}^\mathbb{S} \\
^{U}\downarrow \uparrow^{F} & & ^{G}\uparrow\downarrow^{V} \\
\mathcal{C} & \underoverset{R}{L}{\leftrightarrows} & \mathcal{D}
\end{array}
$$
where
$\mathbb{T}=\langle T,\varepsilon\colon 1_{\mathcal{C}}\Rightarrow
T,\mu \rangle$ is a monad on
$\mathcal{C}$, $\mathbb{S}=\langle S,\zeta\colon 1_{\mathcal{D}}\Rightarrow
S,\eta\rangle$ is a monad on
$\mathcal{D}$, $U$ and $V$ are the forgetful functors, and $F$ and $G$
are the free algebra functors.
Let us write $\tau\colon F U\Rightarrow 1_{\mathcal{C}^{\mathbb{T}}}$ for
the counit of the adjunction $F\dashv U$ and $\sigma\colon
G V\Rightarrow 1_{\mathcal{D}^{\mathbb{S}}}$ for the counit of the
adjunction $G\dashv V$. As usual, we have $T = U F$, $S = V G$,
$\mu=U\tau F$, and $\eta=V\sigma G$.
Finally, let $L$ be a left adjoint to $R$ (which
exists by assumption), and let $\alpha\colon
1_{\mathcal{D}}\Rightarrow R L$ and $\beta\colon LR\Rightarrow
1_{\mathcal{C}}$ be the unit and counit (respectively) of the
adjunction $L\dashv R$.
We would like to construct a functor $K\colon
\mathcal{D}^{\mathbb{S}}\to \mathcal{C}^{\mathbb{T}}$. To get a hint
of what $K$ should look like, let us assume for a moment that such a
$K$ already exists. In this case, we have $K G\dashv V Q(=R U)$. But
$F L$ is also left adjoint to $R U$, and by the uniqueness of a left
adjoint we must have $K G = F L$ (at least up to a natural isomorphism).
From this, we already know how to define $K$ on free algebras. Also,
being a left adjoint, $K$ preserves in particular all coequalizers. But
every $\mathbb{S}$-algebra $\langle D,\xi\colon SD\to D\rangle$ is
the (object part) of a reflexive coequalizer, namely, the [[canonical
presentation]]
$$
G S(D)\underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows}G(D)\overset{\sigma_{\langle
D,\xi\rangle}=\xi}{\rightarrow}\langle D,\xi \rangle
$$
Applying $K$ and using $K G=F L$, we see that $K\langle D,\xi\rangle$
should be the object part of a reflexive coequalizer in
$\mathcal{C}^{\mathbb{T}}$ of the form
$$
F L S(D)\underoverset{F L\xi}{?}{\rightrightarrows}F L(D)\overset{x}{\rightarrow}K\langle
D,\xi \rangle
$$
(recall that we assume that $\mathcal{C}^{\mathbb{T}}$ has
coequalizers of reflexive pairs).
To eventually define $K\langle D,\xi \rangle$ as a coequalizer (as above),
we first need some reasonable guess for the ∞-arrow. For this, we will
need a lemma.
+-- {: .un_lemma}
###### Lemma
There exists a natural transformation $\lambda\colon S R
\Rightarrow R T$ for which the following diagram of functors and
natural transformations is commutative:
$$
\begin{array}{ccccccc} R & \overset{\zeta R}{\to} & S R &
& \overset{\eta R}{\leftarrow}& & S S R \\
& ^{R\varepsilon}\searrow & ^{\lambda}\downarrow & & & & ^{S\lambda}\downarrow\\
& & R T & \overset{R\mu}{\leftarrow} & R T T & \overset{\lambda T}{\leftarrow} & S R T
\end{array}
$$
=--
+-- {: .proof}
###### Proof
Define $\lambda := V\sigma Q F \circ V G R\varepsilon$, so that
$$
\lambda \colon
S R = V G R\overset{V G R\varepsilon}{\to}V G R T = V G R U F = V G V Q
F\overset{V\sigma Q F}{\to} V Q F = R U F = R T.
$$
The required commutativity may be verified by using the commutative
diagrams in the definitions of a monad and an EM-algebra,
naturality, and the triangular identities. For details, see the proof
of Lemma 4.5.1 of [Borceux](#Borceux),
pp. 222-223. (Note that this lemma does not depend on the
existence of a left adjoint for the bottom horizontal arrow, nor on
the existence of coequalizers. Only the commutativity is required.)
=--
We may now return to our task of defining the ∞-arrow in the diagram
preceding the lemma. We would like to get from $F L S(D)$ to $F L(D)$,
and for this, we will construct a natural transformation
$F L S\Rightarrow F L$ in the following way. First, we have
$$
S\overset{S\alpha}{\to} S R L \overset{\lambda L}{\to} R T L.
$$
Applying $L$ and composing with $\beta T L$, we get
$$
LS\overset{L\lambda L\circ LS\alpha}{\longrightarrow}L R T L \overset{\beta TL}{\to} TL.
$$
Applying $F$ and composing with $\tau F L$, we finally get
$$
F L S\overset{F\beta T L\circ F L\lambda L\circ F L
S\alpha}{\longrightarrow}F T L = F U F L\overset{\tau F L}{\to} F L
$$
Let us call the resulting natural transformation $\omega$, that is,
$$
\omega:=\tau F L \circ F\beta T L\circ F L\lambda L\circ F L S\alpha.
$$
Now we take the sought for ∞-arrow to be $\omega_D$, and
*define* $K\langle D,\xi\rangle$ as the object of some fixed
coequalizer of $\omega_D$ and $F L\xi$:
$$
F L S(D)\underoverset{F L\xi}{\omega_D}{\rightrightarrows}F
L(D)\overset{x}{\rightarrow}K\langle D,\xi \rangle.
$$
In order to do this, we must first verify that the parallel
arrows above have a common section (since we only assume that
$\mathcal{C}^{\mathbb{T}}$ has coequalizers of reflexive pairs). To
find a guess for a common section, note that the common section
for the parallel pair in the above canonical presentation in
$\mathcal{D}^{\mathbb{S}}$ is $G\zeta_{V\langle
D,\xi\rangle}=G\zeta_D$, and if $K$ exists, then applying $K$ gives
$K G\zeta_D=F L\zeta_D$. Having this guess, it is now straightforward
to verify that $F L\zeta_D$ is indeed a common section, as required.
So, we have defined an object function of a would be left adjoint $K$.
To make it into a functor left adjoint to $Q$, we will build a
universal arrow from $\langle D,\xi\rangle$ to $Q$, whose object part
is $K\langle D,\xi\rangle$ (Theorem IV.1.2(ii) of [[Categories
Work]]).
To get an arrow $\langle D,\xi\rangle\to Q K\langle D,\xi\rangle$ ,
suppose for a moment that we have a natural
transformation $\varphi\colon G\Rightarrow Q F L$ such that $\varphi\circ
\sigma G = Q\omega\circ \varphi S$. Then the left square in the following
diagram commutes:
$$
\begin{array}{ccccc}G S(D)&
\underoverset{G\xi}{\sigma_{G D}}{\rightrightarrows} & G(D) &\overset{\sigma_{\langle
D,\xi\rangle}=\xi}{\rightarrow}&\langle D,\xi \rangle \\
^{\varphi_{S D}}\downarrow && ^{\varphi_D}\downarrow && ^{\chi}\downarrow\\
Q F L S D &\underoverset{Q F L\xi}{Q\omega_D}{\rightrightarrows} & Q F
L D &\overset{Q x}{\rightarrow}&Q K\langle D,\xi\rangle
\end{array}
$$
Since both rows are forks, it follows that $Q x\circ \varphi_D$ has
the same composition with the arrows of the upper
parallel pair, and hence there exists a unique arrow $\chi\colon
\langle D,\xi\rangle\to Q K\langle D,\xi\rangle$ making the right
square commutative (recall that the upper row is a coequalizer).
It is now possible to prove that the pair $\langle K\langle
D,\xi\rangle,\chi \rangle$ is a universal arrow from $\langle
D,\xi\rangle$ to $Q$, showing that $K$ is indeed (the object
function) of a left adjoint (for details, see the proof of Theorem
4.5.6, pp. 226-227 in [Borceux](#Borceux)).
But we still have to prove the existence of a natural
transformation $\varphi\colon G\Rightarrow Q F L$ such that $\varphi\circ
\sigma G = Q\omega\circ \varphi S$. For this, we define
$\varphi:=\sigma Q F L \circ G R\varepsilon L \circ G\alpha$. Since
$V$ is faithful, to prove the required property of $\varphi$, it is
enough to prove that $V\varphi \circ V\sigma G = V Q\omega\circ
V\varphi S$, and this is a long, yet straightforward,
computation (noting that $V\varphi = \lambda L \circ VG\alpha$ and using
the commutative diagram from Lemma 1; see Lemma 4.5.3, p. 224 of [Borceux](#Borceux)).
## Examples ##
### Forgetful functors between varieties of algebras ###
Since varieties of algebras are [[cocompleteness of varieties of algebras|cocomplete]] and monadic over
$\mathbf{Set}$, the corollary implies that forgetful functors between
varieties of algebras (e.g., the forgetful functor
$\mathbf{Rng}\to\mathbf{Ab}$) are monadic.
### Sufficient conditions for cocompleteness of monadic categories ###
Let $\mathcal{J}$ be an arbitrary category, and consider the commutative diagram
$$
\begin{array}{ccccc}\mathcal{A} & \overset{\Delta}{\to} & \mathcal{A}^\mathcal{J} \\
^{U}\downarrow & & \downarrow^{U^{\mathcal{J}}} \\
\mathcal{C} & \underset{\Delta}{\to} & \mathcal{C}^\mathcal{J}
\end{array}
$$
where $U$ is monadic, $\Delta$ is the [[diagonal functor]] and $U^{\mathcal{J}}=U\circ
-$. If $F$ is left adjoint
to $U$, then $F^\mathcal{J}$ is left adjoint to $U^{\mathcal{J}}$ (using
the unit and counit of the original adjunction, one can construct
appropriate natural transformations that satisfy the triangular
identities, see, e.g., p. 119 of [[Categories Work]]). Also, the
conditions of the monadicity theorem for $U$ imply those for
$U^\mathcal{J}$ (basically because the definition of a split fork involves only
compositions and identities, and because natural transformations are composed
componentwise).
Now, if $\mathcal{C}$ is $\mathcal{J}$-cocomplete (so that the
bottom horizontal functor has a left adjoint) and $\mathcal{A}$ has
coequalizers of reflexive pairs, then the adjoint lifting theorem
implies that $\mathcal{A}$ is $\mathcal{J}$-cocomplete. In particular,
if $\mathcal{A}$ has coequalizers of reflexive pairs and $\mathcal{C}$
is small-cocomplete, then $\mathcal{A}$ is small cocomplete.
## Related pages
* The adjoint lifting theorem is a corollary of the [[adjoint triangle theorem]].
* [[adjoint functor theorem]]
## References ##
* [[Michael Barr]], [[Charles Wells]], section 3.7, pp.131 in: *[[Toposes, Triples, and Theories]]*, Springer (1985), Reprints in Theories and Applications of Categories **12** (2005) 1-287 [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]
* [[Francis Borceux]], _[[Handbook of Categorical Algebra]] II_ , Cambridge UP 1994. (section 4.5, pp.221ff)
{#Borceux}
* {#Johnstone} [[Peter Johnstone]], *Adjoint lifting theorems for categories of algebras*, Bull. London Math. Soc. **7** (1975) 294-297 [[doi:10.1112/blms/7.3.294](https://doi.org/10.1112/blms/7.3.294)]
* [[Peter Johnstone]], section A1.1, p.5 in: *[[Sketches of an Elephant]] I*, Oxford UP 2002
On the dual theorem for [[comonads]]:
* William F. Keigher, _Adjunctions and comonads in differential algebra_, Pacific J. Math. **59,** 1 (1975) 99-112 [[euclid:pjm/1102905501](http://projecteuclid.org/euclid.pjm/1102905501)]
* [[John Power]], _A unified approach to the lifting of adjoints_, [[Cahiers]] **XXIX** 1 (1988) 67-77 [[numdam](http://www.numdam.org/item/CTGDC_1988__29_1_67_0)]
[[!redirects adjoint lifting theorems]]
|
adjoint logic | https://ncatlab.org/nlab/source/adjoint+logic |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Adjoint logic_ or _adjoint type theory_ is [[formal logic]] or [[type theory]] which natively expresses [[adjunctions]] of [[modal operators]], [[adjoint modalities]].
## Related pages
* [temporal logic](temporal+logic#temporal_logic_in_terms_of_adjoints)
## References
* [[Nick Benton]], [[Philip Wadler]], _Linear logic, monads and the lambda calculus_, In IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press, 1996.
* Jason Reed, _A judgemental deconstruction of modal logic_, 2009, ([pdf](http://www.cs.cmu.edu/~jcreed/papers/jdml2.pdf))
* Klaas Pruiksma, William Chargin, [[Frank Pfenning]], and Jason Reed, *Adjoint Logic*, 2018, [[pdf](https://www.cs.cmu.edu/~fp/papers/adjoint18b.pdf), [[PCPR18-AdjointLogic.pdf:file]]]
A framework for ([[homotopy type theory|homotopy]]-)[[type theory|type theoretic]] adjoint logic ([[modal type theory]]) is discussed, in various stages of generality, in
* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
(specifically for [[cohesive homotopy type theory]])
* {#LicataShulman} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_, in _[Logical Foundations of Computer Science 2016](http://lfcs.info/lfcs-2016/)_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf), [slides](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint-lfcs-slides.pdf))
(for [[modal type theory|modal]] [[unary type theory]])
* [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))
(for [[modal type theory|modal]] [[simple type theory]])
Review includes
* {#Licata18} [[Dan Licata]], _Synthetic Mathematics in Modal Dependent Type Theories_, tutorial at _[Types, Homotopy Theory and Verification](https://www.him.uni-bonn.de/programs/current-trimester-program/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)_, 2018 ([pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))
[[!redirects adjoint logics]]
[[!redirects adjoint type theory]]
[[!redirects adjoint type theories]]
|
adjoint modality | https://ncatlab.org/nlab/source/adjoint+modality | [[!redirects adjoint cylinder]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
<div style="float:right;margin:10px 10px 10px 10px;"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Cup_or_faces_paradox.svg/450px-Cup_or_faces_paradox.svg.png" width="200"></div>
The concept of [[adjunction]] as such expresses a [[duality]]. The stronger concept of an _adjoint cylinder_ or _adjoint [[modality]]_ is specifically an [[adjunction]] between [[idempotent monad|idempotent (co-)monads]] and is meant to express specifically a _duality between opposites_.
In terms of the corresponding [[adjoint triple]] of [[reflective subcategory|(co-)reflections]] and [[localizations]] the concept was suggested in ([Lawvere 91, p. 7](#Lawvere91), [Lawvere 94, p. 11](#Lawvere94)) to capture the phenomena of "Unity and Identity of Opposites" as they appear informally in [[Georg Hegel]]'s _[[Science of Logic]]_.
(One might therefore say the notion is meant to capture the idea of "dialectic", though there is some debate as to whether Hegel's somewhat mythical "creation out of [[paradox]]" should really go by this term, see [this Wikipeda entry](#Wikipedia) ).
In terms of [[adjoint pairs]] of [[modal operators]] in the context of [[modal logic]]/[[modal type theory]] and thought of as [[Galois connections]] the concept appears in ([Reyes-Zolfaghari 91](#ReyesZolfaghari91)). Further developments along these lines include ([DJK 14](#DJK14)).
## Definition
In ([Lawvere 94](#Lawvere94)) an _adjoint cylinder_ is defined to be an [[adjoint triple]] such that the outer two adjoints are [[full and faithful functors]]. This means equivalently that the induced [[adjoint pair]] on the codomain of these inclusions consists of an [[idempotent monad|idempotent]] [[monad]] and [[comonad]] ([[adjoint monads]]). One may also consider the situation where the middle functor of the adjoint triple is fully faithful, hence one has adjoint [[modal operators]]
either of the form
$$
U \;\colon\; modality \dashv comodality
\,,
$$
or of the form
$$
U \;\colon\; comodality \dashv modality
\,.
$$
A category equipped with an adjoint modality of the second form is called a _[[category of being]]_ in ([Lawvere 91](#Lawvere91)). If the category is a [[topos]] then this is also called a _[[level of a topos]]_.
Given any such, we may say that the "unity" expressed by the two opposites is exhibited by the canonical [[natural transformation]]
$$
U X
\;\colon\;
\array{
comodal X &\longrightarrow& X &\longrightarrow& modal X
\\
opposite\;1 && unity && opposite\;2
}
$$
which is the composite of the [[counit of a comonad|counit of the comodality]] and the [[unit of a monad|unit of the modality]] ([Lawvere-Rosebrugh 03, p. 245](#LawvereRosebrugh03)).
If for two [[level of a topos|levels]] the next one contains the [[modal types]] of the [[idempotent comonad]] of the former, then [[Lawvere]] speaks of "[[Aufhebung]]" (see there for more).
One can consider longer sequences of such adjoints of co/modalities, but the longer they get, the less likely they are to be non-trivial. The longest that still has good nontrivial models seems to be [[adjoint triples]] of modalities. Of these there is then similarly either the form
$$
modality \dashv comodality \dashv modality
$$
(the "Yin triple") as for instance in the definition of _[[cohesion]]_ and
$$
comodality \dashv modality \dashv comodality
$$
(the "Yang triple") as for instance in the definition of _[[differential cohesion]]_.
Since adjoint triples are equivalently [[adjunctions]] of [[adjunctions]] ([Licata-Shulman, section 5](#LicataShulman)), it is suggestive to denote these as
$$
\array{
\lozenge &\dashv& \bigcirc
\\
\bot && \bot
\\
\bigcirc &\dashv& \Box
}
$$
## Examples
### Simple illustrative examples
{#SimpleIllustrativeExamples}
The following simple illustrative example of an adjunction of the form $\Box \dashv \bigcirc$ has been suggested in ([Lawvere 00](#Lawvere2000)).
+-- {: .num_example #EvenAndOddIntegersAdjointModality}
###### Example
**([[even number|even]] and [[even number|odd]] [[integers]])**
Regard the [[integers]] as a [[preordered set]] $(\mathbb{Z}, \leq)$ in the canonical way, and thus as a [[thin category]].
Consider the [[full subcategory]] inclusions
$$
\array{
(\mathbb{Z}, \leq )
& \overset{even}{\hookrightarrow}&
(\mathbb{Z},\leq)
\\
n &\mapsto & 2 n
}
\phantom{AAAAA}
\array{
(\mathbb{Z}, \leq )
& \overset{odd}{\hookrightarrow}&
(\mathbb{Z},\leq)
\\
n &\mapsto & 2 n + 1
}
$$
of the [[even number|even]] and the [[even number|odd]] [[integers]], as well as the functor
$$
\array{
(\mathbb{Z}, \leq )
& \overset{\lfloor-/2\rfloor}{\longrightarrow}&
(\mathbb{Z},\leq)
\\
n &\mapsto& \lfloor n/2 \rfloor
}
$$
which sends any $n$ to the [[floor]] $\lfloor n/2 \rfloor$ of $n/2$, hence to the largest integer which is smaller or equal to the [[rational number]] $n/2$.
These functors form an [[adjoint triple]]
$$
even \;\dashv\; \lfloor -/2 \rfloor \;\dashv\; odd
$$
and hence induce an adjoint modality
$$
Even \;\dashv\; Odd
$$
on $(\mathbb{Z}, \leq)$ with
1. $Even \coloneqq 2 \lfloor -/2 \rfloor$ sending any integer to its "even [[floor]] value"
1. $Odd \coloneqq 2 \lfloor -/2 \rfloor + 1$ sending any integer to its "odd [[ceiling]] value".
=--
+-- {: .proof}
###### Proof
Observe that for all $n \in \mathbb{Z}$ we have
$$
2 \lfloor n/2 \rfloor
\overset{ \epsilon_n }{\leq}
n
\overset{ \eta_n }{\leq}
2 \lfloor n/2 \rfloor + 1
\,,
$$
where the first inequality is an equality precisely if $n$ is even, while the second is an equality precisely if $n$ is odd. Hence this provides a candidate [[unit of an adjunction|unit]] $\eta$ and [[counit of an adjunction|counit]] $\epsilon$.
Hence by [this characterization](adjoint+functor#UniversalArrow) of [[adjoint functors]]
1. the adjunction $\lfloor -/2 \rfloor \dashv odd$ is equivalent to the condition that
for every $n \leq 2 k + 1$ we have $2 \lfloor n/2 \rfloor + 1 \leq 2 k + 1$;
1. the adjunction $even \dashv \lfloor -/2 \rfloor $ is equivalent to the condition that
for every $2k \leq n$ we have $2k \leq 2 \lfloor n/2 \rfloor $,
which is readily seen to be the case
=--
In the same vein there is an example for an adjunction of the form $\bigcirc \dashv \Box$:
+-- {: .num_example }
###### Example
Consider the inclusion $\iota \colon (\mathbb{Z}, \lt) \hookrightarrow (\mathbb{R}, \lt)$ of the [[integers]] into the [[real numbers]], both regarded as [[linear orders]]. This inclusion has a [[left adjoint]] given by [[ceiling]] and a right adjoint given by [[floor]].
The composite $Ceiling \coloneqq \iota ceiling$ is an [[idempotent monad]] and the composite $Floor \coloneqq \iota floor$ is an [[idempotent comonad]] on $\mathbb{R}$. Both express a _moment of integrality_ in an real number, but in opposite ways, each real number $x\in \mathbb{R}$ sits in between its floor and celling
$$
Floor(x) \leq x \leq Ceiling(x)
\,.
$$
Indeed the moments form an [[adjunction]]
$$
Ceiling \dashv Floor
\,.
$$
=--
### Werden : Nichts $\dashv$ Sein
{#Werden}
For $\mathbf{H}$ a [[topos]]/[[(∞,1)-topos]] consider the "initial topos", the [[terminal category]] $\ast \simeq Sh(\emptyset)$ ([[category of sheaves]] on the empty site).
There is then an [[adjoint triple]]
$$
\mathbf{H}
\stackrel
{\overset{\vdash \varnothing}{\longleftarrow}}
{\stackrel{\overset{}{\longrightarrow}}
{\underset{\vdash \ast}{\longleftarrow}}}
\ast
$$
given by including the [[initial object]] $\varnothing$ and the [[terminal object]] $\ast$ into $\mathbf{H}$, respectively.
In the [[type theory]] of $\mathbf{H}$ this corresponds to the [[adjoint pair]] of [[modalities]]
$$
\varnothing \dashv \ast
$$
which are [[constant functor|constant]] on the [[initial object]]/[[terminal object]], respectively.
The induced [[unit of an adjunction]] transformation is
$$
\array{
\varnothing \longrightarrow X \longrightarrow \ast
}
$$
hence the unique factorization of the unique function $\varnothing \longrightarrow \ast$ through any other [[type]].
Looking through ([Hegel 1812, vol 1, book 1, section 1, chapter 1](#Hegel1812)) one might call $\emptyset$ "nothing", call $\ast$ "being" and then call this unity of opposites "becoming". In particular in §174 of _[[Science of Logic]]_ it says
> there is nothing which is not an intermediate state between being and nothing
which seems to be well-captured by the above unity transformation.
### Quantity : discreteness $\dashv$ continuity
{#Mengen}
The adjoint modality in a [[local topos]] is that given by
[[flat modality]] $\dashv$ [[sharp modality]]
$$
\flat \dashv \sharp
\,.
$$
Capturing [[discrete objects]]/[[codiscrete objects]].
The corresponding unity transformation is
$$
\flat X \longrightarrow X \longrightarrow \sharp X
$$
According to ([Lawvere 94, p. 6](#Lawvere94)) this unity captures the duality that in a [[set]] all [[elements]] are distinct and yet indistinguishable, an apparent [[paradox]] that may be traced back to [[Georg Cantor]].
Looking through Hegel's [[Science of Logic]] at _[On discreteness and repulsion](#Science+of+Logic#OnDiscretenessAndRepulsion)_ one can see that matches with what Hegel calls
> (par 398) Quantity is the unity of these moments of continuity and discreteness
$$
\array{
\flat X &\longrightarrow& X &\longrightarrow& \sharp X
\\
{moment\;of \atop discreteness} && && {moment\;of \atop continuity}
}
$$
### Continuum : repulsion $\dashv$ cohesion
{#ContinuumRepulsionCohesion}
For $\mathbf{H}$ a [[cohesive topos]]/[[cohesive (∞,1)-topos]]
the [[shape modality]] $\dashv$ [[flat modality]] constitute an adjoint cylinder
$$
ʃ \dashv \flat
\,.
$$
The corresponding unity-transformation is the [points-to-pieces transform](cohesive%20topos#CanonicalComparison)
$$
\array{
\flat X \longrightarrow X \longrightarrow ʃ X
}
$$
Looking through ([Hegel 1812, vol 1, book 1, section 2, chapter 1](#Hegel1812)) one might call $\flat$ "repulsion", call $ʃ$ "attraction"/"[[cohesion]]" and then call this unity of opposites "[[continuum]]". Indeed, by the discussion at _[[cohesive topos]]_, this does quite well capture the geometric notion of continuum geometry.
### Infinitesimal Continuum : infin. repulsion $\dashv$ infinit. cohesion
{#ContinuumRepulsionCohesion}
For $\mathbf{H}$ equipped moreover with [[differential cohesion]],
there is the [[infinitesimal object|infinitesimal]] version of
[[shape modality]] $\dashv$ [[flat modality]] namely the adjoint modality
[[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]
$$
\Im \dashv \&
\,.
$$
The corresponding unity-transformation is the
$$
\array{
\& X \longrightarrow X \longrightarrow \Im X
}
$$
maps from the [[coefficients]] for [[crystalline cohomology]] to the [[de Rham space]] of types $X$, where all infinitesimal neighbour points are identified.
In view of the above the unity exhibited here is clearly to be called the "infinitesimal continuum".
### Cohesive sets
The combination of the above two examples of [Continuum](#ContinuumRepulsionCohesion) and [Quantity](#Mengen) is an [[adjoint triple]] of [[modalities]]
$$
ʃ \;\dashv\; \flat \;\dashv\; \sharp
$$
[[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]
characteristic of a [[cohesive topos]].
### Skeleta and Co-Skeleta
[[simplicial skeleton]] $\dashv$ [[simplicial coskeleton]]
### Formal completion $\dashv$ Torsion approximation
{#FormalCompletionAndTorsionApproximation}
For $A$ a [[commutative ring]] or more generally an [[E-∞ ring]] and $\mathfrak{a}\subset \pi_0 A$ a suitable ideal, then $\mathfrak{a}$-[[adic completion]] and $\mathfrak{a}$-[[torsion approximation]] form an adjoint modality on $A MMod$ the [[stable (∞,1)-category|stable]] [[(∞,1)-category of ∞-modules]] $A Mod_\infty$ over $A$.
($\mathfrak{a}$-adic completion) $\dashv$ ($\mathfrak{a}$-torsion approximation)
+-- {: .num_prop #CompletionTorsionAdjointModalityForModuleSpectra}
###### Proposition
Let $A$ be an [[E-∞ ring]] and let $\mathfrak{a} \subset \pi_0 A$ be a [[generators and relations|finitely generated]] ideal in its underlying [[commutative ring]].
Then there is an [[adjoint triple]] of [[adjoint (∞,1)-functors]]
$$
\array{
\underoverset{
A Mod_{\mathfrak{a}comp}^{op}}
{A Mod_{\mathfrak{a}tors}^{op}}
{\simeq}
&\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}&
A Mod^{op}
}
$$
where
* $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$;
* $A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the [[full sub-(∞,1)-categories]] of $\mathfrak{a}$-[[torsion approximation|torsion]] and of $\mathfrak{a}$-[[completion of a module|complete]] $A$-[[∞-modules]], respectively;
* $(-)^{op}$ denotes the [[opposite (∞,1)-category]];
* the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.
=--
+-- {: .proof}
###### Proof
This is effectively the content of ([Lurie "Proper morphisms", section 4](#LurieProper)):
* the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17
* the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there;
* the equivalence of sub-$\infty$-categories is proposition 4.2.5 there.
=--
See at _[[fracture theorem]]_ for more.
### Fermions and supergeometry
On [[super smooth infinity-groupoids]] there is an adjoint modality
deriving from the [adjoint triple relating plain algebra and superalgebra](super+algebra#AdjointsToInclusionOfPlainAlgebra).
The [[right adjoint]] deserves to be called the [[bosonic modality]] ("[[body]]"), hence its [[left adjoint]] the [[fermionic modality]].
This expresses the presence of [[supergeometry]]/[[fermions]], hence ultimately the [[Pauli exclusion principle]]. Following [PN§290](Science+of+Logic#290) this unity of opposites might hence be called "asunderness".
### Totally distributive categories
For $\mathcal{K}$ a [[totally distributive category]]
it induces on its [[category of presheaves]] an adjoint modality whose [[right adjoint]] is the [[Yoneda embedding]] $Y$ postcomposed with its [[left adjoint]] $X$.
### Recollements
See at _[[recollement]]_.
## Related concepts
* [[Aufhebung]]
* [[modal type theory]]
* [[adjoint logic]]
* [[category of being]]
* [[Galois connection]]
## References
### Traditional
The concept of _dialectical reasoning_ is usually attributed to
* [[Plato]], second part of the _[[Parmenides dialogue]]_ .
See
* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Plato -- Dialectic -- Parmenides dialogue](Lectures+on+the+History+of+Philosophy#ParmenidesDialogue)_
Hegel in his _[History of Philosophy](https://www.marxists.org/reference/archive/hegel/works/hp/hpeleatics.htm)_ writes that dialectic begins with [[Zeno]] (one of the characters in that dialogue).
This is much amplified and expanded in
* {#Hegel1812} [[Georg Hegel]], _[[Science of Logic]]_, 1812
The origins of its proposed formalization in [[category theory]] are recalled in
* {#Lambek82} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982)
See also
* {#Wandschneider99} [[Dieter Wandschneider]], _Dialektik als Letztbegründung der Logik_, in Koreanische Hegelgesellschaft (ed.), _Festschrift für Sok-Zin Lim_ Seoul 1999, 255–278 ([pdf](http://www.philosophie.rwth-aachen.de/global/show_document.asp?id=aaaaaaaaaabpltw))
* {#Wikipedia} Wikipedia, _[Hegelian dialectic](http://en.wikipedia.org/wiki/Hegelian_dialectic)_
### In terms of adjoint triples of (co-)reflections and localizations
Conceived of in terms of [[adjoint triples]] of [[reflective subcategory|(co-)reflections]] and [[localization of a category|localizations]] the concept appears in
* {#Lawvere91} [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ in A. Carboni, M. Pedicchio, G. Rosolini, _Category Theory_ , [[Como|Proceedings of the International Conference held in Como]], Lecture Notes in Mathematics 1488, Springer (1991)
* {#Lawvere94} [[William Lawvere]], _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_, Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]])
* {#Lawvere94b} [[William Lawvere]], _[[Tools for the advancement of objective logic]]: closed categories and toposes_, in J. Macnamara and [[Gonzalo Reyes]] (Eds.), _The Logical Foundations of Cognition_, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference "Logic and Cognition"),
pages 43-56, 1994.
* {#Lawvere96} [[William Lawvere]], _[[Unity and Identity of Opposites in Calculus and Physics]]_, Proceedings of ECCT 1994 Tours Conference, Applied Categorical Structures, 4: 167-174 Kluwer Academic Publishers, (1996).
* {#Lawvere00} [[F. W. Lawvere]], _Adjoint Cylinders_, message to catlist November 2000. ([link](https://www.mta.ca/~cat-dist/archive/2000/00-11))
* {#LawvereRosebrugh03} [[William Lawvere]], [[Robert Rosebrugh]], p. 245 of: _[[Sets for Mathematics]]_, Cambridge UP 2003 ([doi:10.1017/CBO9780511755460](https://doi.org/10.1017/CBO9780511755460), [book homepage](http://www.mta.ca/~rrosebru/setsformath/), [GoogleBooks](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics), [pdf](http://patryshev.com/books/Sets%20for%20Mathematics.pdf))
### In terms of adjoint pairs of modal operators
In terms of [[adjoint pairs]] of [[modal operators]] and hence of [[Galois connections]], the concept appears in
* {#ReyesZolfaghari91} [[Gonzalo Reyes]], H. Zolfaghari, _Topos-theoretic approaches to modality_, Lecture Notes in Mathematics 1488 (1991), 359-378.
* {#Reyes91} [[Gonzalo Reyes]], _A topos-theoretic approach to reference and modality_, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 ([Euclid](http://projecteuclid.org/euclid.ndjfl/1093635834))
with further developments in
* M. Sadrzadeh, R. Dyckho, _Positive logic with adjoint modalities: Proof
theory, semantics and reasoning about information_, Electronic Notes in Theoretical
Computer Science 249, 451-470, 2009, in _Proceedings of the 25th Conference on
Mathematical Foundations of Programming Semantics_ (MFPS 2009).
* {#Hermida10} [[Claudio Hermida]], section 3.3. of _A categorical outlook on relational modalities and simulations_, 2010 ([pdf](http://maggie.cs.queensu.ca/chermida/papers/sat-sim-IandC.pdf))
* {#DJK14} Wojciech Dzik, Jouni Järvinen, Michiro Kondo, _Characterising intermediate tense logics in terms of Galois connections_ ([arXiv:1401.7646](http://arxiv.org/abs/1401.7646))
Formalization specifically in [[modal type theory]] is in
* {#LicataShulman} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf))
For an overview of the role of adjunctions in modal logic see:
* [[Matías Menni|M. Menni]], C. Smith, _Modes of Adjointness_ , J. Philos. Logic **43** no.3-4 (2014) pp.365-391.
### Examples
* {#LurieProper} [[Jacob Lurie]], section 4 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_
[[!redirects adjoint cylinders]]
[[!redirects adjoint modality]]
[[!redirects adjoint modalities]]
[[!redirects opposite]]
[[!redirects opposites]]
[[!redirects unity of opposites]]
[[!redirects unities of opposites]]
[[!redirects unity and identity of opposites]]
[[!redirects dialectic]]
[[!redirects dialectics]]
[[!redirects duality of opposites]]
[[!redirects dualities of opposites]] |
adjoint monad | https://ncatlab.org/nlab/source/adjoint+monad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[monad]] $(T,\mu,\eta)$ is [[adjoint functor|adjoint]] to a [[comonad]] $(G,\delta,\epsilon)$, if its underlying endofunctor $T$ is _[[left adjoint]]_ to the underlying [[1-morphism]] $G$ of the comonad, and $\delta$ and $\epsilon$ are conjugate/adjoint/[[mate]] 2-cells to $\mu$ and $\eta$ in the sense explained below.
## Construction
In fact given a [[monad]] $\mathbf{T} = (T,\mu^T,\eta^T)$ which has a _[[right adjoint]]_ $G$, automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$.
Thus there is a bijective correspondence between monads having a right adjoint and comonads having a left adjoint (what [[Alexander Rosenberg]] calls **duality**). This is a little more than a consequence of two general facts:
1. If $T\dashv G$ then $T^k \dashv G^k$ for every [[natural number]] $k$.
2. Given two [[adjunctions]] $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a [[bijection]] between the [[natural transformations]] $\phi:S'\Rightarrow S$ and natural transformations $\psi:T\Rightarrow T'$ such that
$$
\array{
A (S,-) &\to& B(-,T)
\\
{}^{\mathllap{A(\phi,-)}}\downarrow &&\downarrow {}^{\mathrlap{B(-,\psi)}}
\\
A(S',-)&\to & B(-,T')
}
$$
where the horizontal arrows are the natural bijections given by the adjunctions. [Eilenberg & Moore 1965](#EilenbergMoore65) would write $\phi\dashv\psi$ and talk about "adjointness for morphisms" (of functors), which is of course relative to the given adjunctions among functors. MacLane calls the correspondence *conjugation* (p 99-102 in *[[Categories for Working Mathematician]]*). It is a special case, of a general construction of [[mates]].
If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their
[[unit of an adjunction|unit]] and counit of course the upper arrow is $(S M\stackrel{f}\to N)\mapsto T f\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as
$$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ T f\circ\eta_M$$
or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above [[diagram]], as the horizontal arrows are [[isomorphism|invertible]]. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition
$\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e.
$$
T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\phi T}\longrightarrow T' S T \stackrel{T'\epsilon}\longrightarrow T'
$$
and checks that it works. The inverse is similarly given by the composition
$$
S'\stackrel{S'\eta}\longrightarrow S' T S\stackrel{S'\psi S}\longrightarrow S'T'S\stackrel{\epsilon' S}\longrightarrow S
$$
This correspondence now enables in our special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit.
## Examples
\begin{example}\label{AdjointMonadsInducedFromAdjointTriples}
**(adjoint monads induced from adjoint triples of adjoint functors)**
\linebreak
Every [[adjoint triple]] (of [[adjoint functors]]) $F^*\dashv F_* \dashv F^!$ [induces](monad#RelationBetweenAdjunctionsAndMonads) an [[adjoint pair]] $F_* F^*\dashv F_* F^!$. The endofunctor $F_* F^*$ is underlying a monad induced by the adjunction $F^*\dashv F_*$ and $F_* F^!$ is underlying a comonad induced by the adjuntion $F_*\dashv F^!$. This pair of a monad and a comonad are adjoint.
\end{example}
(See also at *[[adjoint modality]]*.)
## Properties
### General
\begin{proposition}
\label{IsomorphismOfEMCategories}
Given an [[adjoint functor|adjoint pair]] of a [[monad]] and comonad
$$
\lozenge \,\dashv\, \Box
$$
on some category $\mathcal{C}$, then there is an [[equivalence of categories|equivalence]] between their [[Eilenberg-Moore categories]] of [[algebra over a monad|algebras]] over $\lozenge$ and [[coalgebra over a comonad|coalgebras]] over $\Box$, compatible with their [[forgetful functors]] to $\mathcal{C}$:
\begin{tikzcd}[column sep=-1pt]
\mathrm{EM}(\lozenge)
\ar[dr, "{ U }"{swap}]
\ar[rr, "{\sim}"]
&&
\mathrm{EM}(\Box)
\ar[dl, "{ U }"]
\\
&
\mathcal{C}
\end{tikzcd}
\end{proposition}
This is due to [Eilenberg & Moore 1965](#EilenbergMoore65), where it is implied by the last part of Prop. 3.3. In the more explicit form above the statement may be found in [MacLane & Moerdijk 1992, Thm. 2 on p. 249](#MacLaneMoerdijk92),
### General adjoint (co)algebras $End(A)$
Given a [[small category]] $A$, the [[endofunctor category]] $End(A)$ (of [[endofunctors]] and [[natural transformations]] between them, with [[vertical composition]] as [[composition]]) is [[monoidal category|monoidal]] with respect to the composition as the tensor product of objects (endofunctor) and Godement product ([[horizontal composition]]) as the [[tensor product]] of morphisms (natural transformations). Hence we can consider [[operads]] and [[algebras over operads]], as well as, dually, coalgebras over cooperads; or some other framework for general algebras and coalgebras (or even props).
In any case, given an adjunction $T\dashv G$, operations $T^n\to T$ dualize to cooperations $G\to G^n$, and more generally multioperations $T^k\to T^l$ dualize to the multioperations $G^l\to G^k$. We would like to sketch the proof that the identities for operations on $T$, correspond to the identities on cooperations on $G$ (and more generally there is a duality among the identities for multioperations). This is essentially the consequence of
__Lemma.__ ([[Zoran Škoda|Zoran]]) Given the adjunction $T\dashv G$ with unit $\eta$ and counit $\epsilon$, and the sequence
$$
T^k \stackrel{\alpha}\longrightarrow T^l\stackrel\beta\longrightarrow T^s
$$
the composition $\alpha^*\circ\beta^*$ of the dual (in the above sense) sequence
$$
G^k \stackrel{\alpha^*}\longleftarrow G^l\stackrel{\beta^*}\longleftarrow G^s
$$
equal to the dual $(\beta\circ\alpha)^*$ of $\beta\circ\alpha$,
*Proof*. [[Mike Shulman]] notices that this is a special case of known contravariant functoriality of [[mate]]s, but here is a direct proof.
We need to prove that the composition
$$
G^s\stackrel{\eta_l G^s}\to G^l T^l G^s\stackrel{G^l\beta G^s}\to G^l T^s G^s\stackrel{G^l \epsilon_s}\to G^l\stackrel{\eta_k G^l}\to G^k T^k G^l\stackrel{G^k\alpha G^l}\to G^k T^l G^l\stackrel{G^k\epsilon_l}\to G^k
$$
equals the composition
$$
G^s\stackrel{\eta_k G^l}\to G^k T^k G^s\stackrel{G^k\alpha G^l}\to G^k T^l G^s\stackrel{G^k\beta G^s}\to G^k T^s G^s\stackrel{G^k\epsilon_s}\to G^k.
$$
Note that in the two compositions there is an opposite order between the expressions involving $\alpha$ and those involving $\beta$. But anyway, their equality reduces to a naturality calculation (which in particular exchanges the order of $\alpha$ and $\beta$ in effect):
$$\array{
G^s &\stackrel{\eta_l G^s}\to & G^l T^l G^s &\stackrel{G^l\beta G^s}\to & G^l T^s G^s&\stackrel{G^l \epsilon_s}\longrightarrow& G^l\\
\eta^k G^s\downarrow &&\downarrow \eta_k G^l T^l G^s&&\downarrow \eta_k G^l T^s G^s&& \downarrow \eta_k G^l \\
G^k T^k G^s &\stackrel{G^k T^k\eta_l G^s}\longrightarrow &G^k T^k G^l T^l G^s &\stackrel{G^k T^k G^l \beta G^s}\longrightarrow & G^k T^k G^l T^s G^s &\stackrel{G^k T^k G^l \epsilon_s}\longrightarrow & G^k T^k G^l\\
G^k \alpha G^s \downarrow && \downarrow G^k \alpha G^l T^l G^s&&\downarrow G^k \alpha G^k T^s G^s&&\downarrow G^k \alpha G^l\\
G^k T^l G^s &\stackrel{G^k T^l\eta_l G^s}\longrightarrow & G^k T^l G^l T^l G^s &\stackrel{G^k T^l G^l \beta G^s}\longrightarrow & G^k T^l G^l T^s G^s &\stackrel{G^k T^l G^l\epsilon_s}\longrightarrow & G^k T^l G^l\\
G^k\beta G^s \downarrow &&\downarrow{\rho} &&\downarrow G^k \epsilon_l T^s G^s &&\downarrow G^k\epsilon_l \\
G^k T^s G^s &=&G^k T^s G^l &=&G^k T^s G^s&\stackrel{G^k \epsilon_s}\longrightarrow& G^k
}$$
where $\rho := G^k \beta G^s \circ G^k \epsilon_l G^s = G^k\epsilon_l T^s G^s\circ G^k T^l G^l \beta G^s$. The commutativity of all small squares in the diagram is evident, except the lower left corner. This one follows by one of the triangle identities for the adjunction $T^l\dashv G^l$. Namely,
$$
G^k \beta G^s = G^k \beta G^s \circ (G^k \epsilon_l T^l G^s\circ G^k T^l \eta_l G^s) = \rho \circ G^k T^l \eta_l G^s
$$
## Examples
### Adjoint modalities
An [[adjoint modality]] is an example of a pair of adjoint monads.
## References
Discussion of adjoint monads originates with
* {#EilenbergMoore65} [[Samuel Eilenberg]], [[John Moore]], Section 3 in: _Adjoint functors and triples_, Illinois J. Math. **9** 3 (1965) 381-398 [[doi:10.1215/ijm/1256068141](https://doi.org/10.1215/ijm/1256068141)]
(there called "adjoint triples", sticking with the old term "triple" for "monad", a terminology that now clashes with the modern use of *[[adjoint triples]]* of [[adjoint functors]]).
Textbook account:
* {#MacLaneMoerdijk92} [[Saunders Mac Lane]], [[Ieke Moerdijk]], pp. 248 in: _[[Sheaves in Geometry and Logic|Sheaves in Geometry and Logic --- A First Introduction to Topos Theory]]_, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://dx.doi.org/10.1007/978-1-4612-0927-0)]
Discussion in the context of [[ambidextrous adjunctions]] and [[Frobenius monads]]:
* {#Street04} [[Ross Street]], *Frobenius monads and pseudomonoids*, J. Math. Phys. **45** 3930 (2004) [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)]
* {#Lauda05} [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, Theory and Applications of Categories **16** 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]
[[!redirects adjoint monads]]
[[!redirects adjoint comonad]]
[[!redirects adjoint comonads]] |
adjoint morphism | https://ncatlab.org/nlab/source/adjoint+morphism |
A [[morphism]] participating in an [[adjunction]].
[[!redirects adjoint morphisms]]
|
adjoint operator | https://ncatlab.org/nlab/source/adjoint+operator | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
Let $A: H\to H$ be an [[unbounded operator]] on a [[Hilbert space]] $H$. An unbounded operator $A^*$ is its __[[adjoint operator|adjoint]]__ if
* $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and
* every $B$ satisfying the above property for $A^*$ is a restriction of $A$.
On [[finite-dimensional Hilbert spaces]], adjoint operators always exists, in [[matrix]]-components with respect to any [[orthonormal linear basis]] given by passage to the [[complex conjugation|complex conjugate]] [[transpose matrix]].
On infinite-dimensional Hilbert spaces an adjoint operator does not need to exist, in general.
## History
{#History}
Recounted by [MacLane 1988](#MacLane88), [p. 330](http://www.ams.org/publicoutreach/math-history/hmath1-maclane25.pdf#page=8):
> Two of [[John von Neumann|von Neummann]]'s papers on this topic [[[Hilbert spaces]]] had been accepted in the Mathematische Annalen, a journal of Springer Verlag. [[Marshall Stone]] had seen the manuscripts, and urged von Neumann to observe that his treatment of linear operators $T$ on a Hilbert space could be much more effective if he were to use the notion of an adjoing $T^ast$ to the linear transformation $T$ --- one for which the now familiar equation
> $\;\;\;\;\; \langle T a, b \rangle \;=\; \langle a, T^\ast b \rangle$
> would hold for all suitable $a$ and $b$. Von Neumann saw the point immediately, as was his wont, and wishes to withdraw the papers before publication. They were already set up in type; Springer finally agreed to cancel them on the condition that von Neumann write for them a book on the subject --- which he soon did [[1932](#vonNeumann1932)].
> This story (told to me by [[Marshall Stone]]) illustrates the important conceptual advance represented by the definition of adjoint operators. &lbrack...] I have written elsewhere [[1970](#MacLane70)] that it is a step toward the subsequent description of a [[functor]] $G$ [[right adjoint]] to a functor $F$, in terms of [a natural isomorphism](adjoint+functor#InTermsOfHomIsomorphism)
> $\;\;\;\;\; hom(F a, b) \;\simeq\; hom(a, G b)$
> between [[hom-sets]] in suitable [[categories]].
(Cf. discussion at [adjoint functor -- idea](adjoint+functor#Idea).)
## Related concepts
* [[self-adjoint operator]]
## References
The notion of adjoint operators is originally due to [[Marshall Stone]], see also the history section [above](#History), as recounted in
* {#MacLane70} [[Saunders MacLane]], *The Influence of M. H. Stone on the Origins of Category Theory*, in *Functional Analysis and Related Fields*, Springer (1970) [[doi:10.1007/978-3-642-48272-4_12](https://doi.org/10.1007/978-3-642-48272-4_12)]
* {#MacLane88} [[Saunders Mac Lane]]: §5 in: *Concepts and Categories in Perspective*, in: P. Duren, *A century of mathematics in America* Part 1, AMS (1988) 323-365. [[pdf](http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf), [ISBN:0-8218-0124-4](https://www.ams.org/publicoutreach/math-history/hmath1-index)]
Original discussion in print is due to:
* {#vonNeumann1932} [[John von Neumann]]:
*Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]
*Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]
Lecture notes:
* [[Bergfinnur Durhuus]], *Operators on Hilbert Space* [[pdf](https://web.math.ku.dk/~durhuus/MatFys/MatFys4.pdf), [[Durhuus-OperatorsOnHilbertSpace.pdf:file]]]
[[!redirects adjoint operators]]
|
adjoint quadruple | https://ncatlab.org/nlab/source/adjoint+quadruple |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An **adjoint quadruple** is a sequence of three [[adjunctions]]
$$
f_! \dashv f^* \dashv f_* \dashv f^!
$$
between a [[quadruple]] of [[morphisms]]. That is, it is an [[adjoint string]] of length 4.
## Properties
### General
Every adjoint quadruple
$$
(f_! \dashv f^* \dashv f_* \dashv f^!)
:
C
\stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}}
D
$$
induces an [[adjoint triple]] on $C$
$$
(f^* f_! \dashv f^* f_* \dashv f^! f_*)
:
C \to C
\,,
$$
(hence a [[monad]] [[left adjoint]] to a [[comonad]] left adjoint to a monad) and an adjoint triple
$$
(f_! f^* \dashv f_* f^* \dashv f_* f^!) : D \to D
$$
on $D$.
Since moreover every [[adjoint triple]] $(F \dashv G \dashv H)$ induces an [[adjoint functor|adjoint pair]] $(G F \dashv G H)$ and an adjoint pair $(F G \dashv H G)$, the adjoint quadruple above induces four adjoint pairs, such as
$$
(f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C
\,.
$$
$\,$
### Canonical natural transformation
Let
$$
(p_! \dashv p^* \dashv p_*\dashv p^!)
\;\colon\;
\mathcal{E}
\longrightarrow
\mathcal{S}
$$
be an [[adjoint quadruple]] of [[adjoint functor|adjoint functors]] such that $p^*$ and $p^!$ are [[full and faithful functor|full and faithful functors]]. We record some general properties of such a setup.
We write
$$
\eta \;\colon\; id \to p^* p_!
$$
etc. for [[unit of an adjunction|units]] and
$$
\epsilon \;\colon\; p_! p^* \to id
$$
etc. for counits.
+-- {: .num_prop #TheCanonicalMorphisms}
###### Proposition/Definition
We have [[commuting diagrams]], [[natural transformation|natural]] in $X \in \mathcal{E}$, $S \in \mathcal{S}$
$$
\array{
p_*X &\underoverset{\simeq}{\epsilon_{p^* X}^{-1}}{\longrightarrow}& p_! p^* p_*X
\\
{}^{\mathllap{p_*(\eta_X)}}\downarrow
&\searrow^{\mathrlap{\theta_X}}&
\downarrow^{\mathrlap{p_!(\epsilon_X)}}
\\
p_* p^* p_! X &\stackrel{\eta_{p_!X}^{-1}}{\longrightarrow}& p_! X
}
$$
and
$$
\array{
p^* S &\stackrel{\eta_{p^* S}}{\longrightarrow}& p^! p_* p^* S
\\
{}^{\mathllap{p^* \epsilon_S^{-1}}}\downarrow
&\searrow^{\mathrlap{\phi_X}}& \downarrow^{\mathrlap{p^!(\eta_S^{-1})}}
\\
p^* p_* p^!S &\stackrel{{\epsilon}_{p_!S }}{\longrightarrow}& p^!S
}
\,.
$$
where the diagonal morphisms
$$
\theta_X : p_* X \to p_! X
$$
and
$$
\phi_S : p^* S \to p^! S
$$
are defined to be the equal composites of the sides of these diagrams.
=--
This appears as ([Johnstone 11, lemma 2.1, corollary 2.2](#Johnstone11)).
+-- {: .num_prop #TheEpiAndTheMono}
###### Proposition
The following conditions are equivalent:
* for all $X \in \mathcal{E}$ the morphism $\theta_X : p_*X \to p_! X$ is an [[epimorphism]];
* for all $S \in \mathcal{S}$,, the morphism $\phi_S : p^*S \to p^! S$
is a [[monomorphism]];
* $p_*$ is [[faithful functor|faithful]] on morphisms of the form $A \to p^* S$.
=--
This appears as ([Johnstone 11, lemma 2.3](#Johnstone11)).
+-- {: .proof}
###### Proof
By the above definition, $\phi_S$ is a [[monomorphism]] precisely if $\eta_{p^* S} : p^* S \to p^! p_* p^* S$ is. This in turn is so (see [[monomorphism]]) precisely if the first [[function]] in
$$
\mathcal{E}(A,p^* X)
\stackrel{(\eta_{p^* X}) \circ (-)}{\longrightarrow}
\mathcal{E}(A, p^! p_* p^* S)
\stackrel{\simeq}{\longrightarrow}
\mathcal{S}(p_* A, p_* p^* S)
$$
and hence the composite is a monomorphism in [[Set]].
By definition of [[adjunct]] and using the $(p_* \dashv p^!)$-[[zig-zag identity]], this is equal to the action of $p_*$ on morphisms
$$
(\eta_{p^* X}) \circ (-) :
(A \to p^* S) \mapsto p_*(A \to p^* S)
\,.
$$
Similarly, by the above definition the morphism $\theta_X$ is an epimorphism precisely if $p_!(\epsilon_X) : p_! p^* p_* X \to p_! X$ is so, which is the case precisely if the top morphism in
$$
\array{
\mathcal{S}(p_! X, S)
&\stackrel{(-) \circ p_!(\epsilon_X)}{\longrightarrow} &
\mathcal{S}(p_! p^* p_* X, S)
\\
{}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
\\
&& \mathcal{E}(p^* p_* X, p^* S)
\\
{}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
\\
\mathcal{E}(X, p^* S) &\stackrel{p_*}{\longrightarrow}& \mathcal{S}(p_* X, p_* p^* S)
}
$$
and hence the bottom morphism is a monomorphism in [[Set]],
where again the commutativity of this diagram follows from the
definition of [[adjunct]] and the
$(p_! \dashv p^*)$-[[zig-zag identity]].
=--
$\,$
## Examples
### Via Kan extension of adjoint pairs
{#ViaKanExtensionOfAdjointPairs}
A rich source of adjoint quadruples arises form [[adjoint pairs]] between [[small categories]] by left/right [[Kan extension]] to their [[categories of presheaves]].
More interesting examples of adjoint quadruples tend to arise from these presheaf constructions when the quadruple ([[corestriction|co]])[[restriction|restricts]] to sub-[[categories of sheaves]].
We spell out two proofs of this fact, the first using [[coend]]-calculus in the generality of [[enriched category theory]], the second using more elementary [[colimit]]-notation.
\begin{prop}\label{KanExtensionOfAdjointPairIsAdjointQuadruple}
**([[Kan extension]] of [[adjoint pair]] is [[adjoint quadruple]])**
\linebreak
For $\mathcal{V}$ a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with all [[limits]] and [[colimits]], let $\mathcal{C}$, $\mathcal{D}$ be two [[small category|small]] $\mathcal{V}$-[[enriched categories]]and let
$$
\mathcal{C}
\underoverset
{\underset{p}{\longrightarrow}}
{\overset{q}{\longleftarrow}}
{\bot}
\mathcal{D}
$$
be a $\mathcal{V}$-[[enriched adjunction]]. Then there are $\mathcal{V}$-[[enriched natural isomorphisms]]
$$
(q^{op})^\ast \;\simeq\; Lan_{p^{op}}
\;\colon\;
[\mathcal{C}^{op},\mathcal{V}]
\longrightarrow
[\mathcal{D}^{op},\mathcal{V}]
$$
$$
(p^{op})^\ast \;\simeq\; Ran_{q^{op}}
\;\colon\;
[\mathcal{D}^{op},\mathcal{V}]
\longrightarrow
[\mathcal{C}^{op},\mathcal{V}]
$$
between the precomposition on [[enriched presheaves]] with one functor and the left/right [[Kan extension]] of the other.
By essential uniqueness of [[adjoint functors]] ([this Prop.](adjoint+functor#UniquenessOfAdjoints)), this means that the two [[Kan extension]] [[adjoint triples]] of $q$ and $p$
$$
\array{
Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}}
\\
&& Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}}
}
$$
merge into an [[adjoint quadruple]]
$$
\array{
Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}}
}
\;\colon\;
[\mathcal{C}^{op},\mathcal{V}]
\leftrightarrow
[\mathcal{D}^{op}, \mathcal{V}]
$$
\end{prop}
\begin{proof}
For every [[enriched presheaf]] $F \;\colon\; \mathcal{C}^{op} \to \mathcal{V}$ we have a sequence of $\mathcal{V}$-[[enriched natural isomorphism]] as follows
$$
\begin{aligned}
(Lan_{p^{op}} F)(d)
& \simeq
\int^{ c \in \mathcal{C} } \mathcal{D}(d,p(c)) \otimes F(c)
\\
& \simeq
\int^{ c \in \mathcal{C} } \mathcal{C}(q(d),c) \otimes F(c)
\\
& \simeq
F(q(d))
\\
& = \left( (q^{op})^\ast F\right) (d)
\,.
\end{aligned}
$$
Here the first step is the [[coend]]-formula for [[left Kan extension]] ([here](Kan+extension#PointwiseByCoEnds)), the second step is the [[enriched adjunction]]-isomorphism for $q \dashv p$ and the third step is the [[co-Yoneda lemma]].
This shows the first statement. By essential uniqueness of adjoints ([this Prop.](adjoint+functor#UniquenessOfAdjoints)), the other statements follow.
\end{proof}
The following is the same argument without using coend-calculus. This argument applies verbatim also, for instance, in [[(infinity,1)-category theory|$\infty$-category theory]] using results from standard sources:
\begin{prop}
\label{AdjointPairInducesAdjointQuadrupleUnderKanExtension}
Given a pair of [[adjoint functors]] between [[small categories]]
\begin{tikzcd}
\mathcal{S}_2
\ar[
rr,
shift left=7pt,
"{ \ell }"{above}
]
&&
\mathcal{S}_1
\ar[
ll,
shift left=7pt,
"{ r }"{below}
]
\ar[
ll,
phantom,
"{ \scalebox{.6}{$\bot$} }"
]
\end{tikzcd}
the induced operations of pre-composition on [[categories of presheaves]] are adjoint to each other, $\ell^\ast \,\dashv\, r^\ast$, and their [[adjoint triples]] of [[Kan extensions]] overlap:
\begin{tikzcd}[column sep=large]
\mathrm{PSh}(\mathcal{S}_2)
\;\;
\ar[
rr,
shift left=-7pt,
"{ \ell_\ast \,\simeq\, r^\ast }"{description}
]
\ar[
rr,
shift left=32pt-7pt,
"{ \ell_! }"{description, pos=.4}
]
&&
\;\;
\mathrm{PSh}(\mathcal{S}_1)
\ar[
ll,
shift right=16pt-7pt,
"{ \ell^\ast \,\simeq\, r_! }"{description}
]
\ar[
ll,
shift right=-16pt-7pt,
"{ r_\ast }"{description, pos=.4}
]
\ar[
ll,
phantom,
shift right=8pt-7pt,
"{\scalebox{.6}{$\bot$}}"
]
\ar[
ll,
phantom,
shift right=24pt-7pt,
"{\scalebox{.6}{$\bot$}}"
]
\ar[
ll,
phantom,
shift right=-8pt-7pt,
"{\scalebox{.6}{$\bot$}}"
]
\end{tikzcd}
\end{prop}
\begin{proof}
{#ProofOfAdjointPairInducesAdjointQuadrupleUnderKanExtension}
We already know that each functor $f$ by itself induces an [[adjoint triple]] $f_! \dashv f^\ast \dashv f_\ast$, by [[Kan extension]]. Due to essential uniqueness of adjoints ([this Prop.](adjoint+functor#UniquenessOfAdjoints)) it is hence sufficient to show that these two adjoint triples "overlap", in that ($\ell^\ast \simeq r_!$ and equivalently) $\ell_\ast \simeq r^\ast$, hence equivalently that $\ell^\ast \dashv r^\ast$.
Now the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) which is characteristic of the latter adjunction $\ell^\ast \dashv r^\ast$ may be obtained as the following sequence of [[natural bijections]]:
$$
\begin{aligned}
&
\mathrm{PSh}(\mathcal{S}_1)
\big(
X_1
,\,
r^\ast(X_2)
\big)
\\
&
\;\simeq\;
\mathrm{PSh}(\mathcal{S}_1)
\Big(
\underset{
\underset{
s_1 \to X_1
}{\longrightarrow}
}{\lim}
\,
y(s_1)
\,
,\,
r^\ast
\big(
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
\,
y(s_2)
\big)
\Big)
\\
&
\;\simeq\;
\underset{
\underset{
s_1 \to X_1
}{\longleftarrow}
}{\lim}
\mathrm{PSh}(\mathcal{S}_1)
\Big(
y(s_1)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
\,
r^\ast
\big(
y(s_2)
\big)
\Big)
\\
&
\;\simeq\;
\underset{
\underset{
s_1 \to X_1
}{\longleftarrow}
}{\lim}
\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
\mathrm{PSh}(\mathcal{S}_1)
\Big(
y(s_1)
,\,
r^\ast
\big(
y(s_2)
\big)
\Big)
\\
& \;\simeq\;
\underset{
\underset{
s_1 \to X_1
}{\longleftarrow}
}{\lim}
\;
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
\mathcal{S}_2
\big(
r(s_1)
,\,
s_2
\big)
\\
& \;\simeq\;
\underset{
\underset{
s_1 \to X_1
}{\longleftarrow}
}{\lim}
\;
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
\mathrm{PSh}(\mathcal{S}_2)
\Big(
y\big(r(s_1)\big)
,\,
y(s_2)
\Big)
\\
& \;\simeq\;
\underset{
\underset{
s_1 \to X_1
}{\longleftarrow}
}{\lim}
\mathrm{PSh}(\mathcal{S}_2)
\Big(
y\big(r(s_1)\big)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
y(s_2)
\Big)
\\
& \;\simeq\;
\mathrm{PSh}(\mathcal{S}_2)
\Big(
\underset{
\underset{
s_1 \to X_1
}{\longrightarrow}
}{\lim}
y\big(r(s_1)\big)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
y(s_2)
\Big)
\\
& \;\simeq\;
\mathrm{PSh}(\mathcal{S}_2)
\Big(
\underset{
\underset{
s_1 \to X_1
}{\longrightarrow}
}{\lim}
\mathcal{S}_2
\big(
(-)
,\,
r(s_1)
\big)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
y(s_2)
\Big)
\\
& \;\simeq\;
\mathrm{PSh}(\mathcal{S}_2)
\Big(
\underset{
\underset{
s_1 \to X_1
}{\longrightarrow}
}{\lim}
\mathcal{S}_2
\big(
\ell(-)
,\,
s_1
\big)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
y(s_2)
\Big)
\\
& \;\simeq\;
PSh(\mathcal{S}_2)
\Big(
\underset{
\underset{
s_1 \to X_1
}{\longrightarrow}
}{\lim}
\ell^\ast
\big(
y(s_1)
\big)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
y(s_2)
\Big)
\\
& \;\simeq\;
PSh(\mathcal{S}_2)
\Big(
\ell^\ast
\big(
\underset{
\underset{
s_1 \to X_1
}{\longrightarrow}
}{\lim}
y(s_1)
\big)
,\,
\underset{
\underset{
s_2 \to X_2
}{\longrightarrow}
}{\lim}
y(s_2)
\Big)
\\
& \;\simeq\;
PSh(\mathcal{S}_2)
\Big(
\ell^\ast
(
X_1
)
,\,
X_2
\Big)
\end{aligned}
$$
Here we used repeatedly
* the [[co-Yoneda lemma]] in the form
$$
X
\;\simeq\;
\underset{
\underset{y(s) \to X}{\longrightarrow}
}{\lim}
\,y(s)
\,,
$$
expressing a [[presheaf]] $X \,\in\, PSh(S)$ as a [[colimit]] of [[representable functors]] $y(s)$ (the colimit is over the [[comma category]] $y \downarrow X$, but that does not even matter in the proof above),
* the fact that any [[hom-functor preserves limits|hom-functor sends colimits in its first argument to limits]],
* the strong [[Yoneda lemma]] which says that $PSh(\mathcal{S})\big(y(s), X\big) \,\simeq\, X(s)$,
* the fact that [[colimits of presheaves are computed objectwise]].
As before, the adjunction $\ell^\ast \dashv r_!$ implies the overlapping adjoint triples by essential uniqueness of [[adjoint functors]] ([this Prop.](adjoint+functor#UniquenessOfAdjoints)).
\end{proof}
### Cohesion
{#CohesiveToposes}
\begin{prop}\label{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple}
**(Kan extension of finite product preserving reflection is cohesive adjoint quadruple)**
\linebreak
Let
\begin{tikzcd}
\mathcal{S}_1
\ar[
r,
shift left=7pt,
"{ p }"{above},
"{ \mathclap{\widehat{\times}} }"{description, pos=.0}
]
&
\mathcal{S}_2
\ar[
l,
shift left=7pt,
hook',
"{ i }"{below}
]
\ar[
l,
phantom,
"{\scalebox{.6}{$\bot$}}"{description}
]
\end{tikzcd}
be a pair of [[adjoint functors]] between [[small categories]] that have [[finite products]] (or at least after passing to their [[free coproduct completion]]), such that
1. the [[right adjoint]] $i$ is [[fully faithful functor|fully faithful]],
1. the [[left adjoint]] $p$ [[preserves limits|preserves]] [[finite products]] (or at least its coproduct-preserving extension to [[free coproduct completions]] does).
Then the induced adjoint quadruple of Kan extensions from Prop. \ref{AdjointPairInducesAdjointQuadrupleUnderKanExtension} is [[cohesive topos|cohesive]] in that
1. the two reverse functors are [[fully faithful functor|fully faithful]].
1. the leftmost adjoint $p_!$ [[preserved limit|preserves]] [[finite products]];
\begin{tikzcd}
\mathrm{PSh}(\mathcal{S}_1)
\ar[
rr,
"{ p_\ast \,\simeq\, i^\ast }"{description}
]
\ar[
rr,
shift left=40pt,
"{ p_! }"{description, pos=.39},
"\mathclap{\times}"{description, pos=0}
]
&&
\mathrm{PSh}(\mathcal{S}_1)
\ar[
ll,
hook',
shift right=20pt,
"{ p^\ast \,\simeq\, i_! }"{description}
]
\ar[
ll,
hook',
shift right=-20pt,
"{ i_\ast }"{description, pos=.39}
]
\ar[
ll,
phantom,
shift right=10pt,
"{\scalebox{.6}{$\bot$}}"
]
\ar[
ll,
phantom,
shift right=30pt,
"{\scalebox{.6}{$\bot$}}"
]
\ar[
ll,
phantom,
shift right=-10pt,
"{\scalebox{.6}{$\bot$}}"
]
\end{tikzcd}
\end{prop}
\begin{proof}
The preservation of finite products by the leftmost adjoint follows by Prop. \ref{LeftKanExtensionOfFinProdPreservingIsFinProdPreserving} below.
The fully faithfulness of $i_!$ follows by Prop. \ref{LeftKanExtensionOfFullyFaithfulIsFullyFaithful} below. This implies that also $i_\ast$ is fully faithful, by [this Prop.](adjoint+triple#FullyFaithful).
\end{proof}
\begin{example}
Consider a [[site]] $\mathcal{S}$ with [[finite products]], in particular with a [[terminal object]]. Then the inclusion of the [[full subcategory]] on this terminal object, which is the [[terminal category]] $\ast$ is an adjunction of the form
$$
\mathcal{S}
\underoverset
{\underset{}{\hookleftarrow}}
{\overset{}{\longrightarrow}}
{\;\;\;\bot\;\;\;}
\ast
\,.
$$
If $\mathcal{S}$ is a *[[cohesive site]]* then the
induced adjoint quadruple from Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} ([[corestriction|co]])[[restriction|restricts]] to the [[category of sheaves]] $Sh(\mathcal{S}) \xhookrightarrow{\;} PSh(\mathcal{S})$ and exhibits it as a [[cohesive topos]].
\end{example}
The proof of Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} as spelled out below applies verbatim also in [[(infinity,1)-category theory|$\infty$-category theory]]. Here, it has interesting examples even without passage to [[(infinity,1)-sheaves|$\infty$-sheaves]], when the [[(infinity,1)-site|$\infty$-sites]] are higher than [[1-sites]]:
\begin{example}\label{COhesionOfGlobalOverGEquivariantHomotopyTheory}
**([[cohesion of global- over G-equivariant homotopy theory]])**
\linebreak
For $G$ a group, there is an adjunction of [[(2,1)-categories]]
\begin{tikzcd}
\mathrm{Sngrt}_{/\prec G}
\ar[
r,
shift left=7pt,
"{ \tau_0 }"{above},
"{ \mathclap{\widehat{\times}} }"{description, pos=.0}
]
&
G \mathrm{Orbt}
\ar[
l,
shift left=7pt,
hook',
"{ i }"{below}
]
\ar[
l,
phantom,
"{\scalebox{.6}{$\bot$}}"{description}
]
\end{tikzcd}
between the [[slice (infinity,1)-category|slice]] of the [[global orbit category]] over the object corresponding to $G$, and the $G$-[[orbit category]]. The adjoint quadruple of [[(infinity,1)-functors|$\infty$-functors]] between [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]] induced by this via Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} exhibits slices of [[global homotopy theory]] as being [[cohesive (infinity,1)-topos|cohesive]] over $G$-[[equivariant homotopy theory]]
\begin{tikzcd}
\phantom{AAAAAAA}
\overset{
\mathclap{
\raisebox{8pt}{
\tiny
\color{blue}
\begin{tabular}{c}
global equivariant
\\
homotopy theory sliced
\\
over $G$-orbi-singularity
\\
\phantom{A}
\end{tabular}
}\;\;\;\;
}
}{
\;\;(\mathrm{Glo} \mathrm{Grpd}_\infty)_{/\prec G}\;\;
}
\ar[
rr,
"{ \mathrm{FixLoc} }"{description}
]
\ar[
rr,
shift left=40pt,
"{ }"{description, pos=.35},
"\mathclap{\times}"{description, pos=0}
]
&&
\overset{
\mathclap{
\;\;
\raisebox{3pt}{
\tiny
\color{blue}
\begin{tabular}{c}
$G$-equivariant
\\
homotopy theory
\\
\phantom{A}
\end{tabular}
}
}
}{
\;\; G \mathrm{Grpd}_\infty \;\;
}
\phantom{AAAAAAA}
\ar[
ll,
hook',
shift right=20pt,
"{ \mathrm{OrbSp} }"{description}
]
\ar[
ll,
hook',
shift right=-20pt,
"{ }"{description, pos=.35}
]
\ar[
ll,
phantom,
shift right=10pt,
"{\scalebox{.5}{$\bot$}}"
]
\ar[
ll,
phantom,
shift right=30pt,
"{\scalebox{.5}{$\bot$}}"
]
\ar[
ll,
phantom,
shift right=-10pt,
"{\scalebox{.5}{$\bot$}}"
]
\end{tikzcd}
For more on this see at *[[cohesion of global- over G-equivariant homotopy theory]]*.
\end{example}
This example is a special case of the following general class:
\begin{example}\label{nTruncationsOfFullSubcategoriesOfInfinityToposes}
For $n \,\in\, \mathbb{N}$ and
for $\mathcal{S} \xhookrightarrow{\;\;} \mathbf{H} $ any [[small (infinity,1)-category|small]] [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of an [[(infinity,1)-topos|$\infty$-topos]] which is
1. has all [[finite products|finite]] [[homotopy products]], these being computed in $\mathbf{H}$,
1. is closed under [[n-truncated object of an (infinity,1)-category|$n$-truncation]]
then the $n$-truncation reflection restricts
\begin{tikzcd}
\mathcal{S}
\ar[
r,
shift left=7pt,
"{ \tau_n }"{above},
"{ \mathclap{\times} }"{description, pos=.0}
]
&
\mathcal{S}_{\tau_n}
\ar[
l,
shift left=7pt,
hook',
"{ i }"{below}
]
\ar[
l,
phantom,
"{\scalebox{.6}{$\bot$}}"{description}
]
\end{tikzcd}
and it preserves finite products (by [this Prop.](n-truncated+object+of+an+infinity1-category#nTruncationInToposPreservesFiniteProducts)).
Moreover, this adjunction [[corestriction|(co)]][[restriction|restricts]] to [[connected objects]] $X \,\in\, \mathcal{S}_{cn} \xhookrightarrow{\;} \mathcal{S}$ (i.e. those for which $\mathcal{S}(X,-)$ preserves coproducts):
\begin{tikzcd}
\mathcal{S}_{\mathrm{cn}}
\ar[
r,
shift left=7pt,
"{ \tau_n }"{above},
"{ \mathclap{\widehat{\times}} }"{description, pos=.0}
]
&
\mathcal{S}_{\mathrm{cn},\tau_n}
\ar[
l,
shift left=7pt,
hook',
"{ i }"{below}
]
\ar[
l,
phantom,
"{\scalebox{.6}{$\bot$}}"{description}
]
\end{tikzcd}
If all objects of $\mathcal{S}$ are coproducts of connected ones then coroduct-preserving extension of the (co)restriced left adjoint is the original left adjoint and hence preserves finite products.
Hence the Kan extensions according to Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple} exhibit the [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-category of $\infty$-presheaves]] $PSh_\infty(\mathcal{S})$ as being [[cohesive (infinity,1)-topos|cohesive]] over $PSh_\infty(\mathcal{S}_{\tau_n})$.
\end{example}
\linebreak
We now spell out the proof of the lemmas used in the proof of Prop. \ref{KanExtensionOfFiniteProductPreservingReflectionIsCohesiveAdjointQuadruple}.
\begin{lemma}\label{LeftKanExtensionIsOriginalFunctorOnRepresentables}
Given a functor $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ between [[small categories]], its [[left Kan extension]]
$f_! \;\colon\; PSh(\mathcal{S}_1) \xrightarrow{\;\;} PSh(\mathcal{S}_1)$ restricts to $f$ on [[representable functor|representables]], in that for $s_1 \,\in\, \mathcal{S}_1$ we have a [[natural isomorphism]]
$$
f_!\big( y(s_1) \big)
\;\simeq\;
y\big(
f(s_1)
\big)
\,.
$$
\end{lemma}
\begin{proof}
For $X \,\in\, PSh(\mathcal{S}_2)$
we have the following sequence of [[natural isomorphism]]:
$$
\begin{aligned}
PSh(\mathcal{S}_2)
\left(
f_!
\left(
y(s_1)
\right)
,\,
X
\right)
& \;\simeq\;
PSh(\mathcal{S}_1)
\left(
y(s_1)
,\,
f^\ast(X)
\right)
\\
&
\;\simeq\;
X\left( f(s_1) \right)
\\
&
\;\simeq\;
PSh(\mathcal{S}_2)
\left(
y\left(f(s_1)\right)
,\,
X
\right)
\,.
\end{aligned}
$$
The first line is the defining adjointness of $f_!$, the second line the [[Yoneda lemma]] over $\mathcal{S}_1$ and the definition of $f^\ast$, while the last line is the Yoneda lemma over $\mathcal{S}_2$.
Since the composite of these isomorphisms is natural, the [[Yoneda lemma]] over $PSh(\mathcal{S}_2)^{op}$ (which is [[large category|large]] but [[locally small category|locally small]], so that the lemma does apply) implies the claim.
\end{proof}
\begin{lemma}\label{LeftKanExtensionOfBinProductPreservingIsBinProductPres}
Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be [[small categories]] with [[binary products]] and $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ a [[functor]] which [[preserved limit|preserves]] these, in that for $s, s' \,\in\, \mathcal{S}_1$ there is a [[natural isomorphism]] $f(s \times s') \,\simeq\, f(X_1) \times f(X_2)$. Then also the [[left Kan extension]] $f_!$ [[preserved limit|preserves]] [[binary products]], in that for $X, X' \,\in\, PSh(\mathcal{S}_1)$ there is a [[natural isomorphism]]
$$
f_!(X \times X')
\;\simeq\;
f_!(X) \times f_!(X')
\,.
$$
\end{lemma}
\begin{proof}
This is the composite of the following sequence of [[natural isomorphisms]]:
$$
\begin{aligned}
f_!(X \times X')
& \;\simeq\;
f_!
\Big(\!
\big(
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\,
y(s)
\big)
\times
\big(
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
y(s')
\big)
\!\Big)
\\
& \;\simeq\;
f_!
\Big(
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
\left(
y(s)
\times
y(s')
\right)
\!\!\Big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\big(
y(s)
\times
y(s')
\big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\big(
y(s \times s')
\big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
y\big( f(s \times s') \big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
y\big( f(s) \times f(s') \big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
y\big( f(s) \big)
\times
y\big(f(s') \big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\left(
y(s)
\right)
\times
f_!
\left(
y(s')
\right)
\\
& \;\simeq\;
\Big(\,
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\,
f_!
\left(
y(s)
\right)
\!\!\Big)
\times
\Big(\,
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\left(
y(s')
\right)
\!\!\Big).
\end{aligned}
$$
Here
* the first line is the [[co-Yoneda lemma]], expressing the [[presheaves]] as [[colimits]] of [[representable functor|representables]].
* The second line uses that in a [[topos]] (like [[Set]] or whatever the [[base topos]] may be) [[universal colimits|colimits are pullback-stable]] and in particular distribute over [[products]].
* The third line uses that $f_!$ is a [[left adjoint]] and that [[left adjoints preserve colimits]].
* Then inside the two colimits we use
1. that the [[Yoneda embedding]] preserves [[limits]], in particular [[products]],
1. Lem. \ref{LeftKanExtensionIsOriginalFunctorOnRepresentables}, to evaluate $f_!$ on representables as $f$,
1. the assumption that $f$ preserves products.
* The last step is the first two steps in reverse.
\end{proof}
The following
Lem. \ref{LeftKanExtensionOfBinProductPreservingOnCoprodComplIsBinProductPres}
is an immediate variant of
Lem. \ref{LeftKanExtensionOfBinProductPreservingIsBinProductPres}
obtained by relaxing the assumptions slightly to a form that is often still readily checked:
\begin{lemma}
\label{LeftKanExtensionOfBinProductPreservingOnCoprodComplIsBinProductPres}
Let
\begin{tikzcd}
\mathcal{S}_1
\ar[
rr,
"{f}"{above},
"{\mathclap{\widehat{\times}}}"{description, pos=0}
]
&&
\mathcal{S}_2
\end{tikzcd}
be a [[functor]] between [[small categories]] such that
1. their [[free coproduct completions]] $PSh_{\sqcup}(-)$ have [[binary products]],
1. the unique coproduct-[[preserved colimit|preserving]] extension $f_!$ of $f$ to these completions [[preserved limit|preserves]] [[binary products]], in that for $s, s' \,\in, \mathcal{S}_1$ there is a [[natural isomorphism]]:
$$
f_!\big( y(s) \times y(s') \big)
\,\simeq\,
f_!\big(y(s)\big) \times f_!\big( y(s') \big)
$$
Then also the left [[Kan extension]] $f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2)$ to the full [[categories of presheaves]] preserves products.
\end{lemma}
\begin{proof}
The proof starts and ends as the proof of Prop. \ref{LeftKanExtensionOfBinProductPreservingIsBinProductPres},
but the main step in between is now more immediate, as it just needs to invoke the assumption:
$$
\begin{aligned}
f_!(X \times X')
& \;\simeq\;
f_!
\Big(\!
\big(
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\,
y(s)
\big)
\times
\big(
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
y(s')
\big)
\!\Big)
\\
& \;\simeq\;
f_!
\Big(
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
\left(
y(s)
\times
y(s')
\right)
\!\!\Big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\big(
y(s)
\times
y(s')
\big)
\\
& \;\simeq\;
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\;
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\left(
y(s)
\right)
\times
f_!
\left(
y(s')
\right)
\\
& \;\simeq\;
\Big(\,
\underset{
\underset{ s \to X}{\longrightarrow}
}{\lim}
\,
f_!
\left(
y(s)
\right)
\!\!\Big)
\times
\Big(\,
\underset{
\underset{ s' \to X}{\longrightarrow}
}{\lim}
\,
f_!
\left(
y(s')
\right)
\!\!\Big).
\end{aligned}
$$
\end{proof}
In conclusion so far:
\begin{prop}\label{LeftKanExtensionOfFinProdPreservingIsFinProdPreserving}
If a pair of [[small categories]] $\mathcal{S}_1$, $\mathcal{S}_2$ has [[finite products]] and a [[functor]] $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ [[preserved limit|preserves]] these, then so does its [[left Kan extension]] $f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2)$.
More generally this is the case if the [[free coproduct completions]] $PSh_{\sqcup}(\mathcal{S}_i)$ have [[finite products]] and the unique coproduct-preserving extension preserves these.
\end{prop}
\begin{proof}
We need to show that $f_!$ preserves (1) the [[terminal object]] and (2) [[binary products]]. With the given assumption on $f$, the first follows with Lem. \ref{LeftKanExtensionIsOriginalFunctorOnRepresentables} while the second follows with Lem. \ref{LeftKanExtensionOfBinProductPreservingIsBinProductPres} or Lem. \ref{LeftKanExtensionOfBinProductPreservingOnCoprodComplIsBinProductPres}, respectively.
\end{proof}
\begin{proposition}\label{LeftKanExtensionOfFullyFaithfulIsFullyFaithful}
The left Kan extension $f_!$ of a [[fully faithful functor]] $f$ between [[small categories]] is itself fully faithful:
\begin{tikzcd}
\mathcal{S}_1
&
\mathcal{S}_2
\ar[
l,
hook',
"{i}"{above}
]
&
\;\;\;\;
\Rightarrow
\;\;\;\;
&
\mathrm{PSh}(\mathcal{S}_1)
&
\mathrm{PSh}(\mathcal{S}_2)
\ar[
l,
hook',
"{i_!}"{above}
]
\end{tikzcd}
\end{proposition}
\begin{proof}
We need to show for $X, X' \,\in\, PSh(\mathcal{S}_2)$ the morphism
$$
PSh(\mathcal{S}_2)
(X,\, X')
\xrightarrow{ \; (i_!)_{X, X'} \; }
PSh(\mathcal{S}_`)
\big(
i_!(X)
,\,
i_!( X')
\big)
$$
is an equivalence. But since $i_!$ is a left adjoint and since $X$ and $X'$ are colimits of representables, this morphism is the unique one which reduces to
\[
\label{ComponentIsomorphismOfAFullyFaithfulFunctor}
\mathcal{S}_2
(s,\, s')
\xrightarrow{ \; i_{s, s'} \; }
\mathcal{S}_1
\big(
i(s)
,\,
i( s')
\big)
\]
on representables, where this is an isomorphism by the assumption that $i$ is fully faithful. If follows that $(i_!)_{X, X'}$, is the compostite of the following isomorphisms, and hence an isomorphism:
$$
\begin{aligned}
&
PSh(\mathcal{S}_1)
\big(
i_!(X_1)
,\,
i_!(X_2)
\big)
\\
&
\;\simeq\;
PSh(\mathcal{S}_1)
\Big(
i_!
\big(
\underset{
\underset{s \to X}{\longrightarrow}
}{\lim}
y(s)
\big)
,\,
i_!
\big(
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
y(s')
\big)
\Big)
\\
&
\;\simeq\;
PSh(\mathcal{S}_1)
\Big(
\underset{
\underset{s \to X}{\longrightarrow}
}{\lim}
i_!
\big(
y(s)
\big)
,\,
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
i_!
\big(
y(s')
\big)
\Big)
\\
&
\;\simeq\;
PSh(\mathcal{S}_1)
\Big(
\underset{
\underset{s \to X}{\longrightarrow}
}{\lim}
y
\big(
i
(s)
\big)
,\,
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
y
\big(
i(s')
\big)
\Big)
\\
&
\;\simeq\;
\underset{
}{\lim}
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
PSh(\mathcal{S}_1)
\Big(
y
\big(
i
(s)
\big)
,\,
y
\big(
i(s')
\big)
\Big)
\\
&
\;\simeq\;
\underset{
\underset{s \to X}{\longleftarrow}
}{\lim}
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
\mathcal{S}_1
\big(
i(s)
,\,
i(s')
\big)
\\
&
\;\simeq\;
\underset{
\underset{s \to X}{\longleftarrow}
}{\lim}
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
\mathcal{S}_2
\big(
s
,\,
s'
\big)
\\
&
\;\simeq\;
\underset{
\underset{s \to X}{\longleftarrow}
}{\lim}
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
PSh(\mathcal{S}_2)
\big(
y(s)
,\,
y(s')
\big)
\\
&
\;\simeq\;
PSh(\mathcal{S}_2)
\Big(
\underset{
\underset{s \to X}{\longrightarrow}
}{\lim}
y(s)
,\,
\underset{
\underset{s' \to X'}{\longrightarrow}
}{\lim}
y(s')
\Big)
\\
&
\;\simeq\;
PSh(\mathcal{S}_2)
\big(
X
,\,
X'
\big)
\end{aligned}
$$
Here the first and last steps are the [[co-Yoneda lemma]] and the preservation of its colimits as in the proofs before. In the middle steps we are using Lem. \ref{LeftKanExtensionIsOriginalFunctorOnRepresentables} to evaluate $i_!$ on representables and then (eq:ComponentIsomorphismOfAFullyFaithfulFunctor) inside the (co)limits.
\end{proof}
## Related concepts
* [[adjunction]], [[adjoint functor]]
* [[adjoint triple]], [[adjoint string]]
## References
The above prop. \ref{TheCanonicalMorphisms} is from:
* {#Johnstone11} [[Peter Johnstone]], _Remarks on punctual local connectedness_, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63. ([tac](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))
[[!redirects adjoint quadruples]] |
adjoint representation | https://ncatlab.org/nlab/source/adjoint+representation |
Every [[Lie group]] $G$ has a canonical [[representation]] on the [[vector space]] underlying its [[Lie algebra]] $\mathfrak{g}$, given by the [[derivative]] of its [[adjoint action]] at the neutral element. This is called the _adjoint representation_.
The [[associated bundles]] via the adjoint representation are called [[adjoint bundles]]. |
adjoint string | https://ncatlab.org/nlab/source/adjoint+string | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Adjoint strings
* table of contents
{: toc}
## Definition
In [[category theory]],
an **adjoint string of length $n$**, **adjoint chain of length $n$**, **adjoint sequence of length $n$**, or an **adjoint $n$-tuple**, is a sequence of $(n-1)$ [[adjunctions]] between $n$ [[functors]] (or more generally [[morphisms]] in a [[2-category]]):
$$f_1 \dashv f_2 \dashv \cdots \dashv f_n $$
## Special cases
* An adjoint $2$-tuple is just an ordinary [[adjunction]].
* An adjoint $3$-tuple is an [[adjoint triple]].
* An adjoint $4$-tuple is an [[adjoint quadruple]].
## Examples
1. There is an adjoint $5$-tuple between $[Set^{op}, Set]$ and $Set$. Indeed, given a [[locally small category]] $B$, and the [[Yoneda embedding]], $y: B \to [B^{op}, Set]$, then $y$ being the rightmost functor of an adjoint $5$-tuple entails that $B$ is equivalent to [[Set]]; see [Rosebrugh-Wood](#RWSets).
1. For any category $C$, there is a functor $ids: C\to Ar(C)$ from $C$ to its [[arrow category]] that assigns the identity morphism of each object. This functor always has both a left and a right adjoint which assign the codomain and domain of an arrow respectively; thus we have an adjoint triple $cod \dashv ids \dashv dom$. If $C$ has an initial object $0$, then $cod$ has a further left adjoint $I$ assigning to each object $x$ the morphism $0\to x$; and dually if $C$ has a terminal object $1$ then $dom$ has a further right adjoint $T$ assigning to $x$ the morphism $x\to 1$. Thus if $C$ has an initial and terminal object, we have an adjoint $5$-tuple.
1. Continuing from the last example, if $C$ is moreover a [[pointed category]] with [[pullbacks]] and [[pushouts]], then $I$ has a further left adjoint that constructs the [[cokernel]] of a morphism $x\to y$, i.e. the pushout of $y \leftarrow x \to 0$; and $T$ has a further right adjoint that constructs the [[kernel]] of a morphism $x \to y$, namely the pullback of $x\to y \leftarrow 0$. Thus we have an adjoint $7$-tuple. In fact, the existence of such an adjoint $7$-tuple characterizes pointed categories among categories with finite limits and colimits.
1. The previous two examples apply also to [[derivators]], and the extension of the analogous adjoint $5$-tuple to a $7$-tuple again characterizes the [[pointed derivators]]. Moreover, the [[stable derivators]] are characterized by the extension of this $7$-tuple to a doubly-infinite adjoint string with period 6 ([GrothShul17](#GrothShul17)).
1. Let $[n]$ denote the [[totally ordered]] $(n+1)$-element set, regarded as a category. For each positive integer $n$, we have $n+1$ order-preserving injections from $[n-1]$ to $[n]$, and $n$ order-preserving surjections from $[n]$ to $[n-1]$. Regarded as functors, these injections and surjections interleave to form an adjoint chain of length $2n + 1$. These categories, functors, and adjunctions form the [[simplex category]] [[simplex category#As2Categories|regarded as a locally posetal 2-category]]; see below.
1. Let $C$ be a category with a [[terminal object]] but no [[initial object]]. Then there are functors
$$
\array{
\delta_i \colon [n+1,C] \to [n,C] & 0\leq i \leq n;
\\
\sigma_i\colon [n,C] \to [n+1,C] & 0\leq i \leq n
}
$$
such that
$$
\delta_0 \dashv \sigma_0 \dashv \cdots \dashv \delta_n \dashv \sigma_n
$$
is a maximal string of adjoint functors (all but $\sigma_n$ are obtained by applying $[-, C]$ to the simplex category example, and $\sigma_n$ exploits the presence of the terminal object of $C$).
1. Generalizing the simplex category example: if $P$ is a [[lax idempotent monad]] with unit $u: 1 \to P$ and multiplication $m: P P \to P$ (so that $m \dashv u P$), then there is an adjoint string
$$P^{n-1} m \dashv P^{n-1} u P \dashv P^{n-2}m P \dashv \ldots \dashv m P^{n-1} \dashv u P^n$$
of length $2 n + 1$, back and forth between $P^{n+1}$ and $P^n$. The example of $[n]$ and $[n+1]$ above is based on the fact that the [[simplex category]] $\Delta$, regarded as a locally posetal [[bicategory]], is the [[walking structure|walking]] lax idempotent monoid.
1. Given an [[ambidextrous adjunction]] (and in particular a [[self-adjoint functor]]), $F \dashv G$ and $G \dashv F$, we of course get an infinite adjoint string
$$\ldots \dashv F \dashv G \dashv F \dashv \ldots$$
of period 2.
## References
A study of adjoint strings, in particular showing that cyclic chains of any length, and adjoint chains of any length exist, may be found in:
* {#Booth72} [[Peter I. Booth]], *Sequences of adjoint functors*, Archiv der Mathematik **23** (1972) 489-493 [[doi:10.1007/BF01304920](https://doi.org/10.1007/BF01304920)]
See also:
* {#Hoffmann80} [[Rudolf-E. Hoffmann]], _Sequences of adjoints for Ens-valued functors_, manuscripta mathematica **32** (1980) [[EuDML;154724](https://eudml.org/doc/154724)]
Characterizing the [[category of sets]] as that whose [[Yoneda embedding]] extends to the [[left adjoint|left]] to an [[adjoint quintuple]] of [[adjoint functors]]:
* {#RWSets} [[Robert Rosebrugh]], [[Richard J. Wood]], *An adjoint characterization of the category of sets*. PAMS **122** 2 (1994) 409-413 [[jstor:2161031](https://www.jstor.org/stable/2161031)]
On [[adjoint quadruples]] with a [[fully faithful functor|fully faithful]] [[right adjoint]]:
* [[Bob Rosebrugh]], [[Richard J. Wood]], *Distributive Adjoint Strings*, Theory and Applications of Categories, **1** 6 (1995) 119-145 [[tac:1-06](http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html)]
See also:
* {#GrothShul17} [[Moritz Groth]], [[Mike Shulman]], _Generalized stability for abstract homotopy theories_, [arXiv:1704.08084](https://arxiv.org/abs/1704.08084).
[[!redirects adjoint string]]
[[!redirects adjoint strings]]
[[!redirects adjoint chain]]
[[!redirects adjoint chains]]
[[!redirects adjoint sequence]]
[[!redirects adjoint sequences]]
[[!redirects adjoint n-tuple]]
[[!redirects adjoint n-tuples]]
[[!redirects distributive adjoint string]]
[[!redirects distributive adjoint strings]]
[[!redirects adjoint quintuple]]
[[!redirects adjoint quintuples]]
|
adjoint triangle theorem | https://ncatlab.org/nlab/source/adjoint+triangle+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
# The adjoint triangle theorem
* automatic table of contents
{: toc}
## Idea
The **adjoint triangle theorem** in [[category theory]] gives conditions under which, given a [[pair]] of [[functors]] and an [[adjoint functor]], further adjoints exist.
Depending on the specific assumptions, the theorem has several variants. The following gives the most common formulation going back to [Dubuc (1968)](#dubuc68).
## Statement ##
+-- {: .num_theorem #ATT}
###### Theorem
Suppose that $U \colon B\to C$ is a [[functor]] which has a [[left adjoint]] $F \,\colon\, C\to B$ with the property that the diagram
$$
F U F U
\;
\underoverset
{\epsilon F U}
{F U \epsilon}
{\rightrightarrows}
\;
F U \xrightarrow{\epsilon} 1_B
$$
is a pointwise [[coequalizer]] (i.e. $U$ is of [[descent type]]). Then for $A$ a category with [[reflexive coequalizer|coequalizers of reflexive pairs]], a functor $R \colon A\to B$ has a [[left adjoint]] if and only if the composite $U R$ does.
=--
+-- {: .proof}
###### Proof
The direction "only if" is obvious since adjunctions compose. For "if", let $F'$ be a left adjoint of $U R$, and define $L:B\to A$ to be the pointwise coequalizer of
$$ F' U F U \xrightarrow{F' U \epsilon} F' U $$
and
$$
F' U F U \xrightarrow{F' U \theta U}
F' U R F' U \xrightarrow{\epsilon' F' U}
F' U
$$
where $\theta:F \to R F'$ is the [[mate]] of the equality $U R = U R$ under the adjunctions $F\dashv U$ and $F'\dashv U R$. One then verifies that this works.
=--
+-- {: .num_remark #Monadic}
###### Remark
The hypotheses on $U$ are satisfied whenever it is [[monadic functor|monadic]].
=--
+-- {: .num_remark #RegMono}
###### Remark
In fact, it suffices to assume that each counit $\epsilon : F U b \to b$ is a [[regular epimorphism]], rather than it is the coequalizer of a specific given pair of maps. See [(Street-Verity), Lemma 2.1](#StreetVerity).
=--
## Ramifications
Similarly, the [[adjoint lifting theorem]] states conditions on a square of functors in order to ensure the existence of certain adjoints. Since a triangle can be viewed as a square with 'two sides composed', it is possible to deduce the adjoint lifting theorem from the adjoint triangle theorem as a corollary.
It is also possible to derive the [[monadicity theorem]] from the adjoint triangle theorem [Dubuc (1968)](#dubuc68).
## Related entries
* [[adjoint functor theorem]]
* [[adjoint lifting theorem]]
## References ##
* [[Michael Barr]], [[Charles Wells]], section 3.7, pp.131 in: *[[Toposes, Triples, and Theories]]*, Springer (1985), Reprints in Theories and Applications of Categories **12** (2005) 1-287 [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]
* {#dubuc68} [[Eduardo Dubuc]], _Adjoint triangles_, pp.69-81 in LNM **61** Springer Heidelberg 1968. [[doi:10.1007/BFb0077118](https://doi.org/10.1007/BFb0077118)]
* I. B. Im, [[Max Kelly|G. M. Kelly]], _Adjoint-Triangle Theorems for Conservative Functors_, Bulletin of the Australian Mathematical Society **36** 1 (1987) pp.133-136. [[doi:10.1017/S000497270002637X](https://doi.org/10.1017/S000497270002637X)]
* [[John Power]], _A unified approach to the lifting of adjoints_, Cahiers de Topologie et Géométrie Différentielle Catégoriques **29** 1 (1988) 67-77. ([numdam](http://www.numdam.org/item/CTGDC_1988__29_1_67_0))
* {#StreetVerity} [[Ross Street]], [[Dominic Verity]], _The comprehensive factorization and torsors_, Theory and Applications of Categories **23** 3 (2010) 42-75. ([TAC](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html))
* [[Walter Tholen]], _Adjungierte Dreiecke, Colimites und Kan-Erweiterungen_, Mathematische Annalen **217** (1975) pp.121-129. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002311682))
Generalizations of the adjoint triangle theorem to [[2-categories]] are considered in
* [[Fernando Lucatelli Nunes]], _On biadjoint triangles_, Theory and Applications of Categories **31** 9 (2016) 217-256. [TAC](http://tac.mta.ca/tac/volumes/31/9/31-09abs.html)
* [[Fernando Lucatelli Nunes]], _On lifting of biadjoints and lax algebras_, General Algebraic Structures with Applications **9** 1 (2018) 29-58. [[doi:10.29252/CGASA.9.1.29](https://doi.org/10.29252/CGASA.9.1.29), [arXiv:1607.03087](https://arxiv.org/abs/1607.03087)]
[[!redirects adjoint triangle]]
[[!redirects adjoint triangles]]
|
adjoint triple | https://ncatlab.org/nlab/source/adjoint+triple |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
> This entry is about the notion of _adjoint triple_ involving three functors. This is not to be confused with the notion of [[adjoint monads]], which were also sometimes called adjoint triples, with "triple" then being a synonym for "monad". However, an adjoint triple in the sense here does induce an [[adjoint monad]]!
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #Defn}
###### Definition
An __adjoint triple__ (of [[functors]] between [[categories]] or generally of [[1-morphisms]] in a [[2-category]])
$$
(F \dashv G \dashv H) \colon C \to D
$$
is a [[triple]] of [[functors]]/morphisms $F,H \colon C \to D$ and $G \colon D \to C$ together with [[adjunction]] data $F\dashv G$ and $G\dashv H$. That is, it is an [[adjoint string]] of length 3.
=--
+-- {: .num_prop #AsAdjunctionOfAdjunctions}
###### Proposition
An adjoint triple $(F\dashv G\dashv H)$, def. \ref{Defn} is equivalently an [[adjoint pair]] in the 2-category whose morphisms are adjoint pairs in the original 2-category, hence an adjunction of adjunctions
$$
(F \dashv G) \dashv (G \dashv H)
\,.
$$
=--
This fact plays an important role in [Licata--Shulman, 5.1](#LicataShulman). Relatedly, it also appears in the characterization of certain kinds of [[geometric morphism]] (e.g. the [[local geometric morphism|local]] ones) in terms of adjunctions in the 2-category [[Topos]].
It may be suggestive to denote this like so
$$
\array{
F &\dashv& G
\\
\bot && \bot
\\
G &\dashv& H
}
$$
such that the two adjoint pairs appear horizontally, while the second order adjunction between them runs vertically.
## Properties
{#Properties}
+-- {: .num_note #GIsBicontinuous}
###### Note
The two adjunctions imply of course that $G$ preserves all [[limit]]s and [[colimit]]s that exist in $D$.
=--
+-- {: .num_note #AdjointPairFromAdjointTriple}
###### Note
Every adjoint triple
$$
(F \dashv G \dashv H) \colon C \to D
$$
gives rise to an [[adjunction|adjoint pair]]
$$
(G F \dashv G H) \colon C \to C
$$
consisting of the [[monad]] $G F$ [[left adjoint]] to the [[comonad]] $G H$ on $C$;
as well as to an adjoint pair
$$
(F G \dashv H G) \colon D \to D
$$
consisting of the comonad $F G$ left adjoint to the monad $H G$ on $D$.
=--
See *[[adjoint monad]]* for more.
In general there is a duality (an [[dual equivalence|antiequivalence of categories]]) between the categories of monads having right adjoints and of comonads having left adjoints.
Note also (see [there](adjoint+monad#IsomorphismOfEMCategories)) that the [[algebra over a monad|algebras]] over a left-[[adjoint monad]] are identified with the [[coalgebra over a comonad|coalgebras]] for its right-adjoint comonad (duel to [Eilenberg & Moore 1965](adjoint+monad#EilenbergMoore65), see eg. [MacLane & Moerdijk 1992, Theorems V.8.1 and V.8.2](#SGL)).
### Fully faithful adjoint triples
{#FullyFaithFulAdjointTriples}
+-- {: .num_prop #FullyFaithful}
###### Proposition
For an adjoint triple $F\dashv G\dashv H$ we have that $F$ is [[full and faithful functor|fully faithful]] if and only if $H$ is fully faithful.
=--
(In this case either [[adjoint pair]] is an [[idempotent adjunction]], see Prop. \ref{Idempotent} below.)
+-- {: .proof}
###### Proof
By a basic [[adjoint functor#FullyFaithfulAndInvertibleAdjoints|property]] of [[adjoint functors]] ([this Prop.](adjoint+functor#FullyFaithfulAndInvertibleAdjoints)), we have that
* the [[left adjoint]] $F$ being full and faithful is equivalent to the [[unit of an adjunction|unit]] $Id \to G F$ being a [[natural isomorphism]];
* the [[right adjoint]] $H$ being full and faithful is equivalent to the counit $G H \to Id$ being a [[natural isomorphism]].
Moreover, by Note \ref{AdjointPairFromAdjointTriple} and the fact that adjoints are unique up to isomorphism, we have that $G F$ is isomorphic to the identity precisely if $G H$ is.
Finally, by a standard fact about [[adjoint functors]] (see for instance ([Elephant, Lemma A1.1.1](#Elephant))), we have that $G H$ is naturally isomorphic to the identity precisely if it is so by the [[unit of an adjunction|counit]].
=--
The preceeding proposition is [[folklore]]; perhaps its earliest appearance in print is ([DT, Lemma 1.3](#DyckhoffTholen)). A slightly shorter proof is in ([KL, Prop. 2.3](#KellyLawvere)). Both proofs explicitly exhibit an inverse to the counit $G H \to Id$ or the unit $Id \to G F$ given an inverse to the other (which could be extracted by [[beta-reduction|beta-reducing]] the above, slightly more abstract argument). It also appears in ([SGL, Lemma 7.4.1](#SGL)).
In the situation of Proposition \ref{FullyFaithful}, we say that $F\dashv G \dashv H$ is a **fully faithful adjoint triple**. This is often the case when $D$ is a category of "spaces" structured over $C$, where $F$ and $H$ construct "discrete" and "codiscrete" spaces respectively.
For instance, if $G\colon D\to C$ is a [[topological concrete category]], then it has both a left and right adjoint which are fully faithful. Not every fully faithful adjoint triple is a topological concrete category (among other things, $G$ need not be [[faithful functor|faithful]]), but they do exhibit certain similar phenomena. In particular, we have the following.
+-- {: .num_prop #FinalLifts}
###### Proposition
Suppose $(F \dashv G \dashv H) \colon C \to D$ is an adjoint triple in which $F$ and $H$ are fully faithful, and suppose that $C$ is [[cocomplete category|cocomplete]]. Then $G$ admits [[final lift|final lifts]] for [[small category|small]] $G$-structured [[sinks]].
=--
+-- {: .proof}
###### Proof
Let $\{G(S_i) \to X\}$ be a small sink in $C$, and consider the diagram in $D$ consisting of all the $S_i$, all the counits $\varepsilon\colon F G(S_i) \to S_i$ (where $F$ is the left adjoint of $G$), and all the images $F G(S_i) \to F(X)$ of the morphisms making up the sink. The colimit of this diagram is preserved by $G$ (since it has a right adjoint as well). But the image of the diagram consists essentially of just the sink itself (since $F$ is fully faithful, $G(\varepsilon)$ is an isomorphism), and its colimit is $X$; hence the colimit of the original diagram is a lifting of $X$ to $D$ (up to isomorphism). It is easy to verify that this lifting has the correct universal property.
=--
Thus, we can talk about objects of $D$ having the [[weak structure]] or [[strong structure]] induced by any small collection of maps.
+-- {: .num_cor #Fibration}
###### Corollary
In the situation of Proposition \ref{FinalLifts}, $G$ is a ([[Street fibration|Street]]) [[Grothendieck fibration|opfibration]]. If it is also an [[isofibration]], then it is a Grothendieck opfibration.
=--
+-- {: .proof}
###### Proof
A final lift of a singleton sink is precisely an opcartesian arrow.
=--
Dually, of course, if $C$ is complete, then $G$ admits initial lifts for small $G$-structured cosinks and is a fibration.
In particular, the proposition and its corollary apply to a [[cohesive topos]], and (suitably categorified) to a [[cohesive (∞,1)-topos]].
### Idempotent adjoint triples
+-- {: .num_prop #Idempotent}
###### Proposition
For an adjoint triple $F\dashv G\dashv H$, the adjunction $F\dashv G$ is an [[idempotent adjunction]] if and only if the adjunction $G\dashv H$ is so.
=--
+-- {: .proof}
###### Proof
The monad $G F$ is left adjoint to the comonad $H G$, with the structure maps being [[mates]]. Therefore, by a standard fact, the category of $G F$-algebras and the category of $H G$-coalgebras are isomorphic over their common base. However, $F\dashv G$ is idempotent precisely when $G F$ is an [[idempotent monad]], hence precisely when the forgetful functor of the category of $G F$-algebras is fully faithful, and dually for $G\dashv H$. Since the categories of algebras are isomorphic respecting their forgetful functors, one forgetful functor is fully faithful if and only if the other is.
=--
A special case of this situation is Prop. \ref{FullyFaithful} above.
## Examples
### Special cases
* If one of the two [[adjoint pairs]] induced from an adjoint triple involving identities, then the other exhibits an _[[adjoint cylinder]]_ / _[[unity of opposites]]_.
* An adjoint triple $F\dashv G\dashv H$ is **Frobenius** if $F$ is naturally isomorphic to $H$. See [[Frobenius functor]].
* An *[[affine morphism]]* is an adjoint triple of functors in which the middle term is [[conservative functor|conservative]]. For example, any [[affine morphism of schemes]] induce an affine triples of functors among the categories of [[quasicoherent module]]s.
* An adjoint triple of functors among $A_\infty$- or [[triangulated functor]]s with certain additional structure is called **spherical** . See e.g. ([Anno](#Anno)). The main examples come from [[Serre functor]]s in a [[Calabi-Yau category]] context.
* A context of [[six operations]] $(f_! \dashv f^!)$, $(f^\ast \dashv f_\ast)$ induces an adjoint triple when either $f^! \simeq f^\ast$ or $f_! = f_\ast$. This is called a _[[Wirthmüller context]]_ or a _[[Grothendieck context]]_, respectively.
### Specific examples
* Given any [[ring]] [[homomorphism]] $f^\circ: R\to S$ (in commutative case dual to an [[affine morphism]] $f: Spec S\to Spec R$ of [[affine schemes]]), there is an adjoint triple $f^!\dashv f_*\dashv f^*$ where $f^*: {}_R Mod\to {}_S Mod$ is an [[extension of scalars]], $f_*: {}_S Mod\to {}_R Mod$ the restriction of scalars and $f^! : M\mapsto Hom_R ({}_R S, {}_R M)$ its [[coextension of scalars|right adjoint]]. This triple is affine in the above sense.
* If $T$ is a [[lax-idempotent 2-monad]], then a $T$-algebra $A$ has an adjunction $a : T A \rightleftarrows A : \eta_A$. If this extends to an adjoint triple with a further left adjoint to $a$, then $A$ is called a [[continuous algebra]].
## Related concepts
* [[adjoint quadruple]], [[adjoint string]]
* [[cohesive topos]]
* [[ambidextrous adjunction]]
* [[affine morphism]], [[affine localization]]
* [[Quillen adjoint triple]]
## References
Some remarks on adjoint triples are in
* {#Johnstone} [[Peter Johnstone]], _Remarks on punctual local connectedness_, [tac/25-03](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html).
The [[modal type theory]] of adjoint triples is discussed in
* {#LicataShulman} [[Dan Licata]] and [[Mike Shulman]], _Adjoint logic with a 2-category of modes_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf)).
On spherical triples see
* {#Anno} Rina Anno, _Spherical functors_, [arxiv/0711.4409](https://arxiv.org/abs/0711.4409).
Generalities are in
* {#Elephant} [[Peter Johnstone]], _[[Sketches of an Elephant]]_.
Proofs of the folklore Proposition \ref{FullyFaithful} can be found in
* {#DyckhoffTholen} Roy Dyckhoff and [[Walter Tholen]], "Exponentiable morphisms, partial products, and pullback complements", JPAA 49 (1987), 103--116.
* {#KellyLawvere} [[G.M. Kelly]] and [[F.W. Lawvere]], "On the complete lattice of essential localizations", Bulletin de la Société Mathématique de Belgique, Série A, v. 41 no 2 (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]
* {#SGL} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic|Sheaves in Geometry and Logic --- A First Introduction to Topos Theory]]_, Springer (1992), [[doi:10.1007/978-1-4612-0927-0](https://dx.doi.org/10.1007/978-1-4612-0927-0)]
Several lemmas concerning adjoint pairs and adjoint triples are included in
* [[Alexander Rosenberg]], _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125, [doi](https://dx.doi.org/10.1023/A:1000479824211)
together with geometric consequences. Note a somewhat nonstandard usage of terminology continuous functor (also flatness in the paper includes having right adjoint).
[[!redirects adjoint triples]]
[[!redirects fully faithful adjoint triple]]
[[!redirects fully faithful adjoint triples]]
[[!redirects adjunction of adjunctions]]
[[!redirects adjunction between adjunctions]] |
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|
Adjointness for 2-Categories | https://ncatlab.org/nlab/source/Adjointness+for+2-Categories |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
This entry is about the book
* [[John Gray]],
_Formal category theory: adjointness for $2$-categories_
Lecture Notes in Mathematics, Vol. 391.
Springer-Verlag, Berlin-New York, 1974. xii+282 pp.
[doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280)
on [[formal category theory]], which is [[category theory]] formulated "formally" via the [[2-category theory]] of the [[2-category]] [[Cat]] of all categories.
This is one of the most influential and comprehensive historical books in low-dimensional [[higher category theory]], following the spirit of:
* {#Law66} [[William Lawvere]], *The Category of Categories as a Foundation for Mathematics*, pp. 1-20 in: [[John W. Gray]], *Fibred and Cofibred Categories*, in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]
More recently, this approach is echoed in [Riehl & Verity 13](homotopy+2-category+of+infinity-categories#RiehlVerity13), where [[Cat]] is enhanced to the [[homotopy 2-category of (∞,1)-categories]] in order to provide [[2-category theory|2-category theoretic]] foundations for [[(∞,1)-category theory]]. At least for [[presentable (infinity,1)-categories|presentable $\infty$-categories]], this is also obtained as the [[2-localization]] of the 2-category of [[combinatorial model categories]] and [[left Quillen functors]] at the [[Quillen equivalences]]: see at *[[2Ho(CombModCat)]]]*.
The book was supposed to be the first part of a four volume work, but unfortunately later volumes/chapters never appeared. It has some parts of 2- and 3-category theory; including the treatment of the famous [[Gray tensor product]] on 2-Cat. See also [[Gray-category]].
Unfortunately, due to changes in terminology, the book may be difficult to read nowadays. Gray uses prefixes such as 'quasi,' 'iso,' and 'weak' to indicate various levels of weakness, but his choice of terminology is not entirely consistent, can be confusing, and is completely different from the standard modern terminology which uses 'lax,' 'oplax,' and 'pseudo' with (mostly) precise and consistent meanings.
The following is a list of some of the definitions given in the book,
along with their modern names and links to nLab entries.
* Section 2
* _2-category_ (see [[strict 2-category]])
* _2-functor/$Cat$-functor_ (i.e. [[strict 2-functor]])
* _Cat-natural transformation_ (see [[enriched functor category]])
* _Quasi-natural transformation_ ([[lax natural transformation]])
* [[modification|Modification]]
* [[2-comma category]]
* [[double category|Double category]]
* Section 3
* [[bicategory|Bicategory]]
* _Pseudo-functor_ ([[lax 2-functor]]), _homomorphic pseudo-functor_
([[pseudofunctor]]), _strict pseudo-functor_ (= strict 2-functor -- see [[enriched functor]])
* Section 4
* $Fun(A,B)$ and $Pseud(A,B)$ (see [[Gray tensor product]])
* Section 6
* _Adjoint morphisms_ in a 2-category ([[adjunction]])
* [[Kan extension]] in a 2-category
* Section 7
* _Transcendental quasi-adjunction_ ([[lax 2-adjunction]])
category: reference
[[!redirects Gray-adjointness-for-2-categories]]
|
Adjointness in Foundations | https://ncatlab.org/nlab/source/Adjointness+in+Foundations |
This page collects links related to the text
* [[William Lawvere]],
_Adjointness in Foundations_,
Dialectica 23 (1969), 281-296,
Reprints in Theory and Applications of Categories, No. 16 (2006), pp 1-16 (revised 2006-10-30)
([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html))
on [[foundations of mathematics]] in [[categorical logic]] and the role of [[adjoints]] in the formalization of [[quantifiers]] ([[base change]]). The main point is that the logical operations on [[propositions]]
[[existential quantifier]] $\dashv$ [[context extension]] $\dashv$ [[universal quantifier]]
constitute an [[adjoint triple]]. In [[type theory]] this is lifted from operations on propositions to operations on [[types]], where it becomes
[[dependent sum]] $\dashv$ [[context extension]] $\dashv$ [[dependent product]].
In [[topos theory]] and [[geometry]] this adjoint triple is often known as _[[base change]]_.
## Related articles
* [[William Lawvere]], _[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14. ([pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf))
* {#Lambek} [[Joachim Lambek]], _The Influence of [[Heraclitus]] on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1981)
## Related $n$Lab entries
* [[interactions of images and pre-images with unions and intersections]]
* [[adjoint modality]]
category: reference
[[!redirects Adjointness in Foundations]]
[[!redirects Adjointness in foundations]]
|
adjoints preserve (co-)limits | https://ncatlab.org/nlab/source/adjoints+preserve+%28co-%29limits |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
One of the basic facts of [[category theory]] is that [[left adjoint|left]]/[[right adjoint|right]] [[adjoint functors]] [[preserved limit|preserves]] [[colimits|co]]/[[limits]], respectively.
## Statement
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ and $\mathcal{D}$ be two [[categories]] and let
$$
(L \dashv R)
\;\colon\;
\mathcal{C}
\underoverset
{\underset{R}{\longrightarrow}}
{\overset{L}{\longleftarrow}}
{\bot}
\mathcal{D}
$$
be a pair of [[adjoint functors]] between them.
Then
1. If $X \colon \mathcal{I} \to \mathcal{C}$ is a [[diagram]] whose [[limit]] $\underset{\longleftarrow}{\lim}_{i} X_i$ exists in $\mathcal{C}$, then this limit is preserved by the [[right adjoint]] $R$ in that there is a [[natural isomorphism]]
$$
R
\left(
\underset{\longleftarrow}{\lim}_i \left(X_i\right)
\right)
\;\simeq\;
\underset{\longleftarrow}{\lim}_i \left( R(X_i) \right)
\,,
$$
where on the right we have the [[limit]] in $\mathcal{D}$ over the [[diagram]] $R \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{R}{\longrightarrow} \mathcal{D}$.
1. If $X \colon \mathcal{I} \to \mathcal{D}$ is a [[diagram]] whose [[colimit]] $\underset{\longrightarrow}{\lim}_{i} X_i$ exists in $\mathcal{D}$, then this colimit is preserved by the [[left adjoint]] $L$ in that there is a [[natural isomorphism]]
$$
L
\left(
\underset{\longrightarrow}{\lim}_i \left(X_i\right)
\right)
\;\simeq\;
\underset{\longrightarrow}{\lim}_i \left( L(X_i) \right)
\,,
$$
where on the right we have the [[colimit]] in $\mathcal{C}$ over the [[diagram]] $L \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{D} \overset{L}{\longrightarrow} \mathcal{C}$.
=--
+-- {: .proof}
###### Proof
We show the first statement, the proof of the second is [[formal dual|formally dual]].
We use the following facts
1. There is a [[natural isomorphism]], $Hom_{\mathcal{C}}(L(d),c) \simeq Hom_{\mathcal{D}}(d,R(c))$; this equivalently characterizes the fact that $(L \dashv R)$ is a pair of [[adjoint functors]];
1. ([[hom-functor preserves limits]]) The [[hom-functor]] sends colimits in the first argument and limits in the second argument to limits of [[hom-sets]]
$$
Hom\left( X, \underset{\longleftarrow}{\lim}_i X_i \right)
\simeq
\underset{\longleftarrow}{\lim}_i Hom\left(X,X_i\right)
$$
and
$$
Hom\left(\underset{\longrightarrow}{\lim}_i X_i, X\right)
\simeq
\underset{\longleftarrow}{\lim} \left(Hom\left(X_i,X\right) \right)
\,.
$$
Again, this is essentially by definition of [[limits]]/[[colimits]].
1. ([[Yoneda lemma]]) If for two objects $X$ and $Y$ in some [[category]] the [[hom-sets]] out of or into these objects (their [[representable functors]]) are [[naturally isomorphic]], then the two objects are isomorphic.
Now using the first two items, we obtain the following chain of [[natural isomorphisms]], for every [[object]] $Y \in \mathcal{D}$:
$$
\begin{aligned}
Hom_{\mathcal{D}}\left( Y, R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \right)
& \simeq
Hom_{\mathcal{C}}\left( L(Y), \underset{\longleftarrow}{\lim}_i X_i\right)
\\
& \simeq
\underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left(L\left(Y\right), X_i\right)\right)
\\
& \simeq
\underset{\longleftarrow}{\lim}_i \left(Hom_{\mathcal{D}}\left(Y, R\left(X_i\right)\right)\right)
\\
& \simeq
Hom_{\mathcal{D}}\left(Y, \underset{\longleftarrow}{\lim}_i \left(R\left(X_i\right) \right) \right)
\end{aligned}
\,.
$$
Hence the third item above, the [[Yoneda lemma]], implies the claim.
=--
## Related entries
* [[hom-functor preserves limits]]
* [[limits preserve limits]]
* [[limits of presheaves are computed objectwise]]
* [[interactions of images and pre-images with unions and intersections]]
* [[limits and colimits by example]]
[[!redirects left adjoints preserve colimits and right adjoints preserve limits]]
[[!redirects right adjoints preserve limits and left adjoints preserve colimits]]
[[!redirects left adjoints preserve colimits]]
[[!redirects right adjoints preserve limits]]
|
adjunct | https://ncatlab.org/nlab/source/adjunct |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A pair
$$
(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D
$$
of [[adjoint functor]]s between [[category|categories]] $C$ and $D$, is characterized by a [[natural isomorphism]]
$$
C(L X,Y) \cong D(X,R Y)
$$
of [[hom-set]]s for [[object]]s $X\in D$ and $Y\in C$. Two [[morphism]]s
$f:L X \to Y$ and $g : X \to R Y$ which correspond under this bijection are said to be **adjuncts** of each other. That is, $g$ is the (right-)adjunct of $f$, and $f$ is the (left-)adjunct of $g$. Sometimes one writes $g = f^\sharp$ and $f = g^\flat$, as in musical notation.
Sometimes people call $\tilde f$ the "adjoint" of $f$, and vice versa, but this is potentially confusing because it is the _functors_ $F$ and $G$ which are adjoint. Other possible terms are _conjugate_, _transpose_, and [[mate]].
## Properties
+-- {: .num_prop #AdjunctionCoUnitGiveesAllAdjuncts}
###### Proposition
**(adjuncts in terms of adjunction (co-)unit)**
Let $\eta_X \colon X \to R L X$ be the [[unit of an adjunction|unit of the adjunction]] and $\epsilon_X \colon L R X \to X$ the [[counit of an adjunction|counit]].
Then
* the adjunct of $f \colon X \to R Y$ in $D$ is the composite
$$
\tilde f
\colon
L X
\overset{L f}{\longrightarrow}
L R Y
\overset{\epsilon_Y}{\longrightarrow}
Y
$$
* the adjunct of $g \colon L X \to Y$ in $C$ is the composite
$$
\tilde g
\colon
X
\stackrel{\eta_X}{\longrightarrow}
R L X
\overset{R g}{\longrightarrow}
R Y
\,.
$$
=--
For proof see [this Prop.](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit) at *[[adjoint functor]]*.
## Examples
* The process of [[currying]] is an instance of passage to adjuncts, specialized to the tensor-hom adjunction of a [[closed monoidal category]].
## Related concepts
* [[adjunction]]
* [[zig-zag law]]/[[triangle identity]]
* [[unit of an adjunction]]
* [[mate]]
* **adjunct**
## References
* [[Categories Work]], second edition, p. 81
* [[Category Theory in Context]], p. 116, 124.
[[!redirects adjuncts]] |
adjunction | https://ncatlab.org/nlab/source/adjunction |
> This page is about adjunctions in general [[2-categories]]. Specifically for the common case of adjunctions in [[Cat]] see at *[[adjoint functors]]*. For the notion of "adjunction of a set to a field" in [[field theory]], see [[field extension]].
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
A [[pair]] of [[1-morphisms]] in a [[2-category]] form an **adjunction** if they are **dual** to each other ([Lambek (1982)](#Lambek82), cf. [here](geometry+of+physics+--+categories+and+toposes#CategoryTheoryIsTheoryOfDuality)) in a precise sense.
There are two archetypical classes of examples:
* If $A$ is a [[monoidal category]] and $\mathbf{B}A$ denotes the [[singleton|one-]][[object]] 2-category whose single [[hom-category]] is $A$ (the [[delooping]] of $A$), then the notion of adjoint morphisms in $\mathbf{B}A$ coincides precisely with the notion of [[dualizable object|dual objects]] in $A$, subsuming, in turn, classical examples such as [[dual vector spaces]] in the case that $A = $ [[FinDimVect]].
* Adjunctions in the [[2-category]] [[Cat]] of [[categories]] are _[[adjoint functors]]_.
* Notice that essentially everything that makes [[category theory]] nontrivial and interesting, beyond [[groupoid]]-theory, is governed by the concept of [[adjoint functors]]. In particular [[universal constructions]] such as [[limit|limits and colimits]], [[Kan extensions]], ([[coend|co]])[[ends]] are examples of adjunctions in [[Cat]].
* Similarly, for $V$ any [[cosmos for enrichment]], adjunctions in the 2-category [[VCat]] of $V$-[[enriched categories]] are equivalently [[enriched adjoint functors]]. Already in simple cases such $V = $ [[truth values]] this subsumes classical concepts such as that of [[Galois connections]].
* Remarkably, even adjunctions in the [[homotopy 2-category]] of [[(infinity,1)-categories|$(\infty,1)$-categories]] are equivalent to [[adjoint (infinity,1)-functors|adjoint $\infty$-functors]], see the examples [below](#AdjointInfinityOneFunctors).
These classes of examples make adjunctions a key notion in *[[formal category theory]]*.
Finally, the notion of *adjunction* may usefully be thought of as a generalization of the notion of *[[equivalence in a 2-category]]*: an adjoint [[1-morphism]] need not be [[invertible morphism|invertible]] (even up to [[2-isomorphism]]) but it does have, in a precise sense, _a left inverse from below_ or _a right inverse from above_.
If an adjoint 1-morphisms happens to be a genuine [[equivalence in a 2-category]], then the adjunction is called an *[[adjoint equivalence]]*.
## Definition
{#Definition}
\begin{defn} \label{DefinitionAdjunction}
An _adjunction_ in a [[2-category]] is
* a [[pair]] of [[objects]] $C$ and $D$
* a [[pair]] of [[1-morphisms]]
$L \colon C \longrightarrow D$ (the *[[left adjoint]]*)
$R \colon D \longrightarrow C$ (the *[[right adjoint]]*)
* a [[pair]] of [[2-morphisms]]
$\eta \colon 1_C \longrightarrow R \circ L$ (the *[[adjunction unit]]*)
$\epsilon \colon L \circ R \longrightarrow 1_D$ (the *[[adjunction counit]]*)
such that the following equivalent conditions hold:
* **[[triangle identity]]** the following [[commuting diagram|diagrams commute]] in the [[hom-categories]] (where "$\cdot$" denotes [[whiskering]]):
\begin{centre}
\begin{tikzcd}
L \ar[r, "L \cdot \eta"]
\ar[dr, swap, "\mathrm{id}"]
& L \circ R \circ L \ar[d, "\epsilon \cdot L"]
\\ & L
\end{tikzcd}
\end{centre}
\begin{centre}
\begin{tikzcd}
R \ar[r, "\eta \cdot R"] \ar[dr, swap, "id"] & R \circ L \circ R \ar[d, "R \cdot \epsilon"]
\\ & R
\end{tikzcd}
\end{centre}
* **[[zig-zag law]]** the following [[equality]] of [[2-morphisms]]:
\begin{tikzcd}[row sep=10pt]
C
\ar[rr, bend left=40, "{\mathrm{id}}",
"{\ }"{name=s1, swap}
]
\ar[r, "L"{description}, "{\ }"{name=t1, pos=.9}]
\ar[
from=s1,
to=t1,
Rightarrow,
"{ \eta }"{swap}
]
&
D
\ar[rr, bend right=40, "{\mathrm{id}}"{swap},
"{\ }"{name=t2}
]
\ar[r, "R"{description}]
&
C
\ar[r, "L"{description}, "{\ }"{swap, name=s2, pos=.1}]
\ar[
from=s2,
to=t2,
Rightarrow,
"{ \epsilon }"
]
&
D
&=&
C
\ar[r, bend left=40, "{L}", "{\ }"{name=s3, swap}]
\ar[r, bend right=40, "{L}"{swap}, "{\ }"{name=t3}]
\ar[from=s3, to=t3, Rightarrow, "{\mathrm{id}}"]
&
D
\end{tikzcd}
\begin{tikzcd}[row sep=10pt]
D
\ar[rr, bend right=40, "{\mathrm{id}}"{swap},
"{\ }"{name=s1}
]
\ar[r, "R"{description}, "{\ }"{name=t1, pos=.9, swap}]
\ar[
from=t1,
to=s1,
Rightarrow,
"{ \epsilon }"{swap, pos=.3}
]
&
C
\ar[rr, bend left=40, "{\mathrm{id}}",
"{\ }"{name=t2, swap}
]
\ar[r, "L"{description}]
&
D
\ar[r, "R"{description}, "{\ }"{name=s2, pos=.1}]
\ar[
from=t2,
to=s2,
Rightarrow,
"{ \eta }"{pos=.8}
]
&
C
&=&
D
\ar[r, bend left=40, "{R}", "{\ }"{name=s3, swap}]
\ar[r, bend right=40, "{R}"{swap}, "{\ }"{name=t3}]
\ar[from=s3, to=t3, Rightarrow, "{\mathrm{id}}"]
&
C
\end{tikzcd}
{#InTermsOfStringDiagrams} In terms of [[string diagrams]] the above data entering the definition looks like
[[adjunction-L.png:pic]]          [[adjunction-R.png:pic]]          [[adjunction-unit.png:pic]]          [[adjunction-co-unit.png:pic]]
(where 1-cells read from right to left and 2-cells from bottom to top), and the [[zig-zag identities]] appear as moves "pulling zigzags straight" (hence the name):
[[adjunction-up-string.png:pic]]          [[adjunction-down-string.png:pic]]
Often, arrows on strings are used to distinguish $L$ and $R$, and most or all other labels are left implicit; so the zigzag identities, for instance, become:
[[adjunction-up-string-minimal.png:pic]]          [[adjunction-down-string-minimal.png:pic]]
\end{defn}
## Properties
### Relation to monads
See at _[monad -- Relation between adjunctions and monads](monad#RelationBetweenAdjunctionsAndMonads)_.
## Examples
### Adjoint functors
{#AdjointFunctors}
\begin{proposition}\label{AdjunctionsInCatAreAdjointFunctors}
An adjunction in the 2-category [[Cat]] of [[categories]], [[functors]] and [[natural transformations]] is equivalently a pair of [[adjoint functors]].
\end{proposition}
See also the proof [here](adjoint+functor#AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat) at *[[adjoint functor]]*.
\begin{proof}
Suppose given functors $L \,\colon\, C \to D$, $R: D \to C$ and the structure of a pair of [[adjoint functors]] in the form of a natural isomorphism of [[hom-sets]] ([here](adjoint+functor#InTermsOfHomIsomorphism))
$$
\Psi_{c, d}
\;\colon\;
\hom_D(L(c), d) \cong \hom_C(c, R(d))
$$
Now the idea is that, in the spirit of the (proof of the) [[Yoneda lemma]], we would like $\Psi$ to be determined by what it does to [[identity morphisms]]. With that in mind, define the [[adjunction unit]] $\eta \colon 1_C \to R L$ by the formula $\eta_c = \Psi_{c, L(c)}(1_{L(c)})$. Dually, define the counit $\varepsilon \,\colon\, L R \to 1_D$ by the formula
$$
\varepsilon_d
\,\coloneqq\, \Psi^{-1}_{R(d), d}(1_{R(d)})
\,.
$$
Then given $g \,\colon\, L(c) \to d$, the claim is that
$$
\Psi_{c, d}(g)
\,=\,
(c \stackrel{\eta_c}{\to} R(L(c)) \stackrel{R(g)}{\to} R(d))
\,.
$$
This may be left as an exercise in the yoga of the Yoneda lemma, applied to $\hom_D(L(c), -) \to \hom_C(c, R(-))$. By [[formal duality]], given $f \,\colon\, c \to R(d)$,
$$
\Psi^{-1}_{c, d}(f) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)
\,.
$$
(We spell out the Yoneda-lemma proof of this dual form [below](#YonedaLemmaArgument).)
Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if $\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)})$, via the recipe just given for $\Psi^{-1}$ we recover
$$1_{L(c)} = (L(c) \stackrel{L(\eta_c)}{\to} L R L(c) \stackrel{\varepsilon_{L(c)}}{\to} L(c))$$
and this is one of the famous [[triangle identities]]: $1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L)$. Here, juxtaposition of functors and natural transformations denotes neither functor application, nor vertical composition, nor horizontal composition, but [[whiskering]]. By duality, we have the other [[triangle identity]] $1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R)$. These two triangular equations are enough to guarantee that the recipes for $\Psi$ and $\Psi^{-1}$ indeed yield mutual inverses.
In conclusion it is perfectly sufficient to define an adjoint pair of functors in $Cat$ as given by unit and counit transformations $\eta: 1_C \to R L$, $\varepsilon: L R \to 1_D$, satisfying the triangle identities above.
\end{proof}
\begin{remark}
The definition of adjunctions via [[unit of an adjunction|units]] and [[counit of an adjunction|counits]] is an "elementary" definition (so that by implication, the formulation in terms of [hom-isomorphisms](adjoint functor#InTermsOfHomIsomorphism) is not elementary) in the sense that while the [[hom-functor]] formulation relies on some notion of [[hom-set|hom-*set*]], the formulation in terms of units and counits is purely in the [[first-order logic|first-order]] [[language]] of [[categories]] and makes no reference to a model of [[set theory]]. The definition via (co)units therefore gives us a definition of adjunctions even if an assumption of [[locally small|local smallness]] is not made.
\end{remark}
\begin{proof}\label{YonedaLemmaArgument}
**(Yoneda-lemma argument)**
\linebreak
We claim that $\Psi^{-1}_{c, d} \,\colon\, \hom_C(c, R(d)) \to \hom_D(L(c), d)$ can be defined by the formula
$$\Psi^{-1}_{c, d}(f: c \to R(d)) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)$$
where $\varepsilon_d \coloneqq \Psi^{-1}_{R(d), d}(1_{R(d)})$. This is by appeal to the proof of the Yoneda lemma applied to the transformation
$$\Psi^{-1}_{-, d}: \hom_C(-, R(d)) \to \hom_D(L(-), d)$$
For the naturality of $\Psi^{-1}$ in the argument $(-)$ would imply that given $f: c \to R(d)$, we have a commutative square
$$\array{
\hom_C(R(d), R(d))
&
\stackrel{\Psi^{-1}_{R(d), d}}{\to}
& \hom_D(L(R(d)), d)
\\
\hom_C(f, R(d))
\big\downarrow
& &
\big\downarrow \hom_D(L(f), d)
\\
\hom_C(c, R(d))
&
\underset{\Psi^{-1}_{c, d}}{\to}
&
\hom_D(L(c), d)
}$$
Chasing the element $1_{R(d)}$ down and then across, we get $f: c \to R(d)$ and then $\Psi^{-1}_{c, d}(f)$. Chasing across and then down, we get $\varepsilon_d$ and then $\varepsilon_d \circ L(f)$. This completes the verification of the claim.
\end{proof}
\begin{corollary}
An adjunction in its [[core]] [[2-groupoid]] $Core(Cat)$ is an [[adjoint equivalence]].
\end{corollary}
### Enriched adjoint functors
Similarly one sees:
\begin{example}\label{EnrichedAdjointFunctors}
For $V$ a [[cosmos for enrichment]], an adjunction in the [[2-category]] [[VCat]] of $V$-[[enriched categories]] is equivalently a pair $V$-[[enriched functors]].
\end{example}
\begin{example}
Let $U \,\colon\, Grp \to Set$ from [[Grp]] to [[Set]] denote the usual [[forgetful functor]] from [[Grp]] to [[Set]]. When we say "$F(X)$ is the [[free group]] generated by a set $X$", we mean there is a function $\eta_X \,\colon\, X \to U(F(X))$ which is universal among functions from $X$ to the underlying set of a group, which means in turn that given a function $f: X \to U(G)$, there is a unique group homomorphism $g \colon F(X) \to G$ such that
$$
f = (X \stackrel{\eta_X}{\to} U(F(X)) \stackrel{U(g)}{\to} U(G))
$$
Here $\eta_X$ is a component of what we call the [[unit of an adjunction|unit of the adjunction]] $F \dashv U$, and the equation above is a recipe for the relationship between the map $g: F(X) \to G$ and the map $f: X \to U(G)$ in terms of the unit.
\end{example}
### Adjoint $(\infty,1)$-functors
{#AdjointInfinityOneFunctors}
\begin{proposition}
\label{AdjointInfinityFunctorsInHomotopy2Category}
An adjunction in the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $(\infty,1)$-categories]] is equivalently a pair of [[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]].
\end{proposition}
([Riehl & Verity 2015, Rem. 4.4.5](adjoint+infinity1-functor#RiehlVerity15); [Riehl & Verity 2022, Prop. F.5.6](adjoint+infinity1-functor#RVElements); see [there](adjoint+infinity1-functor#InTheHomotopy2Category) for more).
\begin{remark}
In view of Prop. \ref{AdjunctionsInCatAreAdjointFunctors},
the remarkable aspect of Prop. \ref{AdjointInfinityFunctorsInHomotopy2Category} is that the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $\infty$-categories]] is sufficient to detect adjointness of [[(infinity,1)-functors|$\infty$-functors]], which would, *a priori*, be defined as a kind of [[higher homotopy]]-[[coherence law|coherent]] adjointness in the full [[(infinity,2)-category|$(\infty,2)$-category]] [[(infinity,1)Cat|$Cat_{(\infty,1)}$]]. For more on this reduction of homotopy-coherent adjunctions to plain adjunctions see [Riehl & Verity 2016, Thm. 4.3.11, 4.4.11](adjoint+infinity1-functor#RiehlVerity16).
\end{remark}
## Related concepts
* [[adjoint functor]], [[adjoint (∞,1)-functor]]
* [[left adjoint]], [[right adjoint]]
* [[duality]]
* [[adjunct]]
* [[dual adjunction]], [[Galois connection]]
* [[monoidal adjunction]]
* [[enriched adjunction]]
* [[derived adjunction]]
* [[adjoint monad]]
* [[heteromorphism]]
* [[adjoint string]]
* [[adjoint triple]], [[adjoint quadruple]]
* [[ambidextrous adjunction]]
* [[adjoint cylinder]]
* [[adjoint logic]]
## References
Adjunctions in 2-categories were introduced (together with the notion of [[strict 2-categories]] itself) in:
* [[Jean-Marie Maranda]], pp. 762 in: *Formal categories*, Canadian Journal of Mathematics **17** (1965) 758-801 [[doi:10.4153/CJM-1965-076-0](https://doi.org/10.4153/CJM-1965-076-0), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf)]
Though see also the following, which uses more modern terminology:
* [[Max Kelly]], §2 in: *Adjunction for enriched categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) [[doi:10.1007/BFb0059145](https://doi.org/10.1007/BFb0059145)]
A thorough 2-categorical account is contained in:
* [[Claude Auderset]], _Adjonctions et monades au niveau des $2 $-catégories_, Cahiers de topologie et géométrie différentielle 15.1 (1974): 3-20.
* [[Stephen Schanuel]] and [[Ross Street]], _The free adjunction_, Cahiers de topologie et géométrie différentielle catégoriques 27.1 (1986): 81-8
Review:
* [[Saunders MacLane]], §XII.4 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
* {#Lack10} [[Steve Lack]], §2.1 in: *A 2-categories companion*, in: *[[Towards Higher Categories]]*, The IMA Volumes in Mathematics and its Applications **152** Springer (2010) [[arXiv:math.CT/0702535](http://arxiv.org/abs/math.CT/0702535), [doi:10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4)]
* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Chapter 6 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))
Fr the special case of [[adjoint functors]] see any text on [[category theory]] (and see the references at _[[adjoint functor]]_), for instance:
* {#Borceux94} [[Francis Borceux]], Vol 1, Section 3 of _[[Handbook of Categorical Algebra]]_
* _[[geometry of physics -- categories and toposes]] -- [Adjunctions](https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes#Adjunctions)_
For some early history and illustrative examples see
* {#Lambek82} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) ([doi:10.1007/978-94-009-8462-2_6](https://link.springer.com/chapter/10.1007/978-94-009-8462-2_6))
(more along these lines at _[[objective logic]]_).
The fundamental role of [[adjoint functors]] in [[logic]]/[[type theory]] originates with the observaiton that [[substitution]] forms an [[adjoint triple]] with [[existential quantification]] and [[universal quantification]]:
* {#Lawvere69} [[William Lawvere]], _Adjointness in Foundations_, ([tac:16](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296
* [[William Lawvere]], _Quantifiers and sheaves_, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf))
Adjunctions in [[programming languages]] (though mainly again just [[adjoint functors]]):
* {#Hinze12} Ralf Hinze, _Generic Programming with Adjunctions_, In: J. Gibbons (ed.) _Generic and Indexed Programming_ Lecture Notes in Computer Science, vol 7470. Springer 2012 ([pdf](http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf), [slides](http://www.cs.ox.ac.uk/ralf.hinze/SSGIP10/Slides.pdf) [doi:10.1007/978-3-642-32202-0_2](https://doi.org/10.1007/978-3-642-32202-0_2))
* Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, _Relational Algebra by Way of Adjunctions_, Proceedings of the ACM on Programming Languages archive Volume 2 Issue ICFP, September 2018 Article No. 86 ([pdf](https://www.cs.ox.ac.uk/jeremy.gibbons/publications/reladj.pdf), [doi:10.1145/3236781](https://dl.acm.org/citation.cfm?doid=3243631.3236781))
See also
* Wikipedia, *[Adjoint Functors](http://en.wikipedia.org/wiki/Adjoint_functors)*
* Catsters, _Adjunctions_ ([YouTube](http://www.youtube.com/watch?v=loOJxIOmShE&feature=channel_page))
-
[[!redirects adjoint pair]]
[[!redirects adjoint pairs]]
[[!redirects adjunctions]] |
adjunction > zigzagepsilon | https://ncatlab.org/nlab/source/adjunction+%3E+zigzagepsilon | <svg xmlns="http://www.w3.org/2000/svg" width="8.5em" height="2em" viewBox="0 0 85 20">
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</svg> |
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adjunction > zigzageta > history | https://ncatlab.org/nlab/source/adjunction+%3E+zigzageta+%3E+history | < [[adjunction > zigzageta]]
[[!redirects adjunction/zigzag]]
[[!redirects adjunction/zigzageta]]
[[!redirects adjunction/zigzag -- history]] |
adjunction between topological spaces and diffeological spaces | https://ncatlab.org/nlab/source/adjunction+between+topological+spaces+and+diffeological+spaces |
+-- {: .num_prop #AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces}
###### Proposition
**([[adjunction between topological spaces and diffeological spaces]])**
There is a pair of [[adjoint functors]]
\[
\label{AdjointFunctorsBetweenTopSpAndDifflgSp}
TopSp
\underoverset{
\underset{
Cdfflg
}{\longrightarrow}
}{
\overset{
Dtplg
}{\longleftarrow}
}{\phantom{AA}\bot\phantom{AA}}
DifflgSp
\]
between the [[categories]] of [[Top|TopologicalSpaces]] and of [[DiffeologicalSpaces]], where
* $Cdfflg$ takes a [[topological space]] $X$ to the **continuous diffeology**, namely the diffeological space on the same underlying set $X_s$ whose plots $U_s \to X_s$ are the [[continuous functions]] (from the underlying [[topological space]] of the [[domain]] $U$).
* $Dtplg$ takes a [[diffeological space]] to the **diffeological topology** ([[D-topology]]), namely the [[topological space]] with the same underlying set $X_s$ and with the [[final topology]] that makes all its plots $U_{s} \to X_{s}$ into [[continuous functions]]: called the _[[D-topology]]_.
Hence a [[subset]] $O \subset \flat X$ is an [[open subset]] in the [[D-topology]] precisely if for each plot $f \colon U \to X$ the [[preimage]] $f^{-1}(O) \subset U$ is an [[open subset]] in the [[Cartesian space]] $U$.
Moreover:
1. the [[fixed point of an adjunction|fixed points of this adjunction]] $X \in$[[Top|TopologicalSpaces]] (those for which the [[counit of an adjunction|counit]] is an [[isomorphism]], hence here: a [[homeomorphism]]) are precisely the [[Delta-generated topological spaces]] ([i.e.](Delta-generated+topological+space#AsDTopologicalSpaces) [[D-topological spaces]]):
$$
X \;\,\text{is}\;\Delta\text{-generated}
\;\;\;\;\;
\Leftrightarrow
\;\;\;\;\;
Dtplg(Cdfflg(X))
\underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow}
X
$$
1. this is an [[idempotent adjunction]], which exhibits $\Delta$-generated/[[D-topological spaces]] as a [[reflective subcategory]] inside [[diffeological spaces]] and a [[coreflective subcategory]] inside all [[topological spaces]]:
\[
\label{DeltaGeneratedSpacesInIdempotentAdjunction}
TopologicalSpaces
\underoverset
{
\underset{
Cdfflg
}{\longrightarrow}
}
{
\overset{
}{\hookleftarrow}
}
{\phantom{AA}\bot\phantom{AA}}
DTopologicalSpaces
\underoverset
{
\underset{
}{\hookrightarrow}
}
{
\overset{
Dtplg
}{\longleftarrow}
}
{\phantom{AA}\bot\phantom{AA}}
DiffeologicalSpaces
\]
Finally, these adjunctions are a sequence of [[Quillen equivalences]] with respect to the:
| | | |
|--|--|--|
| [[classical model structure on topological spaces]] | [[model structure on D-topological spaces]] | [[model structure on diffeological spaces]] |
> Caution: There was a gap in the original proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$. The gap is claimed to be filled now, see the commented references [here](model+structure+on+diffeological+spaces#References).
=--
Essentially these adjunctions and their properties are observed in [Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3](diffeological+space#SYH10), see also [Christensen, Sinnamon & Wu 2014, Sec. 3.2](D-topology#CSW13). The model structures and Quillen equivalences are due to [Haraguchi 13, Thm. 3.3](#model+structure+on+Delta-generated+topological+spaces#Haraguchi13) (on the left) and [Haraguchi-Shimakawa 13, Sec. 7](model+structure+on+diffeological+spaces#HaraguchiShimakawa13) (on the right).
+-- {: .proof}
###### Proof
We spell out the existence of the [[idempotent adjunction]] (eq:DeltaGeneratedSpacesInIdempotentAdjunction):
First, to see we have an [[adjunction]] $Dtplg \dashv Cdfflg$, we check the hom-isomorphism ([here](adjoint+functor#eq:HomIsomorphismForAdjointFunctors)).
Let $X \in DiffeologicalSpaces$ and $Y \in TopologicalSpaces$. Write $(-)_s$ for the underlying sets. Then a [[morphism]], hence a [[continuous function]] of the form
$$
f \;\colon\; Dtplg(X) \longrightarrow Y
\,,
$$
is a [[function]] $f_s \colon X_s \to Y_s$ of the underlying [[sets]] such that for every [[open subset]] $A \subset Y_s$ and every [[smooth function]] of the form $\phi \colon \mathbb{R}^n \to X$ the [[preimage]] $(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n$ is open. But this means equivalently that for every such $\phi$, $f \circ \phi$ is [[continuous function|continuous]]. This, in turn, means equivalently that the same underlying function $f_s$ constitutes a [[smooth function]] $\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y)$.
In summary, we thus have a [[bijection]] of [[hom-sets]]
$$
\array{
Hom( Dtplg(X), Y )
&\simeq&
Hom(X, Cdfflg(Y))
\\
f_s &\mapsto& (\widetilde f)_s = f_s
}
$$
given simply as the [[identity function|identity]] on the underlying [[functions]] of underlying sets. This makes it immediate that this hom-isomorphism is [[natural bijection|natural]] in $X$ and $Y$ and this establishes the [[adjunction]].
Next, to see that the [[D-topological spaces]] are the [[fixed point of an adjunction|fixed points]] of this adjunction,
we apply the above [[natural bijection]] on hom-sets to the case
$$
\array{
Hom( Dtplg(Cdfflg(Z)), Y )
&\simeq&
Hom(Cdfflg(Z), Cdfflg(Y))
\\
(\epsilon_Z)_s &\mapsto& (\mathrm{id})_s
}
$$
to find that the [[counit of the adjunction]]
$$
Dtplg(Cdfflg(X))
\overset{\epsilon_X}{\longrightarrow}
X
$$
is given by the [[identity function]] on the underlying sets $(\epsilon_X)_s = id_{(X_s)}$.
Therefore $\eta_X$ is an [[isomorphism]], namely a [[homeomorphism]], precisely if the open subsets of $X_s$ with respect to the topology on $X$ are precisely those with respect to the topology on $Dtplg(Cdfflg(X))$, which means equivalently that the open subsets of $X$ coincide with those whose pre-images under all continuous functions $\phi \colon \mathbb{R}^n \to X$ are open. This means equivalently that $X$ is a D-topological space.
Finally, to see that we have an [[idempotent adjunction]], it is sufficient to check (by [this Prop.](idempotent+adjunction#EquivalentConditionsForIdempotency)) that the [[comonad]]
$$
Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces
$$
is an [[idempotent comonad]], hence that
$$
Dtplg \circ Cdfflg
\overset{
Dtplg \cdot \eta \cdot Cdfflg
}{\longrightarrow}
Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg
$$
is a [[natural isomorphism]]. But, as before for the adjunction counit $\epsilon$, we have that also the [[adjunction unit]] $\eta$ is the [[identity function]] on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.
=--
|
adjunction of dg-categories | https://ncatlab.org/nlab/source/adjunction+of+dg-categories | ## Idea
...
## References
See section 2.1 in
* [[B. Toen]], _Derived Azumaya algebras and generators for twisted derived categories_, [arXiv:1002.2599](http://arxiv.org/abs/1002.2599).
[[!redirects adjunctions of dg-categories]]
|
adjusted connection | https://ncatlab.org/nlab/source/adjusted+connection |
## Idea
A notion of connection for [[principal ∞-bundles]] that does not impose the fake flatness condition.
## Related concepts
* [[adjusted Weil algebra]]
## References
The construction of non-fake flat [[principal ∞-connections]] originates with
* {#FSS12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], [p. 77](https://arxiv.org/pdf/1011.4735.pdf#page=77) and [p. 80](https://arxiv.org/pdf/1011.4735.pdf#page=80) of: _[[schreiber:Cech cocycles for differential characteristic classes|Čech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 [[arXiv:1011.4735](https://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]
based on the [[adjusted Weil algebras]] discussed earlier in
* {#SSS09} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], Def. 23 on [p. 47](https://arxiv.org/pdf/0801.3480.pdf#page=47) with Prop. 21 on [p. 48](https://arxiv.org/pdf/0801.3480.pdf#page=48) in: *[[schreiber:L-infinity algebra connections|$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport]]*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 [[arXiv:0801.3480](https://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)]
* {#SSS12} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]]: middle boxes on [p. 28](https://arxiv.org/pdf/0910.4001.pdf#page=28) and [p. 32](https://arxiv.org/pdf/0910.4001.pdf#page=32) of: *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012) 169-213 [[arXiv:0910.4001](https://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3)]([arXiv:](https://link.springer.com/article/10.1007/s00220-012-1510-3))]
As the title of [FSS 12](#FSS12) indicates, this procedure constructs [[Čech cohomology]] [[cocycles]] for non-fake flat higher connections in the style of the cocycles in
* {#BrylinskiMcLaughlin96} [[Jean-Luc Brylinski]], D. A. McLaughlin. *Čech cocycles for characteristic classes*, Comm. Math. Phys., **178** 1 (1996) 225–236 [[doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]
for the underlying bundles.
This construction is based on the [[Lie integration]] of [[L-infinity algebras]] by the "[path method](Lie+integration#SmoothIntegration)" and as such works generally but produces very "large" cocycle data, in a sense.
A variant construction tailored towards "smaller" cocycles for low-degree [[Lie n-algebras]] was later proposed in:
* [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], _Explicit Non-Abelian Gerbes with Connections_, [arXiv:2203.00092](https://arxiv.org/abs/2203.00092)
Examples for T-duality:
* [[Hyungrok Kim]], [[Christian Saemann]], _T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections_, [arXiv:2303.16162](https://arxiv.org/abs/2303.16162)
Further references for T-duality:
* [[Konrad Waldorf]], _Geometric T-duality: Buscher rules in general topology_, [arXiv:2207.11799](https://arxiv.org/abs/2207.11799).
* [[Thomas Nikolaus]], [[Konrad Waldorf]], _Higher geometry for non-geometric T-duals_, [arXiv:1804.00677](https://arxiv.org/abs/1804.00677).
[[!redirects adjusted connections]]
|
adjusted Weil algebra | https://ncatlab.org/nlab/source/adjusted+Weil+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The traditional notion of *[[connection on a principal bundle]]* with given [[structure group|structure]] [[Lie algebra]] $\mathfrak{g}$ has a slick [[dg-algebra|dg-algebraic]]-formulation in terms of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ and the [[Weil algebra]] $W(\mathfrak{g})$ of $\mathfrak{g}$ (due to [Henri Cartan 1950](Weil+algebra#Cartan50)).
These concepts of *[[Cartan connection]]* may be generalized to [[L-infinity algebras|$L_\infty$-algebras]] (originally so in [SSS09](#SSS09), [SSS12](#SSS12), [FSS12](#FSS12), which was the starting point for the more comprehensive development in [dcct12](#dcct12)) -- see at *[[connection on a smooth principal infinity-bundle|connection on a smooth principal $\infty$-bundle]]*. In fact, noticing (see [here](L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra)) that the [[Chevalley-Eilenberg algebra]]-construction $CE(-)$ constitutes a [[full subcategory]]-embedding of the 1-category of [[finite type]] [[L-infinity algebras|$L_\infty$-algebras]] into the [[opposite category|opposite 1-category]] of [[dgc-algebras]], there is an immediate $L_\infty$-analog of the notion of [[Weil algebra]] ([SSS09, Def. 16](#SSS09)).
However, this is a [[1-category|1-]][[category theory|category theoretic]] analog, hence is a particularly strict model, defined up to [[isomorphism]], of what the would-be $\infty$-Weil algebra of an $L_\infty$-algebra should be. The latter will only be well-defined up to compatible [[quasi-isomorphism]].
The need to pass to compatible deformations of $L_\infty$-Weil algebras for a satisfactory definition of [[string 2-connections]] ([[1-brane]] connections) and analogous higher notions ([[fivebrane structure|5-brane]]-connection, [[ninebrane structure|9-brane]], etc.) was first discussed around [SSS09, Prop. 21](#SSS09) (and used to exhibit the [[higher gauge theory|higher gauge-theoretic]] nature of the [[Green-Schwarz mechanism]] in [SSS12, §3.2](higher+gauge+theory+of+the+Green-Schwarz+mechanism+--+references#SatiSchreiberStasheff12)), and was justified there by the condition that the canonical [[invariant polynomial]] in that situation does lift as expected.
A more general discussion of the necessary *adjustments* of $L_\infty$-Weil algebras was then given in [Saemann & Schmidt 20](#SaemannSchmidt20), who observe that a good general condition to impose is that the induced [[BRST complex|BRST complexes]] of $L_\infty$-connections ([SSS09, §9.3](#SSS09)) remain well-defined when the [[antifields]] and [[auxiliary fields]] are required to strictly vanish.
These authors introduce the terminology *adjusted Weil algebras* ([Saemann & Schmidt 20, Def. 4.2](#SaemannSchmidt20)) and the resulting *adjusted $L_\infty$-connections* and *adjusted [[higher parallel transport]]* ([Kim & Saemann 2020](KimSaemann20)).
Concretely, the choice of compatible deformation of the Weil algebra determines the [[Bianchi identities]] on the [[curvature characteristic forms]] of corresponding [[L-infinity algebra valued differential forms|$L_\infty$-algebra valued differential forms]] (these [[Bianchi identities]] are embodied by the [[differential]] in the Weil algebra restricted to shifted generators) and without adjustment the Bianchi identities are stronger than (i.e. just special cases of what) they ought to be.
Notice that, while the [[Weil algebra]] by itself is [[contractible homotopy type]], its *adjustments* along [[quasi-isomorphisms]] must satisfy [[horizontal differential form|horizontality]] constraints (such as the vanishing of those [[antifields]]) which makes this a non-trivial procedure.
While the characterization of adjusted Weil algebras for $L_\infty$-algebra in [Saemann & Schmidt 20, Def. 4.2](#SaemannSchmidt20) clearly (generalizes and) conceptually improves on [SSS09, Prop. 21](#SSS09), and while in applications it clearly gives the right answers (the correct higher [[Bianchi identities]]), what is still missing is a purely [[homotopy theory|homotopy theoretic]] justification. Since in practice these adjusted Weil algebras behave and are used much like [[resolutions]] by [[minimal fibrations]] in [[model category]]-theory, it is natural to wonder if there is the structure of a [[homotopical category|homotopical]] [[fibration category]] on the Weil algebra choices for $L_\infty$-algebras, such that this is indeed the case. This question is open.
On the other hand, [Borsten, Kim & Saemann 2021](#BorstenKimSaemann21) argue that adjustment is naturally understood after embedding [[L-infinity algebras|$L_\infty$-algebras]] within [[EL-infinity algebra|$E L_\infty$-algebras]] (and that this is what exhibits [[tensor hierarchies]] as a [[higher gauge theory]]-phenomenon).
## Related entries
* [[principal infinity-connection]]
* [[nonabelian differential cohomology]]
* [[supergravity Lie 6-algebra]]
## References
The original discussion for the special case of [[string 2-connections]] and their higher analogs (such as [[Fivebrane structure|fivebrane]] 6-connections, [[Ninebrane structure|Ninebrane]] 10-connections etc.):
* {#Schreiber07} [[Urs Schreiber]], *[Obstructions to $n$-Bundle Lifts Part II](https://golem.ph.utexas.edu/category/2007/10/obstructions_to_nbundle_lifts.html)* (Oct 2007) [bottom right corner in this hand-drawn diagram: [[Schreiber-PrincipalInfinityConnections-2007.pdf:file]]]
* {#SSS09} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], Def. 23 on [p. 47](https://arxiv.org/pdf/0801.3480.pdf#page=47) with Prop. 21 on [p. 48](https://arxiv.org/pdf/0801.3480.pdf#page=48) in: *[[schreiber:L-infinity algebra connections|$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport]]*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 [[arXiv:0801.3480](https://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)]
* {#SSS12} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]]: middle boxes on [p. 28](https://arxiv.org/pdf/0910.4001.pdf#page=28) and [p. 32](https://arxiv.org/pdf/0910.4001.pdf#page=32) of: *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012) 169-213 [[arXiv:0910.4001](https://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3)]([arXiv:](https://link.springer.com/article/10.1007/s00220-012-1510-3))]
* {#FSS12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], [p. 77](https://arxiv.org/pdf/1011.4735.pdf#page=77) and [p. 80](https://arxiv.org/pdf/1011.4735.pdf#page=80) of: _[[schreiber:Cech cocycles for differential characteristic classes|Čech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 [[arXiv:1011.4735](https://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]
* {#dcct13} [[Urs Schreiber]], Def. 5.2.91 on [p. 645](https://arxiv.org/pdf/1310.7930v1.pdf#page=645) in: *[[schreiber:dcct|differential cohomology in a cohesive topos]]* (2013-)
{#WithHindsight} With hindsight, an early example of such adjustment (namely for the "[[supergravity Lie 6-algebra]]", see [there](supergravity+Lie+6-algebra#ModifiedWeilAlgebra)), is given in:
* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], vol 2, III (8.24) of: *[[Supergravity and Superstrings - A Geometric Perspective]]*, World Scientific (1991) [[doi:10.1142/0224](https://doi.org/10.1142/0224), [epdf](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html)]
(recognized as a "modified Weil algebra" in [revision 8, 2011](https://ncatlab.org/nlab/revision/diff/supergravity+Lie+6-algebra/8), long before the "adjusted"-terminology was proposed).
Characterization of adjusted $\mathfrak{g}$-Weil algebras by requiring well-behaved [[BRST-complexes]] for [[L-infinity algebra valued differential forms|$L_\infty$-algebra valued differential forms]]:
* {#SaemannSchmidt20} [[Christian Saemann]], [[Lennart Schmidt]], *Towards an M5-Brane Model II: Metric String Structures*, Fortschr. Phys. **68** (2020) 2000051 [[arXiv:1908.08086](https://arxiv.org/abs/1908.08086), [doi:10.1002/prop.202000051](https://doi.org/10.1002/prop.202000051)]
Revisiting the special case of the [[string Lie 2-algebra]]:
* {#Schmidt19} [[Lennart Schmidt]], _Twisted Weil Algebras for the String Lie 2-Algebra_, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_ [Durham Symposium](http://www.maths.dur.ac.uk/lms/) 2018, Fortschritte der Physik 2019 [[arXiv:1903.02873](https://arxiv.org/abs/1903.02873), [doi:10.1002/prop.201910016](https://doi.org/10.1002/prop.201910016)]
Application to [[higher parallel transport]]:
* {#KimSaemann20} [[Hyungrok Kim]], [[Christian Saemann]], *Adjusted Parallel Transport for Higher Gauge Theories*, J. Phys. A **52** (2020) 445206 [[arXiv:1911.06390](https://arxiv.org/abs/1911.06390), [doi:10.1088/1751-8121/ab8ef2](https://doi.org/10.1088/1751-8121/ab8ef2)]
Relation to [[EL-infinity algebras|$E L_\infty$-algebras]] and [[tensor hierarchies]]:
* {#BorstenKimSaemann21} [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]]. *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies* [[arXiv:2106.00108](https://arxiv.org/abs/2106.00108)]
Survey and review:
* [[Christian Saemann]], *Adjusted Higher Gauge Theory: Connections and Parallel Transport* Lisbon (2021) [[pdf](https://math.tecnico.ulisboa.pt/seminars/download.php?fid=1485), [[Saemann-AdjustedHGT.pdf:file]]]
* [[Christian Saemann]], *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies*, talk at [SFT@Cloud 2021](https://indico.cern.ch/event/1042834/) (2021) [[pdf](https://indico.cern.ch/event/1042834/contributions/4487418/attachments/2313514/3937748/saemann_generalized_geometry.pdf), [[Saemann-ELInfinityAndTensorHierarchies.pdf:file]]]
* [[Christian Saemann]], *Atiyah Algebroids for Higher and Groupoid Gauge Theories*, [talk at](M-Theory+and+Mathematics#Saemann2024) *[[M-Theory and Mathematics]]*, [[CQTS]] (2024) [[[Saemann-AtiyahAlgebroids.pdf:file]]]
Application to geometric refinement of [[topological T-duality]]:
* [[Hyungrok Kim]], [[Christian Saemann]], *T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections* [[arXiv:2303.16162](https://arxiv.org/abs/2303.16162)]
[[!redirects adjusted Weil algebras]]
|
ADM mass | https://ncatlab.org/nlab/source/ADM+mass |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[general relativity]] the _ADM mass_ is a notion of total [[mass]] contained in [[asymptotically flat spacetimes]].
## Related concepts
* [[positive energy theorem]]
## References
* Dan N. Vollick, _On the Meaning of Various Mass Definitions for Asymptotically Flat Spacetimes_ ([arXiv:2101.12570](https://arxiv.org/abs/2101.12570))
See also
* Wikipedia, _[ADM formalism](https://en.m.wikipedia.org/wiki/ADM_formalism)
|
admin > history | https://ncatlab.org/nlab/source/admin+%3E+history | |
Admir Greljo | https://ncatlab.org/nlab/source/Admir+Greljo |
* [institute page](https://theory.cern/roster/greljo-admir)
## Selected writings
On [[flavour anomalies]] and [[leptoquarks]]:
* {#GST21} [[Admir Greljo]], [[Peter Stangl]], Anders Eller Thomsen, _A Model of Muon Anomalies_ ([arXiv:2103.13991](https://arxiv.org/abs/2103.13991))
category: people |
admissible rule | https://ncatlab.org/nlab/source/admissible+rule |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
=--
=--
\tableofcontents
## Idea
An [[inference rule]] in a [[deductive system]], such as a [[logic]] or [[type theory]], is *admissible* if there is some algorithm for constructing a derivation of the conclusion from derivations of the premises. It is *not* one of the "specified" or [[primitive rules]] of the deductive system, but its admissibility means that it could be added as a primitive rule without changing the set of derivable [[judgments]].
Compare to a *[[derivable rule]]*, which is an admissible rule that is given by a "parametric" proof, or more formally by simply applying a finite number of primitive rules. In contrast, the admissibility algorithm for an admissible rule is allowed to inspect and arbitrarily deconstruct the given derivations of the premises.
Unlike primitive and derivable rules, an admissible rule need *not* be satisfied by all semantics. However, for most deductive systems there is a specified set of "important" admissible rules, and one considers only semantics that do satisfy these rules.
## Examples
* the [[cut rule]] and the identity in [[sequent calculus]]
* [[substitution]] and [[weakening rule|weakening]] in [[natural deduction]]
* [[negation]] in [[classical logic|classical]] [[linear logic]]
## Related Concepts
* [[focusing]]
* [[structural rule]]
[[!redirects admissible rules]] |
admissible subcategory | https://ncatlab.org/nlab/source/admissible+subcategory | A **right-admissible subcategory** (resp. left admissible subcategory) of a [[triangulated category]] $A$ is a strictly [[full]] [[triangulated subcategory]] $B$ such that the inclusion functor $i : B \hookrightarrow A$ admits a right adjoint $i_*$ (resp. a left adjoint $i^*$). It is called simply **admissible** if it is both right- and left-admissible.
## See also
* [[semi-orthogonal decomposition]]
* [[Verdier quotient]] |
Adolf Hurwitz | https://ncatlab.org/nlab/source/Adolf+Hurwitz |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Adolf_Hurwitz)
## Selected writings
On [[branched cover|branched]] [[Riemann surfaces]] and [[braid groups]] as [[fundamental groups]] of [[configuration spaces of points]]:
* [[Adolf Hurwitz]], *Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten*, Mathematische Annalen **39** (1891) 1–60 [[doi:10.1007/BF01199469](https://doi.org/10.1007/BF01199469)]
Proving the [[Hurwitz theorem]]:
* [[Adolf Hurwitz]], _Über die Composition der quadratischen Formen von beliebig vielen Variabeln_, Nachr. Ges. Wiss. Göttingen (1898) pp 309--316 ([GDZ pdf](http://gdz.sub.uni-goettingen.de/download/PPN252457811_1898/LOG_0034.pdf), [EuDML entry](https://eudml.org/doc/58420))
* [[Adolf Hurwitz]], _Über die Komposition der quadratischen Formen_, Math. Ann. 88 (1923) pp 1--25, doi:[10.1007/bf01448439](http://dx.doi.org/10.1007/bf01448439) ([GDZ pdf](http://gdz.sub.uni-goettingen.de/download/PPN235181684_0088/LOG_0005.pdf), [EuDML entry](https://eudml.org/doc/158975))
## Related entries
* [[Hurwitz theorem]]
* [[composition algebra]]
category: people |
Adrian Clingher | https://ncatlab.org/nlab/source/Adrian+Clingher |
* [webpage](http://math.stanford.edu/~clingher/)
## related $n$Lab entries
* [[Sen limit]]
category: people |
Adrian Clough | https://ncatlab.org/nlab/source/Adrian+Clough |
* member of [[CQTS]]
* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/christopher-adrian-clough.html) at NYU Abu Dhabi
* [personal webpage](https://adrianclough.github.io/)
## Selected writings
On pro-algebraic resolutions of [[regular schemes]]:
* [[Adrian Clough]], *Pro-algebraic Resolutions of Regular Schemes*, MSc thesis, ETC Zürich (2014) [[pdf](https://web.ma.utexas.edu/users/adrian.clough/Master_thesis.pdf)]
On [[smooth ∞-groupoids]] (called "[[differentiable stacks]]") and the [[smooth Oka principle]]:
* {#Clough21} [[Adrian Clough]], *A Convenient Category for Geometric Topology*, PhD thesis, UT Austin (2021) [[pdf](https://repositories.lib.utexas.edu/bitstream/handle/2152/114981/CLOUGH-DISSERTATION-2021.pdf), [[Clough-ConvenientCategory.pdf:file]]]
* *The homotopy theory of differentiable sheaves* [[arXiv:2309.01757](https://arxiv.org/abs/2309.01757)]
* [[Adrian Clough]], *The Homotopy Theory of Differentiable Sheaves*, [talk at](CQTS#CloughOct2023) *[Workshop on Homotopy Theory and Applications](CQTS#WorkshopOnHomotopyTheory2023)*, [[CQTS]] (Oct 2023) [video:[YT](https://youtu.be/5NrKo-fPk2A)]
category: people
|
Adrian Norbert Schellekens | https://ncatlab.org/nlab/source/Adrian+Norbert+Schellekens |
* [webpage](http://www.nikhef.nl/~t58/Site/Home.html)
* [articles on arXiv](https://arxiv.org/search/?searchtype=author&query=Schellekens%2C+A)
Bert Schellekens has been working on [[2d CFT]] and specifically [[SCFT]] in relation to [[model building]] in [[string phenomenology]]. In particular he studied [[rational conformal field theory|RCFT]]-[[Gepner model]]-[[orbifold]]-[[KK-compactification|compactifications]] of the [[heterotic string]] and of [[intersecting D-brane models]] (via [[boundary states]]) in the [[type II superstring]].
Recognizing the abundance of semi-[[standard model of particle physics|realistic models]] ([Lerche-Lüst-Schellekens 86](#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](#LercheLustSchellekens87)), Schellekens was maybe the first to recognize and definitely the first to make explicit the existence and relevance of the _[[landscape of string theory vacua]]_ and to coin that term in the first place ([Schellekens 98](#Schellekens98), see [Schellekens 06](#Schellekens06), [Schellekens 08](#Schellekens08), [Schellekens 16](#Schellekens16)) long before this became more widely appreciated.
Schellekens' scans of the "[[landscape of string theory vacua|landscape]]" of [[vertex algebra|vertex-algebraically]] well-defined [[Gepner model]] [[KK-compactification]] (in the sense of _[Spectral Standard Models and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) of the [[heterotic string]] and of [[intersecting D-brane models]] (via [[boundary states]]) of the [[type II superstring]] ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b)) showed that solid and substantial investigation into the [[landscape of string theory vacua]] is possible, worthwhile and probably necessary.
As an example, from [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b), the following is a plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]]. The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.
<center>
<img src="https://ncatlab.org/nlab/files/SchellekensSuperGepnerModelScanII.jpg" width="600">
</center>
## Selected writings
On [[heterotic string theory]]:
* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [<a href="https://doi.org/10.1016/0370-2693(86)91257-8">doi:10.1016/0370-2693(86)91257-8</a>, [inspire:233203](https://inspirehep.net/literature/233203)]
> (relation to [[Niemeier lattices]])
* {#LercheLustSchellekens87} [[Wolfgang Lerche]], [[Dieter Lüst]], [[Bert Schellekens]], _Chiral Four-dimensional Heterotic Strings from Self-dual Lattices_ Nucl. Phys. B 287, 477, 1987 ([pdf](http://lerche.web.cern.ch/lerche/papers/4dhetstrings.pdf))
* {#Schellekens91} [[Bert Schellekens]], _Classification of Ten-Dimensional Heterotic Strings_, Phys.Lett. B277 (1992) 277-284 ([arXiv:hep-th/9112006](http://arxiv.org/abs/hep-th/9112006))
On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", a little later called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:
* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies, characters and strings*, Nuclear Physics B Volume 287, 1987, Pages 317-361 (<a href="https://doi.org/10.1016/0550-3213(87)90108-8">doi:10.1016/0550-3213(87)90108-8</a>)
* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies and modular invariance in string theory*, Physics Letters B 177 (3-4), 317-323, 1986 (<a href="https://doi.org/10.1016/0370-2693(86)90760-4">doi:10.1016/0370-2693(86)90760-4</a>)
* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (<a href="https://doi.org/10.1016/0550-3213(87)90397-X">doi:10.1016/0550-3213(87)90397-X</a>)
* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (<a href="https://doi.org/10.1016/0550-3213(88)90468-3">doi:10.1016/0550-3213(88)90468-3</a>)
On classification of strongly rational holomorphic [[vertex operator algebras]] of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):
* [[Adrian Norbert Schellekens]], *Meromorphic $c = 24$ Conformal Field Theories*, Commun. Math. Phys. **153** (1993) 159-186 [[doi:10.1007/BF02099044](https://doi.org/10.1007/BF02099044)]
On [[orientifold]] [[Gepner model]] [[string phenomenology]]:
* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))
* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))
On the [[landscape of string theory vacua]]:
* {#Schellekens98} [[Bert Schellekens]], _Naar een waardig slot_, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066/18631](https://mobile.repository.ubn.ru.nl/handle/2066/18631))
* {#Schellekens06} [[Bert Schellekens]], _The Landscape "avant la lettre"_ ([arXiv:physics/0604134](http://arxiv.org/abs/physics/0604134))
* {#Schellekens08} [[Bert Schellekens]], _The Emperor's Last Clothes?_, Rept. Prog. Phys.71:072201, 2008 ([arXiv:0807.3249](https://arxiv.org/abs/0807.3249))
* {#Schellekens16} [[Bert Schellekens]], _Big Numbers in String Theory_ ([arXiv:1601.02462](http://arxiv.org/abs/1601.02462))
On [[string phenomenology]] with [[orbifold]]/[[orientifold]] [[intersecting D-brane models]] and [[Gepner models]]:
* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]
## related $n$Lab entries
* [[Gepner model]]
* [[intersecting D-brane model]]
* [[string phenomenology]], [[GUT]]
* [[landscape of string theory vacua]]
* [[discrete torsion]]
category: people
[[!redirects A. N. Schellekens]]
[[!redirects Bert Schellekens]]
|
Adrian Ocneanu | https://ncatlab.org/nlab/source/Adrian+Ocneanu | Adrian Ocneanu is a Romanian origin mathematician working in Penn State. His research is in Operator Algebras.
* [Wikipedia page](http://en.wikipedia.org/wiki/Adrian_Ocneanu)
* [Genealogy project entry](http://genealogy.math.ndsu.nodak.edu/id.php?id=3326)
category: people
[[!redirects A. Ocneanu]] |
Adriano Barenco | https://ncatlab.org/nlab/source/Adriano+Barenco |
## Selected writings
Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:
* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]
On [[quantum computation]]:
* [[Adriano Barenco]], *Quantum Physics and Computers*, Contemp. Phys. **37** (1996) 375-389 [[arXiv:quant-ph/9612014](https://arxiv.org/abs/quant-ph/9612014), [doi:10.1080/00107519608217543](https://doi.org/10.1080/00107519608217543)]
category: people
|
Adrien Bouhon | https://ncatlab.org/nlab/source/Adrien+Bouhon |
* [GoogleScholar page](https://scholar.google.com/citations?user=me2JMkUAAAAJ&hl=en)
* [institute page](https://www.su.se/english/profiles/adbo4636-1.383815)
## Selected writings
On [[braid group statistics|anyonic braiding]] of nodal points in the [[Brillouin zone]] of [[semi-metals]] ("braiding in momentum space"):
* [[Adrien Bouhon]], QuanSheng Wu, [[Robert-Jan Slager]], Hongming Weng, Oleg V. Yazyev, [[Tomáš Bzdušek]], *Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe*, Nature Physics **16** (2020) 1137--1143 ([arXiv:1907.10611](https://arxiv.org/abs/1907.10611), [doi:10.1038/s41567-020-0967-9](https://doi.org/10.1038/s41567-020-0967-9))
* Bin Jiang, [[Adrien Bouhon]], Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, [[Robert-Jan Slager]], Jian-Hua Jiang *Observation of non-Abelian topological semimetals and their phase transitions*, Nature Physics **17** (2021) 1239-1246 [[arXiv:2104.13397](https://arxiv.org/abs/2104.13397), [doi:10.1038/s41567-021-01340-x](https://doi.org/10.1038/s41567-021-01340-x)]
* Siyu Chen, [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions* (formerly: *Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions*),
Phys. Rev. B **105** (2022) L081117 $[$[arXiv:2108.10330](https://arxiv.org/abs/2108.10330), [doi:10.1103/PhysRevB.105.L081117](https://doi.org/10.1103/PhysRevB.105.L081117)$]$
* [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Multi-gap topology and non-Abelian braiding of phonons from first principles*, Phys. Rev. B **105** (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi:10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115))
* *[[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], *Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates*, Nature Communications volume 13, Article number: 423 (2022) ([doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9))
* [[Adrien Bouhon]], [[Robert-Jan Slager]], *Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs* $[$[arXiv:2203.16741](https://arxiv.org/abs/2203.16741)$]$
* [[Robert-Jan Slager]], [[Adrien Bouhon]], [[Fatma Nur Ünal]], *Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase* [[arXiv:2208.12824](https://arxiv.org/abs/2208.12824)]
* Wojciech J. Jankowski, Mohammedreza Noormandipour, [[Adrien Bouhon]], [[Robert-Jan Slager]], *Disorder-induced topological quantum phase transitions in Euler semimetals* [[arXiv:2306.13084](https://arxiv.org/abs/2306.13084)]
category: people
|
Adrien Brochier | https://ncatlab.org/nlab/source/Adrien+Brochier |
* [webpage](http://abrochier.org/index.php)
## Selected writings
On [[horizontal chord diagrams]] and [[Vassiliev braid invariants]]:
* [[Adrien Brochier]], _Cyclotomic associators and finite type invariants for tangles in the solid torus_, Algebr. Geom. Topol. 13 (2013) 3365-3409 ([arXiv:1209.0417](https://arxiv.org/abs/1209.0417))
On construction of [[4d TQFT]] via [[factorization homology]] from [[braided monoidal category|braided]] [[tensor categories]] (with relation to [[double affine Hecke algebras]]):
* [[David Ben-Zvi]], [[Adrien Brochier]], [[David Jordan]], _Integrating quantum groups over surfaces: quantum character varieties and topological field theory_ ([arXiv:1501.04652](http://arxiv.org/abs/1501.04652))
On [[4d TQFTs]] from [[fully dualizable object|fully dualizable]] [[braided monoidal category|braided]] [[tensor categories]], via the [[cobordism hypothesis]]:
* [[Adrien Brochier]], [[David Jordan]], [[Pavel Safronov]], [[Noah Snyder]], _Invertible braided tensor categories_ ([arXiv:2003.13812](https://arxiv.org/abs/2003.13812))
category: people |
Adrien Douady | https://ncatlab.org/nlab/source/Adrien+Douady |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Adrien_Douady)
category: people |
Adrien-Marie Legendre | https://ncatlab.org/nlab/source/Adrien-Marie+Legendre |
* [St. Andrews History of mathematics entry](http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Legendre.html)
## related $n$Lab entries
* [[Legendre transform]]
category:people
[[!redirects Legendre]]
|
AdS-CFT correspondence | https://ncatlab.org/nlab/source/AdS-CFT+correspondence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
### General idea
The _AdS-CFT correspondence_ at its heart is the observation ([Witten 98, Section 2.4](#Witten98)) that the [[classical field theory|classical]] [[action functionals]] for various [[field (physics)|fields]] coupled to [[Einstein gravity]] on [[anti de Sitter spacetime]] are, when expressed as [[functions]] of the [[asymptotic boundary]]-values of the [[field (physics)|fields]], of the form of [[generating functions]] for [[correlators]]/[[n-point functions]] of a [[conformal field theory]] on that [[asymptotic boundary]], in a [[large N limit]].
{#HolographicPrincipleInIntroduction} This is traditionally interpreted as a concrete realization of a vague "[[holographic principle]]" according to which [[quantum gravity]] in [[bulk]] spacetimes is controlled, in one way or other, by "[[boundary field theories]]" on effective spacetime boundaries, such as [[event horizons]]. The original and main motivation for the [[holographic principle]] itself was the fact that the apparent [[black hole entropy]] in [[Einstein gravity]] scales with the [[area]] of the [[event horizon]] instead of the black hole's bulk volume (which is not even well-defined), suggesting that gravity encodes or is encoded by some [[boundary field theory]] associated with horizons; an idea that, in turn, seems to find a concrete realization in [[open/closed string duality]] in the [[near horizon geometry|vicinity]] of, more generally, [[black branes]]. The original intuition about holographic black hole entropy has meanwhile found remarkably detailed reflection in (mathematically fairly rigorous) analysis of [[holographic entanglement entropy]], specifically via holographic [[tensor networks]], which turn out to embody key principles of the AdS/CFT correspondence in the guise of [[quantum information theory]], with concrete applications such as to [[quantum error correcting codes]].
The AdS/CFT correspondence itself crucially involves the exceptional [[isomorphism]] between the [[isometry group]] of [[anti de Sitter spacetime]] $AdS_{d+1}$ (the [[anti de Sitter group]]) and the [[conformal group]] of [[Minkowski spacetime]] of dimension $d$: the [[connected component]] of both is the [[special orthogonal group]] $SO(d,2)$. But the AdS/CFT correspondence is deeper and more subtle than this [[group theory]] underlying it, in particular in how it puts [[field (physics)|fields]] and [[states]] on the gravity side in correspondence with [[sources]] and [[correlators]] on the field theory side, respectively.
In extrapolation of these elementary computations, the AdS/CFT correspondence [[conjecture|conjecturally]] extends to a more general identification of [[states]] of [[gravity]] ([[quantum gravity]]) on asymptotically [[anti de Sitter spacetimes]] of [[dimension]] $d+1$ with [[correlators]]/[[n-point functions]] of [[conformal field theories]] on the [[asymptotic boundary]] of dimension $d$ ([Gubser-Klebanov-Polyakov 98 (12)](#GubserKlebanovPolyakov98), [Witten 98, (2.11)](#Witten98)), such that [[perturbative quantum field theory|perturbation theory]] on one side of the correspondence relates to [[non-perturbative effects|non-perturbation]] on the other side.
While this works to some extent quite generally (see e.g. [Natsuume 15](#Natsuume15) for review), allowing applications such as [[AdS/CFT in condensed matter physics]] and [[AdS/QCD|AdS/CFT in quantum chromodynamics]], the tightest form of the correspondence relates the [[1/N expansion]] of [[superconformal field theories]] ([[super Yang-Mills theories]]) on the [[asymptotic boundaries]] of [[near-horizon limits]] of $N$ coincident [[black brane|black]] [[M2-branes]]/[[D3-branes]]/[[M5-branes]] to corresponding sectors of the [[string theory]]/[[M-theory]] [[quantum gravity]] in the [[bulk]] [[spacetime]] away from the brane.
Before the proposal for the actual matching rule of ADS/CFT ([Gubser-Klebanov-Polyakov 98 (12)](#GubserKlebanovPolyakov98), [Witten 98, (2.11)](#Witten98)) it was by matching of [[BPS-states]] in these situations that the existence of an AdS/CFT correspondence was proposed in [Maldacena 97a](#Maldacena97a), [Maldacena 97b](#Maldacena97b); these articles are now widely regarded as the origin of the idea of the AdS/CFT correspondence.
A quick way to see that the [[supersymmetric]]-cases of AdS/CFT for [[near horizon geometries]] of [[M2-branes]], [[D3-branes]] and [[M5-branes]] must be special is to observe that these are the only dimensions in which there are [[super anti de Sitter spacetime]]-enhancements of [[anti de Sitter spacetime]], matching the classification of simple [[superconformal geometry|superconformal symmetries]], see [there](supersymmetry#ClassificationSuperconformal):
[[!include superconformal symmetry -- table]]
It had already been observed in ([Duff-Sutton 88](#DuffSutton88), see [Duff 98](#Duff98), [Duff 99](#Duff99)) that the field theory of small perturbation of a [[Green-Schwarz sigma-model]] for a [[fundamental brane]] stretched over the [[asymptotic boundary]] of the [[AdS spacetime|AdS]] [[near horizon geometry]] of its own [[black brane]]-incarnation is, after [[diffeomorphism]] [[gauge fixing]], a [[conformal field theory]]. This was further developed in [Claus-Kallosh-Proeyen 97](#ClausKalloshProeyen97), [DGGGTT 98](#DGGGTT98), [Claus-Kallosh-Kumar-Townsend 98](#ClausKalloshKumarTownsend98), [Pasti-Sorokin-Tonin 99](#PastiSorokinTonin99). See also at _[super p-brane -- As part of the AdS-CFT correspondence](Green-Schwarz+action+functional#AsPartOfTheAdSCFTCorrespodence)
More recently, for the archetypical case of AdS/CFT relating [[N=4 D=4 super Yang-Mills theory]] to [[type IIB string theory]] on [[super anti de Sitter spacetime]] $AdS_5 \times S^5$, fine detailed checks of the correspondence have been performed ([Beisert et al. 10](#BeisertEtAl10), [Escobedo 12](#Escobedo12)), see the section _[Checks](#Checks)_ below.
Thus regarded as a [[duality in string theory]], the AdS/CFT correspondence is an incarnation of [[open/closed string duality]], reflecting the fact that the physics on [[D-branes]] has two equivalent descriptions:
1) as a [[Yang-Mills theory|Yang-Mills]]-[[gauge theory]] coming from [[open strings]] attached to the [[brane]]
2) as a [[gravity]] theory coming from [[closed strings]] emitted/absorbed by the brane.
<center>
<img src="https://ncatlab.org/nlab/files/OpenClosedCylinderWorldsheetEndingOnBranes.jpg" width="400">
</center>
> graphics grabbed from [Schomerus 07, Figure 4](open/closed+string+duality#Schomerus07), see also e.g.
[Peschanskia 09, Figure 1](open/closed+string+duality#Peschanskia09)
This gives a vivid intuitive picture of the mechanism underlying the correspondence: An excitation of the gauge field on the brane goes along with an excitation of the field of gravity around the brane, and either is faithfully reflected in the other; at least in the suitable limits.
### Small-$N$ corrections
{#SmallNCorrections}
The AdS/CFT correspondence has been widely discussed and is mostly understood by default only in the *[[large N limit|large $N$ limit]]* and for large [['t Hooft coupling]], where the given gauge theory is dual to plain [[classical field theory|classical]] [[supergravity]], which stands out as being particularly tractable and well-understood.
But it is expected ([AGMOO99](#AharonyGubserMaldacenaOoguriOz99), [p. 60](https://arxiv.org/pdf/hep-th/9905111.pdf#page=61)) that the [[duality in string theory|duality]] still applies in the opposite [[large 1/N limit]], now involving on the [[gravity]]-side corrections
1. from [[perturbative string theory]] (for small [['t Hooft coupling]], there are some checks of such stringy corrections) and
1. from putative [[M-theory]] (for the full [[non-perturbative effect|non-perturbative]] [[large 1/N limit]], which remains largely unexplored):
{#AdSCFTForSmallN} $\,$
<center>
<div style="margin:-40px 10px 10px 10px;"><img src="https://ncatlab.org/nlab/files/AdSCFTForSmallN-230627.jpg" width="750">
</div>
</center>
> (graphics adapted from *[[schreiber:Anyonic topological order in TED K-theory|SS22]]*)
Notice that for real-world applications such as to the [[confinement]]/[[mass gap]]-problem of [[quantum chromodynamics]], the value of $N$ typically is indeed small (the number of [[color charge|colors]] in [[quantum chromodynamics]] is $N_c = 3$) so that the [[string theory]]/[[M-theory]]-corrections to the [[AdS/QCD correspondence]] are going to be crucial for the full discussion of these applications:
<center>
<a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf#page=8">
<img src="https://ncatlab.org/schreiber/files/ProblemQCDToProblemM230120.jpg" width="670">
</a>
</center>
In lack of a full formulation of [[M-theory]] (see [M-theory -- The open problem](M-theory#TheOpenProblem)) approximate forms of the AdS/CFT correspondence away from the case of [[conformal invariance]], [[supersymmetry]], [[large N limit]] and/or exact [[anti de Sitter spacetime|anti de Sitter geometry]] are being argued to be of use for understanding [[quantum chromodynamics]] (for instance the [[quark-gluon plasma]] ([Policastro-Son-Starinets 01](#PolicastroSonStarinets01), but most notably [[confinement|confined]] [[hadron]]-spectra -- the _[[AdS/QCD correspondence]]_) and for various models in [[solid state physics]] (the _[[AdS-CFT in condensed matter physics]]_, see e.g. [Hartnoll-Lucas-Sachdev 16](#HartnollLucasSachdev16)).
More in detail, since the [[near horizon geometry]] of [[BPS state|BPS]] [[black branes]] is conformal to the [[Cartesian product]] of [[anti de Sitter spaces]] with the unit $n$-sphere around the brane, the [[cosmology]] of [[intersecting D-brane models]] realizes the [[observable universe]] on the [[asymptotic boundary]] of an _approximately_ [[anti de Sitter spacetime]] (see for instance [Kaloper 04](#Kaloper04), [Flachi-Minamitsuji 09](#FlachiMinamitsuji09)). The basic structure is hence that of _[[Randall-Sundrum models]]_, but details differ, such as notably in _warped throat_ geometries, see [Uranga 05, section 18](#Uranga05).
These warped throat models go back to [Klebanov-Strassler 00](#KlebanovStrassler00) which discusses aspects of [[confinement]] in [[Yang-Mills theory]] on conincident ordinary and _[[fractional D-brane|fractional]]_ [[D3-branes]] at the [[singularity]] of a warped [[conifold]]. See also [Klebanov-Witten 98](#KlebanovWitten98)
<center>
<img src="https://ncatlab.org/nlab/files/KlebanovStrassler.jpg" width="640">
</center>
> snippet grabbed from [Uranga 05, section 18](#Uranga05)
> here: "RS"=[[Randall-Sundrum model]]; "KS"=[Klebanov-Strassler 00](#KlebanovStrassler00)
In particular this means that AdS-CFT duality applies in _some approximation_ to [[intersecting D-brane models]] (e.g. [Soda 10](#Soda10), [GHMO 16](#GHMO16)), thus allowing to compute, to some approximation, [[non-perturbative effects]] in the [[Yang-Mills theory]] on the intersecting branes in terms of [[gravity]] on the ambient warped throat $\sim$ [[anti de Sitter spacetime|AdS]] ([Klebanov-Strassler 00, section 6](#KlebanovStrassler00))
### Matching single trace observables to string excitation
The [[single trace operators]]/observables in [[conformal field theories]] such as [[super Yang-Mills theories]] play a special role in the [[AdS-CFT correspondence]]: They correspond to single [[string]] excitations on the [[AdS spacetime|AdS]]-[[supergravity]] side of the correspondence, where, curiously, the "string of characters/letters" in the argument of the trace gets literally mapped to a [[superstring]] in [[spacetime]] (see the references [below](#ReferencesRelationToStringExcitations)).
From [Polyakov 02](#Polyakov02), referring to gauge fields and their [[single trace operators]] as _letter_ and _words_, respectively:
> The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.
> These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.
From [Berenstein-Maldacena-Nastase 02](#BerensteinMaldacenaNastase02), who write $Z$ for the elementary [[field observables]] ("letters") $\mathbf{\Phi}$ above:
> In summary, the "string of $Z$s" becomes the physical string and that each $Z$ carries one unit of $J$ which is one unit of $p_+$. Locality along the worldsheet of the string comes from the fact that planar diagrams allow only contractions of neighboring operators. So the Yang Mills theory gives a [[string bit model]] where each bit is a $Z$ operator.
On the [[CFT]] side these _[[BMN operators]]_ of fixed length (of "letters") are usefully identified as [[spin chains]] which, with the [[dilatation operator]] regarded as their [[Hamiltonian]], are [[integrable systems]] ([Minahan-Zarembo 02](#MinahanZarembo02), [Beisert-Staudacher 03](#BeisertStaudacher03)).
This integrability allows a detailed matching between
* [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]]
* the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] $\times$ [[5-sphere|S5]]
under [[AdS/CFT duality]] ([Beisert-Frolov-Staudacher-Tseytlin 03](#BeisertFrolovStaudacherTseytlin03), ...). For review see [BBGK 04](#BBGK04), [Beisert et al. 10](#BeisertEtAl10).
(...)
\linebreak
[[!include Polyakov gauge-string duality -- section]]
## Checks
{#Checks}
At the heart of the duality is the observation that the classical [[action functionals]] for various [[field (physics)|fields]] coupled to [[Einstein gravity]] on [[anti de Sitter spacetime]] are, when expressed as [[functions]] of the [[asymptotic boundary]] values of the [[field (physics)|fields]], equal to the [[generating functions]] for the [[correlators]]/[[n-point functions]] of a [[conformal field theory]] on that asymptotic boundary.
These computations were laid out in [Witten 98, section 2.4 "Some sample computation"](#Witten98). These follow from elementary manipulation in [[differential geometry]] (involving neither [[supersymmetry]] nor [[string theory]]). A good exposition is in [Hartnoll-Lucas-Sachdev 16, Section 1.6](#HartnollLucasSachdev16)
For the more ambitious matching of the spectrum of the dilatation operator of [[N=4 D=4 super Yang-Mills theory]] to the corresponding spectrum of the [[Green-Schwarz superstring]] on the [[super anti de Sitter spacetime]] $AdS_5 \times S^5$ detailed checks are summarized in [Beisert et al. 10](#BeisertEtAl10), [Escobedo 12](#Escobedo12)
<center>
<img src="https://ncatlab.org/nlab/files/AdSCFTFrolovTseytlinLimitCheck.jpg" width="600">
</center>
> graphics grabbed from [Escobedo 12](#Escobedo12)
Comparison to [[string scattering amplitudes]] beyond the planar SCFT limit: [ABP 18](#ABP18).
Numerical checks using [[lattice gauge theory]] are reviewed in [Joseph 15](#Joseph15).
Exact duality checks pertaining to the full stringy regime for [[AdS3-CFT2|$AdS_3/CFT_2$]]: [Eberhardt-Gaberdiel 19a](#EberhardtGaberdiel19a), [Eberhardt-Gaberdiel 19b](#EberhardtGaberdiel19b), [Eberhardt-Gaberdiel-Gopakumar 19](#EberhardtGaberdielGopakumar19). For more see the references [there](AdS3-CFT2+and+CS-WZW+correspondence#ReferencesAdS3CFT2).
See also
* Umut Gursoy, Guim Planella Planas, *Worldsheet from worldline* [[arXiv:2311.10142](https://arxiv.org/abs/2311.10142)]
## Examples
The solutions to [[supergravity]] that preserve the maximum of 32 [[supersymmetries]] are (e.g. [HEGKS 08 (1.1)](BPS+state#HEGKS08))
* $AdS_5 \times S^5$ in [[type II supergravity]]
* $AdS_7 \times S^4$ in [[11-dimensional supergravity]]
* $AdS_4 \times S^7$ in [[11-dimensional supergravity]]
as well as their [[Minkowski spacetime]] and plane wave limits. These are the main [[KK-compactifications]] for the following examples-
### $AdS_5 / CFT_4$ -- Horizon limit of D3-branes
{#AdS5CFT4}
[[type II string theory]] on 5d [[anti de Sitter spacetime]] (times a 5-sphere) is dual to [[N=4 D=4 super Yang-Mills theory]] on the [[worldvolume]] of a [[D3-brane]] at the [[asymptotic boundary]]
([Maldacena 97, section 2](#Maldacena97a))
([Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 3 and 4](#AharonyGubserMaldacenaOoguriOz99))
### $AdS_7 / CFT_6$ -- Horizon limit of M5-branes
{#AdS7CFT6}
We list some of the conjectured statements and their evidence concerning the case of $AdS_7/CFT_6$-duality.
The hypothesis ([Maldacena 97, section 3.1](#Maldacena97a)) (see ([Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.1](#AharonyGubserMaldacenaOoguriOz99)) for a review)
is that
* the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of $N$ coincident [[M5-branes]]
is holographically related to
* [[11-dimensional supergravity]] [[Kaluza-Klein mechanism|reduced on]] the 4-[[sphere]] $S^4$ with $N$ units of [[flux]] of the [[supergravity C-field]] to [[7d supergravity]] on an asymptotically [[anti de Sitter spacetime]].
In
* {#WittenI} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ J. Geom. Phys.22:103-133,1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
effectively this relation was already used to computed the 5-brane [[partition function]] in the abelian case from the states of abelian [[7d Chern-Simons theory]]. (The quadratic refinement of the [[supergravity C-field]] necessary to make this come out right is what led to [[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]] and hence to the further mathematical development of [[differential cohomology]] and its application in physics.)
In ([Witten 98, section 4](#Witten98)) this construction is argued for from within the framework of AdS/CFT, explicitly identifying the [[7d Chern-Simons theory]] here with the compactification of the 11-dimensional Chern-Simons term of the [[supergravity C-field]] in [[11-dimensional supergravity]], which locally is
$$
\begin{aligned}
S_{11d SUGRA, CS}(C_3)
&=
\int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge G_4
\\
& = N \, \int_{AdS_7} C_3 \wedge G_4
\end{aligned}
\,.
$$
But in fact the [[quantum anomaly]] cancellation ([[Green-Schwarz mechanism|GS-type mechanism]]) for 11d sugra introduces a quantum correction to this Chern-Simons term ([DLM, equation (3.14)](\DLM)), making it locally become
$$
\begin{aligned}
S(\omega,C_3)
&=
\int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge (G_4 + I_8(\omega))
\\
& = N \, \int_{AdS_7}
\left(
C_3 \wedge G_4
+
\frac{1}{48} CS_{p_2}(\omega) -
\frac{1}{12} CS_{\frac{1}{2}p_1}(\omega)
\wedge tr(F_\omega \wedge \omega)
\right)
\end{aligned}
\,,
$$
where now $\omega$ is the local 1-form representative of a [[spin connection]] and where $CS_{p_2}$ is a [[Chern-Simons form]] for the second [[Pontryagin class]] and $CS_{\frac{1}{2}p_1}$ for the first.
That therefore not an abelian, but this _nonabelian_ [[higher dimensional Chern-Simons theory]] should be dual to the nonabelian [[6d (2,0)-superconformal QFT]] was maybe first said explicitly in ([LuWang 2010](#LuWang)).
Its gauge field is hence locally and ignoring the flux quantization subtleties a pair consisting of the abelian 3-form field $C$ and a [[Spin group]] $Spin(6,1)$-valued [[connection on a bundle|connection]] (see _[[supergravity C-field]] for global descriptions of such pairs). Or maybe rather $Spin(6,2)$ to account for the constraint that the configurations are to be asymptotic [[anti de Sitter spacetimes]] (in analogy to the well-understood situation in [[3d quantum gravity]], see there for more details).
{#CommentSezginSundell} Indeed, in ([SezginSundell 2002, section 7](#SezginSundell)) more detailed arguments are given that the 7-dimensional dual to the free 6d theory is a [[higher spin gauge theory]] for a higher spin [[gauge group]] extending the ([[superconformal group|super]]) [[conformal group]] $SO(6,2)$.
A [[non-perturbative quantum field theory|non-perturbative]] description of this nonabelian [[7d Chern-Simons theory]] as a [[local prequantum field theory]] (hence defined [[non-perturbative quantum field theory|non-perturbatively]] on the global [[moduli stack]] of [[field (physics)|fields]] ([[twisted differential string structures]], in fact)) was discussed in ([FSS 12a](#FSS12a), [FSS 12b](#FSS12b)).
General discussion of [[boundary field theory|boundary]] [[local prequantum field theories]] relating higher Chern-Simons-type and higher WZW-type theories is in ([[schreiber:differential cohomology in a cohesive topos|dcct 13, section 3.9.14]]). Specifically, a characterization along these lines of the [[Green-Schwarz action functional]] of the [[M5-brane]] as a holographic [[infinity-Wess-Zumino-Witten theory - contents|higher WZW-type]] boundary theory of a 7d Chern-Simons theory is found in ([FSS 13](#FSS13)).
Analogous discussion of the 6d theory as a higher WZW analog of a 7d Chern-Simons theory phrased in terms of [[extended quantum field theory]] is ([[4-3-2 8-7-6|Freed 12]]).
### $AdS_4 / CFT_3$ --Horizon limit of M2-branes
{#AdS4CFT3}
[[11d supergravity]]/[[M-theory]] on the asymptotic $AdS_4$
spacetime of an [[M2-brane]].
([Maldacena 97, section 3.2](#Maldacena97a), [Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.2](#AharonyGubserMaldacenaOoguriOz99), [Klebanov-Torri 10](KlebanovTorri10))
### $AdS_3 / CFT_2$ -- Horizon limit of D$p$-D($p+k$) brane bound states
{#AdS3CFT2}
(for more see at _[[AdS3-CFT2 and CS-WZW correspondence]]_)
[[D1-D5 brane system]] in [[type IIB string theory]]
([Maldacena 97, section 4](#Maldacena97a))
([Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 5](#AharonyGubserMaldacenaOoguriOz99))
[[D6-D8 brane bound state]] with [[D2-D4 brane bound state]] [[defect QFT|defects]] in [[massive type IIA string theory]]
([Dibitetto-Petri 17](#DibitettoPetri17), ...)
[[D4-D8 brane bound state]] with [[D2-D6 brane bound state]] [[defect QFT|defects]] in [[massive type IIA string theory]]
([Dibitetto-Petri 18](#DibitettoPetri18), ...)
### $AdS_2 / CFT_1$
{#AdS2CFT1}
see at _[[nearly AdS2/CFT1]]_
### Non-conformal duals
#### Horizon limit of $Dp$-branes for arbitrary $p$
([Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.3](#AharonyGubserMaldacenaOoguriOz99))
#### Horizon limit of NS5-brane
([Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.1.4](#AharonyGubserMaldacenaOoguriOz99))
#### QCD models
While all of the above horizon limits product [[super Yang-Mills theory]], one can consider certain limits of these in which they look like plain [[QCD]], at least in certain sectors. This leads to a discussion of holographic description of QCD properties that are actually experimentally observed.
([Aharony-Gubser-Maldacena-Ooguri-Oz 99, section 6.2](#AharonyGubserMaldacenaOoguriOz99))
See the _[References -- Applications -- In condensed matter physics](#ToCondensedMatterPhysics)_.
### Further gauge theories induced by compactification and twisting
[[!include gauge theory from AdS-CFT -- table]]
## Formalizations
The full formalization of AdS/CFT is still very much out of reach, but maybe mostly for lack of trying.
But see [Anderson 04](#Anderson04).
One proposal for a formalization of a toy version in the context of [[AQFT]] is [[Rehren duality]]. However, it does not seem that this actually formalizes AdS-CFT, but something else.
## Related concepts
* [[near-horizon geometry]]
* [[Fefferman-Graham ambient construction]]
* [[holographic entanglement entropy]], [[ER = EPR]]
* [[single trace operator]]
* [[duality in physics]], [[duality in string theory]]
* [[T-duality]], [[S-duality]], [[U-duality]]
* [[open/closed string duality]]
* [[KLT relations]]
* [[black hole in anti de Sitter spacetime]]
* [[S-matrix theory]]
* [[anti de Sitter gravity]]
* [[Randall-Sundrum model]]
* [[Sachdev-Ye-Kitaev model]]
* [[flat space holography]]
* [[p-adic AdS-CFT]]
* [[dS/CFT correspondence]]
[[!include table of branes]]
\linebreak
## References
[[!include Polyakov gauge-string duality -- references]]
### Original articles on AdS/CFT duality
The rough conjecture originates in:
* {#Maldacena97a} [[Juan Maldacena]], _The Large $N$ limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys. 2:231, 1998 ([hep-th/9711200](http://arxiv.org/abs/hep-th/9711200))
* {#Maldacena97b} [[Juan Maldacena]], _Wilson loops in Large $N$ field theories_, Phys. Rev. Lett. __80__ (1998) 4859 ([hep-th/9803002](http://arxiv.org/abs/hep-th/9803002))
The actual rule for matching [[bulk field theory|bulk]] [[states]] to [[generating functions]] for [[boundary field theory|boundary]] [[correlators]]/[[n-point functions]] is due to
* {#GubserKlebanovPolyakov98} [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], around (12) of: *Gauge theory correlators from non-critical string theory*, Physics Letters B **428** 105-114 (1998) [[hep-th/9802109](http://arxiv.org/abs/hep-th/9802109), <a href="https://doi.org/10.1016/S0370-2693(98)00377-3">doi:10.1016/S0370-2693(98)00377-3</a>]
* {#Witten98} [[Edward Witten]], around (2.11) of _Anti-de Sitter space and holography_, Advances in Theoretical and Mathematical Physics 2: 253–291, 1998 ([hep-th/9802150](http://arxiv.org/abs/hep-th/9802150))
See also:
* [[Tom Banks]], [[Michael Douglas]], [[Gary Horowitz]], [[Emil Martinec]], _AdS Dynamics from Conformal Field Theory_ ([arXiv:hep-th/9808016](https://arxiv.org/abs/hep-th/9808016), [spire:474214](http://inspirehep.net/record/474214))
* {#Giraldo12} Carlos Andrés Cardona Giraldo, _Correlation functions in AdS/CFT correspondence_, 2012 ([spire:1652794](inspirehep.net/record/1652794), [pdf](https://digital.bl.fcen.uba.ar/download/tesis/tesis_n5179_CardonaGiraldo.pdf))
Discussion of how [[Green-Schwarz action functionals]] for super $p$-branes in AdS target spaces induce, after [[diffeomorphism]] [[gauge fixing]], superconformal field theory on the [[worldvolumes]] (see _[[singleton representation]]_) goes back to
* {#DuffSutton88} [[Mike Duff]], C. Sutton, _The Membrane at the End of the Universe_, New Sci. 118 (1988) 67-71 ([spire:268230](http://inspirehep.net/record/268230?ln=en))
and was further developed in
* {#ClausKalloshProeyen97} Piet Claus, [[Renata Kallosh]], [[Antoine Van Proeyen]], _M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions_, Nucl.Phys. B518 (1998) 117-150 ([arXiv:hep-th/9711161](http://arxiv.org/abs/hep-th/9711161))
* {#DGGGTT98} [[Gianguido Dall'Agata]], Davide Fabbri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, _The $Osp(8|4)$ singleton action from the supermembrane_, Nucl.Phys.B542:157-194, 1999 ([arXiv:hep-th/9807115](http://arxiv.org/abs/hep-th/9807115))
* {#ClausKalloshKumarTownsend98} Piet Claus, [[Renata Kallosh]], J. Kumar, [[Paul Townsend]], [[Antoine Van Proeyen]], _Conformal Theory of M2, D3, M5 and 'D1+D5' Branes_, JHEP 9806 (1998) 004 ([arXiv:hep-th/9801206](http://arxiv.org/abs/hep-th/9801206))
* {#PastiSorokinTonin99} [[Paolo Pasti]], [[Dmitri Sorokin]], Mario Tonin, _Branes in Super-AdS Backgrounds and Superconformal Theories_ ([arXiv:hep-th/9912076](http://arxiv.org/abs/hep-th/9912076))
Review is in
* {#Duff98} [[Mike Duff]], _Anti-de Sitter space, branes, singletons, superconformal field theories and all that_, Int.J.Mod.Phys.A14:815-844,1999 ([arXiv:hep-th/9808100](https://arxiv.org/abs/hep-th/9808100))
* {#Duff99} [[Mike Duff]], _TASI Lectures on Branes, Black Holes and Anti-de Sitter Space_ ([arXiv:hep-th/9912164](https://arxiv.org/abs/hep-th/9912164))
The resulting [[superconformal algebra|super-conformal]] [[Green-Schwarz sigma-model|brane scan]]:
* {#BlencoweDuff88} [[Miles P. Blencowe]], [[Mike Duff]], _Supersingletons_, Physics letters B, **203** 3(1988) 229-236 ([cds:184143](http://cds.cern.ch/record/184143), <a href="http://dx.doi.org/10.1016/0370-2693(88)90544-8">doi:10.1016/0370-2693(88)90544-8</a>)
* {#Duff09} [[Michael Duff]], _Near-horizon brane-scan revived_, Nucl. Phys. B 810:193-209, 2009 ([arXiv:0804.3675](http://arxiv.org/abs/0804.3675))
* [[Michael Duff]], *The conformal brane-scan: an update* ([arXiv:2112.13784](https://arxiv.org/abs/2112.13784))
See also at _[super p-brane -- As part of the AdS-CFT correspondence](Green-Schwarz+action+functional#AsPartOfTheAdSCFTCorrespodence)_.
Sketch of a derivation of AdS/CFT:
* {#Nastase18} [[Horatiu Nastase]], _Towards deriving the AdS/CFT correspondence_ ([arXiv:1812.10347](https://arxiv.org/abs/1812.10347))
* [[Ofer Aharony]], [[Shai Chester]], [[Erez Urbach]], _A Derivation of AdS/CFT for Vector Models_ ([arXiv:2011.06328](https://arxiv.org/abs/2011.06328))
Further references include:
* [[Edward Witten]], _Three-dimensional gravity revisited_, [arxiv/0706.3359](http://arxiv.org/abs/0706.3359)
* C.R. Graham, [[Edward Witten]], _Conformal anomaly of submanifold observables in AdS/CFT correspondence_, [hepth/9901021](http://arxiv.org/abs/hep-th/9901021).
* [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))
Via [[exceptional field theory]]:
* Oscar Varela, _Super-Chern-Simons spectra from Exceptional Field Theory_ ([arXiv:2010.09743](https://arxiv.org/abs/2010.09743))
### Introductions and surveys
Surveys and introductions:
* [[Edward Witten]], *Baryons and Branes in Anti de Sitter Space* talk at *[String98](https://online.kitp.ucsb.edu/online/strings98/)* (1998) [[web](https://online.kitp.ucsb.edu/online/strings98/witten/)]
> (by the title, apparently originally intended to touch on [[AdS-QCD]], but de facto ending with focus on the problem of [[flat space holography]])
* [[Alexander Polyakov]], *The wall of the cave*, Int. J. Mod. Phys. A **14** (1999) 645-658 [[arXiv:hep-th/9809057](https://arxiv.org/abs/hep-th/9809057), [doi:10.1142/S0217751X99000324](https://doi.org/10.1142/S0217751X99000324)]
> (early account with focus on [[AdS-QCD duality]])
* [[John Schwarz]], *Introduction to M Theory and AdS/CFT Duality*, in: *Quantum Aspects of Gauge Theories, Supersymmetry and Unification* Lecture Notes in Physics **525**, Springer (1999) [[arXiv:hep-th/9812037](https://arxiv.org/abs/hep-th/9812037), [doi:10.1007/BFb0104239](https://doi.org/10.1007/BFb0104239)]
> (early review with an eye towards [[M-theory]])
* [[Jens L. Petersen]], _Introduction to the Maldacena Conjecture on AdS/CFT_, Int. J. Mod. Phys. A **14** (1999) 3597-3672 [[hep-th/9902131](http://arxiv.org/abs/hep-th/9902131), [doi:10.1142/S0217751X99001676](http://dx.doi.org/10.1142/S0217751X99001676)]
* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. **323** 183-386 (2000) $[$<a href="https://doi.org/10.1016/S0370-1573(99)00083-6">doi:10.1016/S0370-1573(99)00083-6</a>, [arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111)$]$
* {#Anderson04} Michael T. Anderson, _Geometric aspects of the AdS/CFT correspondence_ ([arXiv:hep-th/0403087](https://arxiv.org/abs/hep-th/0403087))
* [[Horatiu Nastase]], _Introduction to AdS-CFT_ ([arXiv:0712.0689](http://arxiv.org/abs/0712.0689))
* [[Horatiu Nastase]], _Introduction to AdS/CFT correspondence_, Cambridge University Press, 2015 ([cds:1984145](http://cds.cern.ch/record/1984145), [doi:10.1017/CBO9781316090954](https://doi.org/10.1017/CBO9781316090954))
* [[Jan de Boer]], _Introduction to AdS/CFT correspondence_, [pdf](http://www-library.desy.de/preparch/desy/proc/proc02-02/Proceedings/pl.6/deboer_pr.pdf)
* [[Gary Horowitz]], [[Joseph Polchinski]], _Gauge/gravity duality_ ([gr-qc/0602037](http://arxiv.org/abs/gr-qc/0602037))
* [[Joseph Polchinski]], _Introduction to Gauge/Gravity Duality_ ([arXiv:1010.6134](https://arxiv.org/abs/1010.6134))
* {#Natsuume15} [[Makoto Natsuume]], _AdS/CFT Duality User Guide_, Lecture Notes in Physics 903, Springer 2015 ([arXiv:1409.3575](https://arxiv.org/abs/1409.3575))
* Sebastian De Haro, Daniel R. Mayerson, [[Jeremy Butterfield]], _Conceptual Aspects of Gauge/Gravity Duality_, Foundations of Physics (2016), 46 (11), pp. 1381-1425 ([arXiv:1509.09231](https://arxiv.org/abs/1509.09231))
* Johanna Erdmenger, _Introduction to Gauge/Gravity Duality_, PoS (TASI2017) 001 ([arXiv:1807.09872](https://arxiv.org/abs/1807.09872))
* Nirmalya Kajuri, _ST4 Lectures on Bulk Reconstruction_ ([arXiv:2003.00587](https://arxiv.org/abs/2003.00587))
See also
* Wikipedia, _[AdS/CFT correspondence](http://en.wikipedia.org/wiki/AdS/CFT_correspondence)_
Review of [[Yangian]] symmetry:
* [[Alessandro Torrielli]], _Yangians, S-matrices and AdS/CFT_, J. Phys. A44: 263001, 2011 ([arXiv:1104.2474](http://arxiv.org/abs/1104.2474))
### Lattice gauge theory computations
Review of [[lattice gauge theory]]-numerics for the [[AdS-CFT correspondence]]:
* {#Joseph15} Anosh Joseph, _Review of Lattice Supersymmetry and Gauge-Gravity Duality_
([arXiv:1509.01440](https://arxiv.org/abs/1509.01440))
Using the [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] to the [[BMN matrix model]] for [[lattice gauge theory]]-computations in [[D=4 N=4 SYM]] and for numerical checks of the [[AdS-CFT correspondence]]:
* {#HIKNT13} Masazumi Honda, [[Goro Ishiki]], Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, _Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory_, JHEP 1311 (2013) 200 ([arXiv:1308.3525](https://arxiv.org/abs/1308.3525))
### On single trace operators
The correspondence of [[single trace operators]] to [[superstring]] excitations under the [[AdS-CFT correspondence]] originates with these articles:
* {#Polyakov02} [[Alexander Polyakov]], _Gauge Fields and Space-Time_, Int. J. Mod. Phys. A17S1 (2002) 119-136 [[arXiv:hep-th/0110196](https://arxiv.org/abs/hep-th/0110196), [doi:10.1142/S0217751X02013071](https://doi.org/10.1142/S0217751X02013071)]
* {#BerensteinMaldacenaNastase02} [[David Berenstein]], [[Juan Maldacena]], [[Horatiu Nastase]], _Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills_, JHEP 0204 (2002) 013 ([arXiv:hep-th/0202021](https://arxiv.org/abs/hep-th/0202021))
(whence "[[BMN operators]]")
* [[Steven Gubser]], [[Igor Klebanov]], [[Alexander Polyakov]], *A semi-classical limit of the gauge/string correspondence*, Nucl. Phys. B **636** (2002) 99-114 [[arXiv:hep-th/0204051](https://arxiv.org/abs/hep-th/0204051), <a href="https://doi.org/10.1016/S0550-3213(02)00373-5">doi:10.1016/S0550-3213(02)00373-5</a>]
* {#Kruczenski04} [[Martin Kruczenski]], _Spiky strings and single trace operators in gauge theories_, JHEP 0508:014, 2005 ([arXiv:hep-th/0410226](https://arxiv.org/abs/hep-th/0410226))
The identification of the relevant [[single trace operators]] with [[integrable system|integrable]] [[spin chains]] is due to
* {#MinahanZarembo02} J. A. Minahan, [[Konstantin Zarembo]], _The Bethe-Ansatz for $N=4$ Super Yang-Mills_, JHEP 0303 (2003) 013 ([arXiv:hep-th/0212208](https://arxiv.org/abs/hep-th/0212208))
* {#BeisertStaudacher03} [[Niklas Beisert]], [[Matthias Staudacher]], _The $\mathcal{N}=4$ SYM Integrable Super Spin Chain_,
Nucl. Phys. B670:439-463, 2003 ([arXiv:hep-th/0307042](https://arxiv.org/abs/hep-th/0307042))
which led to more detailed matching of [[single trace operators]] to rotating string excitations in
* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))
Review includes
* {#BBGK04} A. V. Belitsky, [[Volker Braun]], A. S. Gorsky, G. P. Korchemsky, _Integrability in QCD and beyond_, Int. J. Mod. Phys. A19:4715-4788, 2004 ([arXiv:hep-th/0407232](https://arxiv.org/abs/hep-th/0407232))
* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))
[[!include AdS2-CFT1 -- references]]
### On $AdS_3 / CFT_2$
{#ReferencesAdS3CFT2}
(For more see the references at _[[AdS3/CFT2]]_.)
An exact correspondence of the symmetric [[orbifold]] [[CFT]] of [[Liouville theory]] with a string theory on $AdS_3$ is claimed in:
* {#EberhardtGaberdiel19a} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _String theory on $AdS_3$ and the symmetric orbifold of Liouville theory_ ([arXiv:1903.00421](https://arxiv.org/abs/1903.00421))
* {#EberhardtGaberdiel19b} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Strings on $AdS_3 \times S^3 \times S^3 \times S^1$_ ([arXiv:1904.01585](https://arxiv.org/abs/1904.01585))
* {#EberhardtGaberdielGopakumar19} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _Deriving the $AdS_3/CFT_2$ Correspondence_ ([arXiv:1911.00378](https://arxiv.org/abs/1911.00378))
* Andrea Dei, [[Lorenz Eberhardt]], _Correlators of the symmetric product orbifold_ ([arXiv:1911.08485](https://arxiv.org/abs/1911.08485))
based on
* Shouvik Datta, [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$_ JHEP 1801 (2018) 146 ([arXiv:1709.06393](https://arxiv.org/abs/1709.06393))
See also
* Stefano Speziali, _Spin 2 fluctuations in 1/4 BPS AdS3/CFT2_ ([arxiv:1910.14390](https://arxiv.org/abs/1910.14390))
* [[Lorenz Eberhardt]], _$AdS_3/CFT_2$ at higher genus_ ([arXiv:2002.11729](https://arxiv.org/abs/2002.11729))
* [[Lorenz Eberhardt]], _Summing over Geometries in String Theory_ ([arXiv:2102.12355](https://arxiv.org/abs/2102.12355))
On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:
* {#DibitettoPetri17} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arxiv:1707.06154](https://arxiv.org/abs/1707.06154))
* [[Nicolò Petri]], section 6.5 of: _Supersymmetric objects in gauged supergravities_ ([arxiv:1802.04733](https://arxiv.org/abs/1802.04733))
* {#Petri18} [[Nicolò Petri]], _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))
* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))
* Yolanda Lozano, Niall T. Macpherson, Carlos Nunez, Anayeli Ramirez, $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))
* Yolanda Lozano, Niall T. Macpherson, Carlos Nunez, Anayeli Ramirez, _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))
* Yolanda Lozano, Niall T. Macpherson, Carlos Nunez, Anayeli Ramirez, _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))
* Kostas Filippas, _Non-integrability on $AdS_3$ supergravity_ ([arxiv:1910.12981](https://arxiv.org/abs/1910.12981))
On [[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:
* {#DibitettoPetri18} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))
* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))
### On $AdS_4 / CFT_3$
* {#KlebanovTorri10} [[Igor Klebanov]], Giuseppe Torri, _M2-branes and AdS/CFT_, Int.J.Mod.Phys.A25:332-350,2010 ([arXiv:0909.1580](http://arxiv.org/abs/0909.1580))
* Kazuo Hosomichi, _M2-branes and AdS/CFT: A Review_ ([arXiv:2003.13914](https://arxiv.org/abs/2003.13914))
* Silvia Penati, _Exact Results in AdS4/CFT3_ ([arXiv:2004.00841](https://arxiv.org/abs/2004.00841))
### On $AdS_5 / CFT_4$
* [Beisert et. al. 10](#BeisertEtAl10)
* {#Escobedo12} Jorge Escobedo, _Integrability in AdS/CFT: Exact Results for Correlation Functions_, 2012 ([spire:1264432](http://inspirehep.net/record/1264432))
Computing dual [[string scattering amplitudes]] by AdS/CFT beyond the [[planar limit]]:
* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))
### On $AdS_7 / CFT_6$
We list references specific to $AdS_7/CFT_6$.
In
* {#WittenI} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ J. Geom. Phys.22:103-133,1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
* {#Witten98} [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ JHEP 9812:012,1998 ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))
it is argued that the [[conformal blocks]] of the [[6d (2,0)-superconformal QFT]] are entirely controled just by the effective [[higher dimensional Chern-Simons theory|7d Chern-Simons theory]] inside [[11-dimensional supergravity]], but only the abelian piece is discussed explicitly.
The fact that this Chern-Simons term is in fact a _nonabelian_ [[higher dimensional Chern-Simons theory]] in $d = 7$, due the [[quantum anomaly]] cancellation, is clear from the original source, equation (3.14) of
* {#DLM} [[Michael Duff]], [[James Liu]], [[Ruben Minasian]], _Eleven Dimensional Origin of String/String Duality: A One Loop Test_ ([arXiv:hep-th/9506126](http://arxiv.org/abs/hep-th/9506126))
but seems not to be noted explicitly in the context of $AdS_7/CFT_6$ before the references
* H. Lü, Yi Pang, _Seven-Dimensional Gravity with Topological Terms_ Phys.Rev.D81:085016 (2010) ([arXiv:1001.0042](http://arxiv.org/abs/1001.0042))
* {#LuWang} H. Lu, Zhao-Long Wang, _On M-Theory Embedding of Topologically Massive Gravity_ Int.J.Mod.Phys.D19:1197 (2010) ([arXiv:1001.2349](http://arxiv.org/abs/1001.2349))
More on the relation between the [[M5-brane]] and supergravity on $AdS_7 \times S^4$ and arguments for the $SO(5)$ [[R-symmetry]] group on the 6d theory from the 7d theory are given in
* [[Alexei Nurmagambetov]], I. Y. Park, _On the M5 and the AdS7/CFT6 Correspondence_, Phys. Lett. B524 (2002) 185-191 ([arXiv:hep-th/0110192](http://arxiv.org/abs/hep-th/0110192))
See also
* M. Nishimura, Y. Tanii, _Local Symmetries in the $AdS_7/CFT_6$ Correspondence_, Mod. Phys. Lett. A14 (1999) 2709-2720 ([arXiv:hep-th/9910192](http://arxiv.org/abs/hep-th/9910192))
Discussion of the $CFT_6$ in $AdS_7/CFT_6$ via [[conformal bootstrap]]:
* [[Shai Chester]], [[Eric Perlmutter]], _M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture_, J. High Energ. Phys. (2018) 2018: 116 ([arXiv:1805.00892](https://arxiv.org/abs/1805.00892))
* [[Luis Alday]], [[Shai Chester]], Himanshu Raj, _6d $(2,0)$ and M-theory at 1-loop_ ([arXiv:2005.07175](https://arxiv.org/abs/2005.07175))
In
* {#SezginSundell} [[Ergin Sezgin]], P. Sundell, _Massless Higher Spins and Holography_ ([hep-th/0205131](http://arxiv.org/abs/hep-th/0205131))
arguments are given that the 7d theory is a [[higher spin gauge theory]] extension of $SO(6,2)$.
### Generalization beyond exact AdS / exact CFT
Discussion for [[cosmology]] of [[intersecting D-brane models]] (ambient $\sim$ [[anti de Sitter spacetimes]] with the $\sim$ conformal intersecting branes at the [[asymptotic boundary]]) includes the following (see also at _[[Randall-Sundrum model]]_):
* {#KlebanovStrassler00} [[Igor Klebanov]], [[Matthew Strassler]], _Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi^{SB}$-Resolution of Naked Singularities_, JHEP 0008:052, 2000 ([arXiv:hep-th/0007191](https://arxiv.org/abs/hep-th/0007191))
* {#KlebanovWitten98} [[Igor Klebanov]], [[Edward Witten]], _Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity_, Nucl.Phys.B536:199-218, 1998 ([arXiv:hep-th/9807080](https://arxiv.org/abs/hep-th/9807080))
* {#Kaloper04} Nemanja Kaloper, _Origami World_, JHEP 0405 (2004) 061 ([arXiv:hep-th/0403208](https://arxiv.org/abs/hep-th/0403208))
* {#Uranga05} [[Angel Uranga]], section 18 of _TASI lectures on String Compactification, Model Building, and Fluxes_, 2005 ([pdf](http://cds.cern.ch/record/933469/files/cer-002601054.pdf))
* {#FlachiMinamitsuji09} Antonino Flachi, Masato Minamitsuji, _Field localization on a brane intersection in anti-de Sitter spacetime_, Phys.Rev.D79:104021, 2009 ([arXiv:0903.0133](https://arxiv.org/abs/0903.0133))
* {#Soda10} Jiro Soda, _AdS/CFT on the brane_, Lect.Notes Phys.828:235-270, 2011 ([arXiv:1001.1011](https://arxiv.org/abs/1001.1011))
* {#Teraguchi07} Shunsuke Teraguchi, around slide 21 _String theory and its relation to particle physics_, 2007 ([pdf](http://phys.cts.ntu.edu.tw/ppp7/talks/PPP7_Shunsuke_Teraguchi.pdf))
* {#GHMO16} Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, _The Born-Infeld/Gravity Correspondence_, Phys. Rev. D 94, 066009 (2016) ([arXiv:1602.01640](https://arxiv.org/abs/1602.01640))
[[!include pp-waves as Penrose limits of AdS spacetimes -- references]]
### Applications to physics
{#Appications}
#### To gravity
Discussion of [[event horizons]] of [[black holes]] in terms of AdS/CFT (the "[[firewall problem]]") is in
* Kyriakos Papadodimas, Suvrat Raju, _An Infalling Observer in AdS/CFT_ ([arXiv:1211.6767](http://arxiv.org/abs/1211.6767))
To [[black hole]] interiors:
* [[Juan Maldacena]], _Toy models for black holes II_, talk at PiTP 2018 _From QBits to spacetime_ ([recording](https://video.ias.edu/PiTP/2018/0726-JuanMaldacena))
> The [[SYK model]] gives us a glimpse into the interior of an [[extremal black hole]]...That's the feature of SYK that I find most interesting...It is a feature this model has, that I think no other model has
To [[symmetries]] in gravity:
* {#HarlowOoguri18} [[Daniel Harlow]], [[Hirosi Ooguri]], _Constraints on symmetry from holography_, Phys. Rev. Lett. 122, 191601, 2019 ([arXiv:1810.05337](https://arxiv.org/abs/1810.05337), [doi:10.1103/PhysRevLett.122.191601](https://doi.org/10.1103/PhysRevLett.122.191601))
#### To the quark-gluon plasma
Applications of AdS-CFT to the [[quark-gluon plasma]] of [[QCD]]:
Expositions and reviews include
* Pavel Kovtun, _Quark-Gluon Plasma and String Theory_, RHIC news (2009) ([blog entry](http://www.bnl.gov/rhic/news/091107/story2.asp))
* Makoto Natsuume, _String theory and quark-gluon plasma_ ([arXiv:hep-ph/0701201](http://arxiv.org/abs/hep-ph/0701201))
* [[Steven Gubser]], _Using string theory to study the quark-gluon plasma: progress and perils_ ([arXiv:0907.4808](http://arxiv.org/abs/0907.4808))
* {#Biagazzi12} Francesco Biagazzi, A. l. Cotrone, _Holography and the quark-gluon plasma_, AIP Conference Proceedings 1492, 307 (2012) ([doi:10.1063/1.4763537]( https://doi.org/10.1063/1.4763537), [slides pdf](http://cp3-origins.dk/content/movies/2013-01-14-bigazzi.pdf))
* {#Brambilla14} Brambilla et al., section 9.2.2 of _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981 ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))
Holographic discussion of the [[shear viscosity]] of the quark-gluon plasema goes back to
* {#PolicastroSonStarinets01} [[Giuseppe Policastro]], D.T. Son, A.O. Starinets, _Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma_, Phys. Rev. Lett.87:081601, 2001 ([arXiv:hep-th/0104066](http://arxiv.org/abs/hep-th/0104066))
Other original articles include:
* Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, _Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma_ ([arXiv:1003.1138](http://arxiv.org/abs/1003.1138))
* Brett McInnes, _Holography of the Quark Matter Triple Point_ ([arXiv:0910.4456](http://arxiv.org/abs/0910.4456))
#### To particle physics
* [[Joseph Polchinski]], [[Matthew Strassler]], _Hard Scattering and Gauge/String Duality_, Phys. Rev. Lett. 88:031601, 2002, ([arXiv:hep-th/0109174](http://lanl.arxiv.org/abs/hep-th/0109174))
For more see at _[[AdS/QCD correspondence]]_.
#### To fluid dynamics
Application to [[fluid dynamics]] -- see also at _[[fluid/gravity correspondence]]_:
* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))
#### To condensed matter physics
{#ToCondensedMatterPhysics}
On [[AdS-CFT in condensed matter physics]]:
Textbook account
* {#HartnollLucasSachdev16} [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], _Holographic quantum matter_, MIT Press 2018 ([arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [publisher](https://mitpress.ublish.com/book/holographic-quantum-matter))
Further reviews include the following:
* A S T Pires, _Ads/CFT correspondence in condensed matter_ ([arXiv:1006.5838](http://arxiv.org/abs/1006.5838))
* [[Subir Sachdev]], _Condensed matter and AdS/CFT_ ([arXiv:1002.2947](http://arxiv.org/abs/1002.2947))
* Yuri V. Kovchegov, _AdS/CFT applications to relativistic heavy ion collisions: a brief review_ ([arXiv:1112.5403](http://arxiv.org/abs/1112.5403))
* Alberto Salvio, _Superconductivity, Superfluidity and Holography_ ([arXiv:1301.0201](http://arxiv.org/abs/1301.0201))
* _[Holography and Extreme Chromodynamics](http://igfae.usc.es/~holoquark2018/)_, Santiago de Compostela, July 2018
### Applications in mathematics
#### To the volume conjecture
Suggestion that the statement of the [[volume conjecture]] is really [[AdS-CFT duality]] combined with the [[3d-3d correspondence]] for [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:
* {#GangKimLee14b} Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, Section 3.2_Holography of 3d-3d correspondence at Large $N$_, JHEP04(2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))
* {#GangKim18} Dongmin Gang, [[Nakwoo Kim]], around (21) of: _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))
#### To deep learning in neural networks
On the [[deep learning]] algorithm on [[neural networks]] as analogous to the [[AdS/CFT correspondence]]:
* Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, _Machine Learning Spatial Geometry from Entanglement Features_, Phys. Rev. B 97, 045153 (2018) ([arxiv:1709.01223](https://arxiv.org/abs/1709.01223))
* W. C. Gan and F. W. Shu, _Holography as deep learning_, Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) ([arXiv:1705.05750](https://arxiv.org/abs/1705.05750))
* J. W. Lee, _Quantum fields as deep learning_ ([arXiv:1708.07408](https://arxiv.org/abs/1708.07408))
* [[Koji Hashimoto]], Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, _Deep Learning and AdS/CFT_, Phys. Rev. D 98, 046019 (2018) ([arxiv:1802.08313](https://arxiv.org/abs/1802.08313))
### Philosophy of AdS/CFT
* Radin Dardashti, Richard Dawid, Sean Gryb, Karim Thébault (2020). *On the Empirical Consequences of the AdS/CFT Duality*. In Nick Huggett, Keizo Matsubara, and Christian Wüthrich (Eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 284-303). Cambridge: Cambridge University Press. ([doi:10.1017/9781108655705.016](https://doi.org/10.1017/9781108655705.016), [arXiv:1810.00949](https://arxiv.org/abs/1810.00949))
[[!redirects AdS/CFT]]
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|
AdS-CFT in condensed matter physics | https://ncatlab.org/nlab/source/AdS-CFT+in+condensed+matter+physics |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[AdS-CFT correspondence]] applies _exactly_ only to a few highly symmetric [[quantum field theories]], notably to [[N=4 D=4 super Yang-Mills theory]]. However, it still applies in adjusted form when moving away from these special points in field theory space (e.g [Park 2022](#Park22)).
For instance [[quantum chromodynamics]] (one sector of the [[standard model of particle physics]]) is crucially different from but still similar enough to [[N=4 D=4 super Yang-Mills theory]] that some of its [[observables]], in particular otherwise intractable [[non-perturbative effects]], have been argued to be usefully approximated by [[AdS-CFT]]-type dual [[gravity]]-observables, for instance the [[shear viscosity]] of the [[quark-gluon plasma]]. This is hence also called the _[[AdS/QCD correspondence]]_.
Similarly, as far as systems in [[condensed matter physics]] are described well by some [[effective field theory]], one may ask whether this, in turn, is usefully related to [[gravity]] on some [[anti de-Sitter spacetime]] and use this to study the solid state system, notably its [[non-perturbative effects]]. This area goes under the name _AdS/CMT_.
[[Andreas Karch]] writes [here](http://www.math.columbia.edu/~woit/wordpress/?p=9426#comment-226376):
> These anomalous transport coefficients have first been calculated in AdS/CFT. The relevant references are [8], [9] and [10] in the Son/Surowka paper. In the AdS/CFT calculations these particular transport coefficients only arise due to Chern-Simons terms, which are the bulk manifestation of the field theory anomalies. At that point it was obvious to many of us that there should be a purely field theory based calculation, only using anomalies, that can derive these terms. Son and Surowka knew about this. They were sitting next door to me when they started these calculations. Many of us tried to find these purely field theory based arguments and failed. Son and Surowka succeeded.
> If you ask anyone serious about applying AdS/CFT to strongly coupled field theories why they are doing this, they would (hopefully) give you an answer along the lines of "AdS/CFT provides us with toy models of strongly coupled dynamics. While the field theories that have classical AdS duals are rather special, we can still learn important qualitative insights and find new ways to think about strongly coupled field theories." Once AdS/CFT stumbles on a new phenomenon in these solvable toy models, we want to go back to see whether we can understand it without the crutch of having to rely on AdS/CFT. Any result that only applies in theories with holographic dual is somewhat limited in its applications. In this sense, anomalous transport is a poster child for what AdS/CFT can be used for: a new phenomenon that had been missed completely by people studying field theory gets uncovered by studying these toy models. Once we knew what to look for, a purely field theoretic argument was found that made the AdS/CFT derivation obsolete.
> This is applied AdS/CFT as it should be. Solvable examples exhibit new connections which then can be proven to be correct more generally and are not limited to the toy models that were used to uncover them.
Similarly [Hartnoll-Lucas-Sachdev 16, p. 8](#HartnollLucasSachdev16):
> Our main objective here
is to make clear that explicit examples of the duality are
known in various dimensions and that they are found by
using string theory as a bridge between quantum field
theory and gravity.
## Properties
### Description by tensor networks
Discussion of [[renormalization]] and [[entangled states]] of [[non-perturbative effects]] in [[solid state physics]] proceeds via [[tensor networks]] ([Swingle 09](#Swingle09), [Swingle 13](#Swingle13)) and the resulting discovery of the relation to [[holographic entanglement entropy]].
In this context, a _[[tensor network]]_ is a _[[string diagram]] [[concept with an attitude|with an attitude]], in that it is (just) a [[string diagram]], but with
1. the [[tensor product]] of all its external [[objects]] regarded as a [[space of states]] of a [[quantum system]];
1. the [[element]] in that [[tensor product]] defined by the string diagram regarded a a [[state]] ([[wave function]]) of that quantum system.
For instance, if $\mathfrak{g}$ is a [[metric Lie algebra]] (with [[string diagram]]-notation as shown [there](metric+Lie+algebra#Definition)), and with each [[tensor product]]-power of its [[dual vector space]] regarded as [[Hilbert space]], hence as a [[space of quantum states]], via the given [[inner product]] on $\mathfrak{g}$, then an example of a _tree tensor network state_ is the following:
<center>
<img src="https://ncatlab.org/nlab/files/TensorNetworkStateFromMetricLieAlgebra.jpg" width="300">
</center>
The [[quantum states]] arising this way are generically highly [[entangled state|entangled]]: roughly they are the more entangled the more [[vertices]] there are in the corresponding tensor network.
Tree tensor network states in the form of [[Bruhat-Tits trees]] play a special role in the [[AdS/CFT correspondence]], either as
1. a kind of [[lattice QFT]]-approximation to an actual [[boundary field theory|boundary]] [[CFT]] [[quantum state]],
1. as the [[p-adic geometry|p-adic geometric]] incarnation of [[anti de Sitter spacetime]] in the [[p-adic AdS/CFT correspondence]],
1. as a reflection of actual [[crystal]]-site quantum states in [[AdS/CFT in solid state physics]]:
<center>
<img src="https://ncatlab.org/nlab/files/BruhatTitsTreeTensorNetworkStateFromMetricLie.jpg" width="300">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
For [[Bruhat-Tits tree]] [[tensor network states]] one finds that the [[holographic entanglement entropy]] of the tensor subspace associated with an [[interval]] on the [[boundary]] becomes proportional, for large number of [[vertices]], to the [[hyperbolic space|hyperbolic]] bulk boundary [[length]] of the segment of the tree network that ends on this interval, according to the [[Ryu-Takayanagi formula]] ([PYHP 15, Theorem 2](#PYHP15)). For more on this see [below](#ForHolographicEntanglementEntropy).
## Related concepts
* [[strange metal]] (as opposed to [[Landau's Fermi liquid theory]])
* [[superconductor]]
* [[Sachdev-Ye-Kitaev model]]
* [[quark-gluon plasma]]
## References
### General
Textbooks:
* {#ZaanenLiuSunSchalm15} [[Jan Zaanen]], [[Yan Liu]], Ya-Wen Sun, [[Koenraad Schalm]], *Holographic Duality in Condensed Matter Physics*, Cambridge University Press 2015 [[doi:10.1017/CBO9781139942492](https://doi.org/10.1017/CBO9781139942492)]
* {#HartnollLucasSachdev16} [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], _Holographic quantum matter_, MIT Press (2018) [[arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [ISBN:9780262348010](https://mitpress.ublish.com/book/holographic-quantum-matter)]
Reviews and lectures:
* [[Sean Hartnoll]], _Lectures on holographic methods for condensed matter physics_, Class. Quant. Grav. **26** 224002, 2009 [[arXiv:0903.3246](https://arxiv.org/abs/0903.3246)]
* [[John McGreevy]], *Holographic duality with a view toward many-body physics*, Adv. High Energy Phys. 723105 (2010) [[arXiv:0909.0518](https://arxiv.org/abs/0909.0518), [doi:10.1155/2010/723105](https://doi.org/10.1155/2010/723105)]
* A. Pires, _AdS/CFT correspondence in condensed matter_ ([arXiv:1006.5838](http://arxiv.org/abs/1006.5838))
* [[Subir Sachdev]], _Condensed matter and AdS/CFT_ ([arXiv:1002.2947](http://arxiv.org/abs/1002.2947))
* K. Schalm and R. Davison, _A simple introduction to AdS/CFT
and its application to condensed matter physics_, D-ITP Advanced Topics in Theoretical Physics Fall 2013, ([[SchalmDavisonAdSCFT.pdf:file]])
* [[Matteo Baggioli]], *Gravity, holography and applications to condensed matter* $[$[arXiv:1610.02681](https://arxiv.org/abs/1610.02681)$]$
* {#Amoretti17} [[Andrea Amoretti]], _Condensed Matter Applications of AdS/CFT: Focusing on strange metals_, 2017 ([spire:1610363](http://inspirehep.net/record/1610363), [[AmorettiStrangeMetals.pdf:file]])
* [[Horatiu Nastase]], *String Theory Methods for Condensed Matter Physics*, Cambridge University Press (2017) [[doi:10.1017/9781316847978](https://doi.org/10.1017/9781316847978) ]
* [[Hong Liu]], [[Julian Sonner]], _Quantum many-body physics from a gravitational lens_, Nature Rev. Phys. **2** (2020) 615-633 [[arXiv:2004.06159](https://arxiv.org/abs/2004.06159), [doi:10.1038/s42254-020-0225-1](https://doi.org/10.1038/s42254-020-0225-1)]
* Mike Blake, Yingfei Gu, [[Sean A. Hartnoll]], Hong Liu, [[Andrew Lucas]], Krishna Rajagopal, [[Brian Swingle]], [[Beni Yoshida]], *Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes* [[arXiv:2203.04718](https://arxiv.org/abs/2203.04718)]
* [[Subir Sachdev]], *Statistical mechanics of strange metals and black holes* ([arXiv:2205.02285](https://arxiv.org/abs/2205.02285))
* [[Umut Gürsoy]], *Recent developments in gauge-gravity duality applied to quantum many-body systems*, talk at *[[Strings 2022]]* [[indico:4940863](https://indico.cern.ch/event/1085701/contributions/4940863), [pdf](https://indico.cern.ch/event/1085701/contributions/4940863/attachments/2483542/4263813/Slides_Gursoy.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=69bc5a22-b77a-4830-97b8-49c7d6aa1c29) ]
See also:
* [[Jan Zaanen]], _[Anti-de-Sitter/Condensed Matter Theory](https://www.lorentz.leidenuniv.nl/zaanen/wordpress/research/anti-de-sittercondensed-matter/)_
On Lifshitz holography relevant fordescribing disorder systems:
* {#Park22} Chanyong Park, *Holographic two-point functions in a disorder system* [[arXiv:2209.07721](https://arxiv.org/abs/2209.07721)]
On holographic description of [[phonon]] gases in non-merallic [[crystals]]:
* Xiangqing Kong, Tao Wang, Liu Zhao, *High temperature AdS black holes are low temperature quantum phonon gases* [[arXiv:2209.12230](https://arxiv.org/abs/2209.12230)]
On holographic description of quantum [[spin chains]]:
* Naoto Yokoi, Yasuyuki Oikawa, Eiji Saitoh, *Holographic Dual of Quantum Spin Chain as Chern-Simons-Scalar Theory* [[arXiv:2310.01890](https://arxiv.org/abs/2310.01890)&rbrack:
### Via supergravity
{#ReferencesViaSupergravity}
Usage of full [[supergravity]] (retaining the [[gravitino]]) for [[AdS-CMT]], with application to [[graphene]]-like systems:
* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], [[Mario Trigiante]], *Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity*, J. High Energ. Phys. **2018** 7 (2018) [[arXiv:1801.08081](https://arxiv.org/abs/1801.08081), <a href="https://doi.org/10.1007/JHEP04(2018)007">doi:10.1007/JHEP04(2018)007</a>]
* {#ACDGNT20} [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], A. Gallerati, R. Noris, [[Mario Trigiante]], [[Jorge Zanelli]], *$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene*, JHEP 01 (2020) 084 [[arXiv:1910.03508](https://arxiv.org/abs/1910.03508), <a href="https://doi.org/10.1007/JHEP01(2020)084">doi:10.1007/JHEP01(2020)084</a>]
* Antonio Gallerati, *Supersymmetric theories and graphene*, in *40th International Conference on High Energy physics* (ICHEP2020), PoS **390** (2021) [[arXiv:2104.07420](https://arxiv.org/abs/2104.07420), [doi:10.22323/1.390.0662](https://doi.org/10.22323/1.390.0662)]
Exposition:
* [[Bianca L. Cerchiai]], *Supergravity in a pencil*, [talk at](M-Theory+and+Mathematics#Cerchiai2020) *[[M-Theory and Mathematics]] [2020](M-Theory+and+Mathematics#2020)*, NYU Abu Dhabi [[[CerchiaiSlidesAtMTheoryAndMathematics2020.pdf:file]], video: [YT](https://youtu.be/xE7TmwyqqaU)]
* [[Bianca L. Cerchiai]], *Holography, Supergravity and Graphene*, talk at *106th online SIF Congress* (2020) [[pdf](https://agenda.infn.it/event/23656/contributions/120378/attachments/75347/96340/cerchiai_sif2020.pdf), [[Cerciai-SIF2020.pdf:file]]]
Background discussion of supergravity with [[asymptotic boundaries]] (in the [[D'Auria-Fré formulation of supergravity|D'Auria-Fré formulation]]):
* [[Laura Andrianopoli]], [[Lucrezia Ravera]], *On the geometric approach to the boundary problem in supergravity*, Universe **7** 12 (2021) 463 [[arXiv:2111.01462](https://arxiv.org/abs/2111.01462), [doi:10.3390/universe7120463](https://doi.org/10.3390/universe7120463)]
See also
* *[New supergravity tools to study strongly coupled physical systems](https://cordis.europa.eu/article/id/202912-new-supergravity-tools-to-study-strongly-coupled-physical-systems)*
### Application to topological phases of matter
{#ApplicatinToTopologicalPhases}
On holographic descriptions of [[topological semimetals]] via the [[AdS-CMT correspondence]]:
* [[Karl Landsteiner]], [[Yan Liu]], *The holographic Weyl semi-metal*, Physics Letters B **753** (2016) 453-457 [[arXiv:1505.04772](https://arxiv.org/abs/1505.04772), [doi:10.1016/j.physletb.2015.12.052](https://doi.org/10.1016/j.physletb.2015.12.052)]
* [[Karl Landsteiner]], [[Yan Liu]], Ya-Wen Sun, *Quantum phase transition between a topological and a trivial semimetal from holography*, Phys. Rev. Lett. **116** 081602 (2016) [[arXiv:1511.05505](https://arxiv.org/abs/1511.05505), [doi:10.1103/PhysRevLett.116.081602](https://doi.org/10.1103/PhysRevLett.116.081602)]
* Ling-Long Gao, [[Yan Liu]], Hong-Da Lyu, *Black hole interiors in holographic topological semimetals* [[arXiv:2301.01468](https://arxiv.org/abs/2301.01468)]
### Application to band structure
Application to [[electron band structure]] of multi-layer [[graphene]]:
* Jeong-Won Seo, Taewon Yuk, Young-Kwon Han, Sang-Jin Sin, *ABC-stacked multilayer graphene in holography* [[arXiv:2208.14642](https://arxiv.org/abs/2208.14642)]
### Application to quantum chromodynamics
Discussion of [[quantum chromodynamics]] via [[AdS/CFT in condensed matter physics]] (see also at [[AdS/QCD]]):
* Yuri V. Kovchegov, _AdS/CFT applications to relativistic heavy ion collisions: a brief review_ ([arXiv:1112.5403](http://arxiv.org/abs/1112.5403))
* {#SantiagodeCompostela18} _[Holography and Extreme Chromodynamics](http://igfae.usc.es/~holoquark2018/)_, Santiago de Compostela, July 2018
### Application to BEC and superfluidity
Application to [[Bose-Einstein condensates]], [[superfluidity]] and [[vortices]]:
* Yu-Kun Yan, Shanquan Lan, Yu Tian, Peng Yang, Shunhui Yao, Hongbao Zhang, *Towards an effective description of holographic vortex dynamics* [[arXiv:2207.02814](https://arxiv.org/abs/2207.02814)]
* Aristomenis Donos, Polydoros Kailidis, *Dissipative effects in finite density holographic superfluids* [[arXiv:2209.06893](https://arxiv.org/abs/2209.06893)]
* [[Mario Flory]], Sebastian Grieninger, Sergio Morales-Tejera, *Critical and near-critical relaxation of holographic superfluids* [[arXiv:2209.09251](https://arxiv.org/abs/2209.09251)]
### Application to superconductivity
Discussion of [[superconductivity]] via [[AdS/CFT in condensed matter physics]]:
* [[Sean Hartnoll]], [[Christopher Herzog]], [[Gary
Horowitz]], _Building an AdS/CFT superconductor_, Phys. Rev. Lett. 101:031601, 2008 ([arXiv:0803.3295](https://arxiv.org/abs/0803.3295))
* Alberto Salvio, _Superconductivity, Superfluidity and Holography_ ([arXiv:1301.0201](http://arxiv.org/abs/1301.0201))
* Rong-Gen Cai, Li Li, Li-Fang Li, Run-Qiu Yang, _Introduction to Holographic Superconductor Models_, Sci China-Phys Mech Astron, 2015, 58(6):060401 ([arXiv:1502.00437](https://arxiv.org/abs/1502.00437))
* Chuan-Yin Xia, Hua-Bi Zeng, Yu Tian, Chiang-Mei Chen, [[Jan Zaanen]],
*Holographic Abrikosov lattice: vortex matter from black hole* ([arXiv:2111.07718](https://arxiv.org/abs/2111.07718))
* Dong Wang, Xiongying Qiao, Qiyuan Pan, Chuyu Lai, Jiliang Jing, *Holographic entanglement entropy and complexity for the excited states of holographic superconductors* [[arXiv:2301.00513](https://arxiv.org/abs/2301.00513)]
> (relation to [[holographic entanglement entropy]])
On [[strange metals]], high $T_c$-[[superconductors]] and [[AdS/CMT duality]]:
* [[Jan Zaanen]], _Planckian dissipation, minimal viscosity and the transport in cuprate strange metals_, SciPost Phys. 6, 061 (2019) ([arXiv:1807.10951](https://arxiv.org/abs/1807.10951))
* [[Jan Zaanen]], *Lectures on quantum supreme matter* ([arXiv:2110.00961](https://arxiv.org/abs/2110.00961))
### Application to chiral magnets
* Yuki Amari, Muneto Nitta, *Chiral Magnets from String Theory* [[arXiv:2307.11113](https://arxiv.org/abs/2307.11113)]
### Application to quasicrystals
Discussion of [[asymptotic boundaries]] of [[hyperbolic space|hyperbolic]] [[tensor networks]] as conformal [[quasicrystals]]:
* Latham Boyle, Madeline Dickens, Felix Flicker, _Conformal Quasicrystals and Holography_, Phys. Rev. X 10, 011009 (2020) ([arXiv:1805.02665](https://arxiv.org/abs/1805.02665))
* Alexander Jahn, Zoltán Zimborás, Jens Eisert, _Central charges of aperiodic holographic tensor network models_ ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))
### Relation to $p$-adic AdS/CFT correspondence
Proposed realization of aspects of [[p-adic AdS/CFT correspondence]] in [[solid-state physics]]:
* Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, [[Steven Gubser]], Monika Schleier-Smith, _Treelike interactions and fast scrambling with cold atoms_, Phys. Rev. Lett. 123, 130601 (2019) ([arXiv:1905.11430](https://arxiv.org/abs/1905.11430))
[[!include topological phases of matter via K-theory -- references]]
[[!redirects AdS/CFT in condensed matter physics]]
[[!redirects AdS/CFT correspondence in codensed matter physics]]
[[!redirects AdS-CFT correspondence in codensed matter physics]]
[[!redirects AdS/CMT]]
[[!redirects AdS-CMT]]
[[!redirects AdS/CMT duality]]
[[!redirects AdS-CMT duality]]
[[!redirects AdS/CMT correspondence]]
[[!redirects AdS-CMT correspondence]]
[[!redirects AdS/CFT in solid state physics]]
[[!redirects AdS-CFT in solid state physics]]
[[!redirects AdS/CFT correspondence in solid state physics]]
[[!redirects AdS-CFT correspondence in solid state physics]]
[[!redirects AdS/CFT duality in solid state physics]]
[[!redirects AdS/CFT correspondence in condensed matter physics]]
[[!redirects holographic CMT]]
|
AdS-QCD correspondence | https://ncatlab.org/nlab/source/AdS-QCD+correspondence | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
\begin{imagefromfile}
"file_name": "HardWallModelPredictions.jpg",
"float": "right",
"width": 440,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "From [Erlich 09, section 1.2](#Erlich09)"
\end{imagefromfile}
What is called _holographic QCD_ or _AdS/QCD correspondence_ or similar (review includes [Aharony 02](#Aharony02), [Erlich 09](#Erlich09), [Kim-Yi 11](#KimYi11), [Erlich 14](#Erlich14), [Rebhan 14](#Rebhan14), [Rho-Zahed 16](#RhoZahed16)) is a quantitatively predictive [[model (in theoretical physics)|model]] for [[quantum chromodynamics]] ("[[QCD]]", the [[strong nuclear force]]-sector of the [[standard model of particle physics]]) via "holography" (as in the [[AdS/CFT correspondence]]), hence regarding it as the [[boundary field theory]] of an (at least) [[D=5 super Yang-Mills theory|5-dimensional]] [[Yang-Mills theory]] ("[[bottom-up model building|bottom-up]] holographic QCD"), specifically one [[geometric engineering|geometrically engineered]] on [[intersecting D-brane model|intersecting D-branes]] ("[[top-down model building|top-down]] holographic QCD") and here specifically on [[D4-D8 brane intersections]] (the _[[Witten-Sakai-Sugimoto model]]_ due to [Witten 98](#Witten98), [Karch-Katz 02](#KarchKatz02), [Sakai-Sugimoto 04](#SakaiSugimoto04), [Sakai-Sugimoto 05](#SakaiSugimoto05)).
{#QCDNotManifestlyAnSCFT} While [[QCD]], taken at face value, is not a [[quantum field theory]] of the kind considered in the plain [[AdS-CFT correspondence]] -- since it is not a [[conformal field theory]] (but see [KPV22](confinement#KPV22)) and not [[supersymmetry|supersymmetric]] (hence not an [[SCFT]], but see at *[[hadron supersymmetry]]*) and does not have a large number $N$ of [[color charge|colors]] (but see [below](#SmallNOpenProblem)), it is thought that approximations/deformations of [[AdS-CFT]] may still usefully apply.
Holographic QCD captures the [[non-perturbative effect|non-perturbative]] [[confinement|confined]] regime of [[QCD]], which is otherwise elusive (the [[mass gap problem|Mass Gap Millennium Problem]]), where the would-be [[quarks]] are all [[bound state|bound]]/[[confinement|confined]] inside [[color charge|color]]-less [[hadrons]], with the [[meson]] [[field (physics)|fields]] instead being the [[gauge field]] of a [[flavour (particle physics)|flavour]]-[[chiral perturbation theory|gauge theory]] (holographic dictionary, e.g. [Kim-Yi 11 (3.1)](#KimYi11), see also "[[hidden local symmetry]]") and the [[baryons]] being [[solitons]] of this flavour/meson field, namely _[[skyrmions]]_.
\begin{imagefromfile}
"file_name": "IntersectingBranesSSWModel.jpg",
"float": "right",
"width": 600,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "From [Rebhan 14](#Rebhan14)"
\end{imagefromfile}
This [[duality in string theory|dual]] description of the [[color charge|color]] [[gauge theory]] of [[quarks]] and [[gluons]] instead as [[flavour (particle physics)|flavour]] [[gauge theory]] of [[baryons]] and [[mesons]] is [[geometric engineering of QFT|geometrically brought out]] by the [[D4-D8 brane intersections]] of the [[Witten-Sakai-Sugimoto model|Witten-Sakai-Sugimoto]] [[intersecting D-brane model]]: Here the [[open strings]] on the [[D4-branes|D4]] _[[color branes]]_ give the [[color charge|color]]/[[gluon]] gauge field, while those on the [[D8-brane|D8]] _[[flavor branes]]_ give the [[flavour (particle physics)|flavour]]/[[meson]] gauge field, those stretching between D4 and D8 give the [[quarks]] and the [[closed strings]] give the [[glueballs]].
(See at _[WSS brane configuration](#WSSBraneConfiguration)_ below.) This way [[color charge|color]]/[[flavour (particle physics)|flavor]] [[duality in physics|duality]] is mapped to [[open/closed string duality]] (as the [[D8-branes]] are treated as probe branes).
Notice that the [[flavour physics|flavour sector]] is where most of the open problems regarding the [[standard model of particle physics]] are located ([[flavour problem]], [[flavour anomalies]]).
\begin{imagefromfile}
"file_name": "SakaiSugimotoModel.jpg",
"float": "right",
"width": 570,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "From [Erlich 09, section 1.1](#Erlich09)"
\end{imagefromfile}
Various fundamental characteristics of [[QCD]] that remain mysterious in the [[color charge|colored]]-[[quark]] model readily find a conceptual explanation in terms of this [[geometric engineering of QFT|geometric engineering]] of [[flavour physics]], notably the phenomena of [[confinement]] and of [[chiral symmetry breaking]], but also for instance [[vector meson dominance]] and the [[Cheshire cat principle]].
\begin{imagefromfile}
"file_name": "AdSCFTForQCD.jpg",
"float": "right",
"width": 600,
"unit": "px",
"margin": {
"top": -40,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "From [Aoki-Hashimoto-Iizuka 12](#AHI12)"
\end{imagefromfile}
Indeed, holographic QCD gives accurate quantitative predictions of [[confinement|confined]] [[hadron]] spectra, hence of the physics of ordinary [[atomic nuclei]] (see comparison between [[experiment]] and predictions of holographic QCD [below](#BottomUpModels)) which is out of reach for [[perturbative quantum field theory|perturbation theory]] and otherwise computable, at best, via the blind numerics of [[lattice QCD]]. This means ([Witten 98](#Witten98)) that holographic QCD provides a conceptual solution to the _[[mass gap problem]]_ (not yet a rigorous proof, but a proof strategy).
Concretely, much of the [[phenomenology|phenomenological]] success of [[holographic QCD]] is (review in [Rho-Zahed 16, Chapter III](#RhoZahed16)) due to the holographic emergence of the time-honored but _ad hoc_ [[Skyrmion]]-[[model (in theoretical physics)|model]] of [[baryons]], as [[solitons]] in the [[meson]] [[flavour (particle physics)|flavour]]-[[gauge field]].
\begin{imagefromfile}
"file_name": "FirstEightSkyrmions.jpg",
"width": 450,
"unit": "px",
"caption": "From [Manton 11](Skyrmion#Manton11)"
\end{imagefromfile}
Moreover, in holographic QCD this [[Skyrmion]] model of [[baryons]] emerges in its modern improved form, where the [[pion]] [[field (physics)|field]] is accompanied by the whole tower of [[vector mesons]] (the [[rho meson]] etc.): these [[meson]] species are holographically unified as the transversal [[KK-mechanism|KK-modes]] in the holographic theory. Already just adjoining the [[rho meson]] to the [[pion]] makes the resulting [[Skyrmions]], and hence holographic QCD, give accurate results for light [[nuclei]] all the way up to [[carbon]] ([Naya-Sutcliffe 18a](Skyrmion#NayaSutcliffe18a), [Naya-Sutcliffe 18b](Skyrmion#NayaSutcliffe18b)).
\begin{imagefromfile}
"file_name": "SkyrmionsWithRho.jpg",
"width": 540,
"unit": "px",
"caption": "From [Naya-Sutcliffe 18](Skyrmion#NayaSutcliffe18)"
\end{imagefromfile}
The mechanism behind this description of [[baryons]] and [[nuclei]] via [[holographic QCD]] is the theorem of [Atiyah-Manton 89](skyrmion#AtiyahManton89) (highlighted as such in [Sutcliffe 10](skyrmion#Sutcliffe10)) which identifies [[Skyrmions]] in 3+1-dimensional [[Yang-Mills theory]] with [[Kaluza-Klein
mechanism|KK modes]] (transversal [[holonomies]]) of [[instantons]] in 4+1-dimensional YM theory:
$$
\text{baryons}
\;\;
\overset{\text{Skyrme}}{\leftrightarrow}
\;\;
{
{\text{Skyrmions}}
\atop
{\text{in}\; d=3+1}
}
\;\;
\overset{\text{Atiyah-Manton}}{\leftrightarrow}
\;\;
{
{\text{Instantons}}
\atop
{\text{in}\; d=4+1}
}
$$
This fact (of [[experiment]]/[[phenomenology]] on the left and of [[mathematics]] on the right) combined with the emergence of [[strings]] in the [['t Hooft limit]] of [[QCD]] reveals a _de facto_ [[AdS/CFT correspondence|holographic]] nature of [[QCD]]. The task in [[holographic QCD]] is to sort out the fine-print.
{#SmallNOpenProblem} A key open problem here is that the [[AdS/CFT correspondence]] is currently well understood only in the [[large N limit]], where the number $N_c$ of [[color charge|colors]] and the [['t Hooft coupling]] $\lambda$ are both large. But for [[QCD]] the number of colors is small, $N_c = 3$. While the correspondence is thought to hold also in the [[small N limit]], here the [[classical field theory|classical]] [[supergravity|super]]-[[gravity]]-computations on the dual (AdS) side will receive [[small N limit|small-N corrections]] (highlighted for holographic QCD e.g. in [Sugimoto 16](#Sugimoto16), see references [below](#StringAndMTheoryCorrection)) from [[perturbative string theory]] (for small [['t Hooft coupling]]) which are hard to compute, and then from [[M-theory]] (for small $N_c$) which are largely unknown, as [formulating M-theory remains an open problem](M-theory#TheOpenProblem). Hence from the perspective of [[small N limit|small-N corrected]] [[holographic QCD]], the [[mass gap problem]]/[confinement problem](confinement#OpenProblem) translates to the problem of formulating [[M-theory]]:
<center>
<a href="https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf#page=8">
<img src="https://ncatlab.org/schreiber/files/ProblemQCDToProblemM230120.jpg" width="670">
</a>
</center>
\linebreak
\linebreak
From [Yi 09](#Yi09):
> [[QCD]] is a challenging theory. Its most interesting aspects, namely the [[confinement]] of [[color charge|color]] and the [[chiral symmetry breaking]], have defied all analytical approaches. While there are now many data accumulated from the [[lattice gauge theory]], the methodology falls well short of giving us insights on how one may understand these phenomena analytically, nor does it give us a systematic way of obtaining a low energy theory of [[QCD]] below the [[confinement]] [[scale]].
> $[...]$
> it has been proposed early on that [[baryons]] are topological [[solitons]], namely [[Skyrmions]] $[$but$]$ the usual [[Skyrmion]] picture of the [[baryon]] has to be modified significantly in the context of full [[QCD]].
> $[...]$
> the [[AdS/CFT|holographic picture]] naturally brings a gauge principle in the [[bulk spacetime|bulk]] description of the [[flavor physics|flavor]] dynamics in such a way that all spin one [[mesons]] as well as [[pions]] would enter the $[$ [[Skyrmion|skyrmionic]]-$]$construction of [[baryons]] on the equal footing.
> $[...]$
> [[holographic QCD]] is similar to the [[chiral perturbation theory]] in the sense that we deal with exclusively [[gauge invariance|gauge-invariant]] [[observables|operators]] of the theory. The huge difference is, however, that this new approach tends to treat all [[gauge invariance|gauge-invariant]] objects together. Not only the light [[meson]] [[field (physics)|fields]] like [[pions]] but also heavy [[vector mesons]] and [[baryons]] appear together, at least in principle. In other words, a [[holographic QCD]] deals with all [[color charge|color]]-[[trivial representation|singlets]] simultaneously, giving us a lot more predictive power.
> $[...]$
> The expectation that there exists a more intelligent theory consisting only of [[gauge invariance|gauge-invariant]] objects in the [[large N limit|large Nc limit]] is thus realized via [[string theory]] in a somewhat surprising manner that the master fields, those truly physical degrees of freedom, actually live not in four dimensional Minkowskian world but in five or
higher dimensional curved geometry. This is not however completely unanticipated, and was heralded in the celebrated work by [Eguchi and Kawai in early 1980's](#EguchiKawai82) which is all the more remarkable in retrospect.
> $[...]$
> To compare against actual [[QCD]], we must fix $[$the [['t Hooft coupling]] and the [[Kaluza-Klein compactification|KK]]-[[scale]]$]$ to fit
both the [[pion]] decay constant $f_\pi$ and the [[mass]] of the first [[vector meson]]. After this fitting, all other infinite number of [[masses]] and [[coupling constants]] are fixed. This version $[$the holographic [[WSS model]]$]$ of the [[holographic QCD]] is extremely predictive.
> $[...]$
> $[$this$]$ elevates the classic [[skyrmion|Skyrme picture]] based on [[pions]] to a unified model involving all [[vector meson|spin one mesons]] in addition to [[pions]]. This is why the picture is extremely predictive.
> As we saw in this note, for low momentum processes, such as soft [[pion]] [[scattering amplitude|processes]], soft [[rho meson]] exchanges, and soft elastic scattering of [[photons]], the $[$holographic [[WSS model|WSS]]-$]$model's predictions compare extremely well with experimental data. It is somewhat mysterious that the [[baryon]] sector works out almost as well as the [[meson]] sector
\linebreak
{#FromSNM16} From [Suganuma-Nakagawa-Matsumoto 16, p. 1](#SuganumaNakagawaMatsumoto16):
> Since 1973, [[quantum chromodynamics]] (QCD) has been established as the fundamental theory of the [[strong nuclear force|strong interaction]]. Nevertheless, it is very difficult to solve QCD directly in an analytical manner, and many effective models of QCD have been used instead of QCD, but most models cannot be derived from QCD and its connection to QCD is unclear.
> To analyze nonperturbative QCD, the [[lattice QCD]] Monte Carlo simulation has been also used as a first-principle calculation of the strong interaction. However, it has several weak points. For example, the [[quantum state|state]] information (e.g. the [[wave function]]) is severely limited, because [[lattice QCD]] is based on the [[path-integral]] formalism. Also, it is difficult to take the [[chiral fermion|chiral]] limit, because zero-mass [[pions]] require [[large volume limit|infinite volume lattices]]. There appears a notorious "[sign problem](lattice+gauge+theory#SignProblem)" at finite density.
> On the other hand, [[holographic QCD]] has a direct connection to [[QCD]], and can be derived from QCD in some limit. In fact, [[holographic QCD]] is equivalent to infrared QCD in [[large N limit|large Nc]] and strong [['t Hooft coupling]] $\lambda$, via [[gauge/gravity correspondence]]. Remarkably, [[holographic QCD]] is successful to reproduce many [[hadron]] [[phenomenology]] such as [[vector meson dominance]], the KSRF relation, hidden local symmetry, the GSW model and the [[skyrmion|Skyrme soliton picture]]. Unlike [[lattice QCD]] simulations, holographic QCD is usually formulated in the chiral limit, and does not have the [sign problem](lattice+gauge+theory#SignProblem) at finite density.
\linebreak
From [Rho et a. 16](#RhoEtAl16):
> One can make $[$[[chiral perturbation theory]]$]$ consistent with [[QCD]] by suitably matching the [[correlators]] of the [[effective field theory|effective theory]] to those of [[QCD]] at a [[scale]] near $\Lambda$. Clearly this procedure is not limited to only one set of [[vector mesons]]; in fact, one can readily generalize it to an infinite number of hidden [[gauge fields]] in an [[effective field theory|effective]] [[Lagrangian density|Lagrangian]]. In so doing, it turns out that a fifth dimension is "deconstructed" in a (4+1)-dimensional (or 5D) [[Yang-Mills theory|Yang–Mills]] type form. We will see in Part III that such a structure arises, [[top-down model building|top-down]], in [[string theory]].
> $[...]$
> $[$this [[holographic QCD]]$]$ model comes out to describe — unexpectedly well — low-energy properties of both [[mesons]] and [[baryons]], in particular those properties reliably described in quenched [[lattice QCD]] simulations.
> $[...]$
> One of the most noticeable results of this [[AdS/QCD|holographic model]] is the first derivation of [[vector meson dominance|vector dominance]] (VD) that holds both for [[mesons]] and for [[baryons]]. It has been somewhat of an oddity and a puzzle that [[Jun John Sakurai|Sakurai's]] [[vector meson dominance|vector dominance]] — with the lowest [[vector mesons]] [[rho meson|ρ]] and ω — which held very well for [[pion|pionic]] [[form factors]] at low momentum transfers famously failed for [[nucleon]] [[form factors]]. In this [[AdS/QCD|holographic model]], the [[vector meson dominance|VD]] comes out automatically for both the [[pion]] and the [[nucleon]] provided that the infinite $[$[[Kaluza-Klein mechanism|KK-]]$]$tower is included. While the [[vector meson dominance|VD]] for the [[pion]] with the infinite tower is not surprising given the successful Sakurai VD, that the [[vector meson dominance|VD]] holds also for the [[nucleons]] is highly nontrivial. $[...]$ It turns out to be a consequence of a [[AdS/QCD|holographic]] [[Cheshire cat principle|Cheshire Cat phenomenon]]
\linebreak
[[!include Polyakov gauge-string duality -- section]]
\linebreak
## Models
{#Models}
In approaches to $AdS/QCD$ one distinguishes [[top-down model building]] -- where the ambition is to first set up a globally consistent ambient [[intersecting D-brane model]] where a [[Yang-Mills theory]] at least similar to [[QCD]] arises on suitable [[D-branes]] ([[geometric engineering of gauge theories]]) -- from [[bottom-up model building]] approaches which are more cavalier about global consistency and first focus on accurately fitting the intended [[phenomenology]] of [[QCD]] as the [[asymptotic boundary|asymptotic]] [[boundary field theory]] of [[gravity]]+[[gauge theory]] on some [[anti de Sitter spacetime]]. (Eventually both these approaches should meet "in the middle" to produce a [[model (in theoretical physics)|model]] which is both [[standard model of particle physics|realistic]] as well as globally consistent as a [[string vacuum]]; see also at _[[string phenomenology]]_.)
<center>
<img src="https://ncatlab.org/nlab/files/BottomUpAndTopDownIntersDBraneModelBuilding.png" width="700"/>
</center>
> graphics from [Aldazabal-Ibáñez-Quevedo-Uranga 00](bottom-up+and+top-down+model+building#AldazabalIbanezQuevedoUranga00)
### Top-down models
{#TopDownModels}
#### Witten-Sakai-Sugimoto model
{#WittenSakaiSugimotoModel}
A good [[top-down model building]]-approach to AdS/QCD is due to [Sakai-Sugimoto 04](#SakaiSugimoto04), [Sakai-Sugimoto 05](#SakaiSugimoto05) based on [Witten 98](#Witten98), see [Rebhan 14](#Rebhan14), [Sugimoto 16](#Sugimoto16) for review.
##### Brane configuration
{#WSSBraneConfiguration}
The [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineers]] something at least close to [[QCD]]: on the [[worldvolume]] of [[intersecting D-brane models|coincident]] [[black brane|black]] [[M5-branes]] with [[near horizon geometry]] a [[KK-compactification]] of $AdS_7 \times S^4$ in the decoupling limit where the [[worldvolume]] theory becomes the [[6d (2,0)-superconformal SCFT]]. Here the [[KK-compactification]] is on a [[torus]] with anti-periodic boundary conditions for the [[fermions]] in one direction, thus [[spontaneous symmetry breaking|breaking]] all [[supersymmetry]] ([[Scherk-Schwarz mechanism]]). Here the first circle [[KK-compactification|reduction]] realizes, under [[duality between M-theory and type IIA string theory]], the [[M5-branes]] as [[D4-branes]], hence the model now looks like 5d [[Yang-Mills theory]] further [[KK-compactification|compactified]] on a circle. ([Witten 98, section 4](#Witten98)).
The further introduction of [[intersecting D-brane model|intersecting]] [[D8-branes]] and [[anti D-brane|anti]] [[D8-branes]] to [[D4-D8 brane bound states]] makes a sensible sector of [[chiral fermions]] appear in this model ([Sakai-Sugimoto 04](#SakaiSugimoto04), [Sakai-Sugimoto 05](#SakaiSugimoto05))
{#BraneConfigurationDiagram} The following diagram indicates the Witten-Sakai-Sugimoto [[intersecting D-brane model]] that [[geometric engineering of QFT|geometrically engineers]] [[QCD]]:
<center>
<img src="https://ncatlab.org/nlab/files/WSSBraneConfigurationEngineeringQCDII.jpg" width="740"/>
</center>
<div style="float:right;margin:0 10px 10px 0;">
<img src="https://ncatlab.org/nlab/files/D8D6NS5.jpg" width="380"/>
</div>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
Here we are showing:
1. the [[color brane|color]] [[D4-branes]];
1. the [[flavor brane|flavor]] [[D8-branes]];
with
1. the [[5d Chern-Simons theory]] on their [[worldvolume]]
1. the corresponding [[4d WZW model]] on the boundary
exhibiting the [[vector meson fields]] in the [[Skyrmion model]];
1. the [[baryon]] [[D4-branes]]
(see below at _[Baryons](#Baryons)_);
1. the [[Yang-Mills monopole]] [[D6-branes]]
(see at _[[D6-D8-brane bound state]]_);
1. the [[NS5-branes]] (often not considered here).
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
\linebreak
\linebreak
\linebreak
##### Glueballs
{#WSSModelGlueballs}
Already before adding the D8-branes (hence already in the pure Witten model) this produces a pure [[Yang-Mills theory]] whose [[glueball]]-spectra may usefully be compared to those of [[QCD]]:
<center>
<img src="https://ncatlab.org/nlab/files/GlueballSpectrumSSWModel.jpg" width="700">
</center>
> graphics from [Rebhan 14](#Rebhan14)
##### Hadrons
{#Hadrons}
In this [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in the [[bulk field theory]]:
###### Mesons
The [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two quarks
###### Baryons
{#Baryons}
The [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent ([Sugimoto 16, 15.4.1](#Sugimoto16)) guises:
1. as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]
([Witten 98b](#Witten98b), [Gross-Ooguri 98, Sec. 5](#GrossOoguri98), [BISY 98](#BISY98), [CGS98](#CGS98))
1. as [[skyrmions]]
([Sakai-Sugimoto 04, section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](#SakaiSugimoto05), see [Bartolini 17](#Bartolini17)).
For review see [Sugimoto 16](#Sugimoto16), [Yi 09](#Yi09), [Yi 11](#Yi11), [Yi 13](#Yi13), also [Rebhan 14, around (18)](#Rebhan14).
<center>
<img src="https://ncatlab.org/nlab/files/BaryonsAsD4Branes.jpg" width="700">
</center>
> graphics from [Sugimoto 16](#Sugimoto16)
Equivalently, these baryon states are the [[Yang-Mills instantons]] on the [[D8-brane]] giving the [[D4-D8 brane bound state]] ([Sakai-Sugimoto 04, 5.7](#SakaiSugimoto04)) as a special case of the general situation for [[Dp-D(p+4)-brane bound states]] (e.g. [Tong 05, 1.4](Dp-D%28p%2B4%29-brane+bound+state#Tong05)).
<center>
<img src="https://ncatlab.org/nlab/files/BaryonVertexDefectInAdSQCD.jpg" width="490">
</center>
> graphics from [Cai-Li 17](#CaiLi17)
<center>
<img src="https://ncatlab.org/nlab/files/D6InD8InAdSQCD.jpg" width="700">
</center>
> graphics from [ABBCN 18](#ABBCN18)
This already produces [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):
<center>
<img src="https://ncatlab.org/nlab/files/BaryonSpectrumInSakaiSugimoto.jpg" width="700">
</center>
> graphics from [Sugimoto 16](#Sugimoto16)
Moreover, the above 4-brane model for baryons is claimed to be equivalent to the old **[[Skyrmion]] model** (see [Sakai-Sugimoto 04, section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](#SakaiSugimoto05), [Sugimoto 16, 15.4.1](#Sugimoto16), [Bartolini 17](#Bartolini17)).
But the Skyrmion model of baryons has been shown to apply also to [[bound states]] of [[baryons]], namely the [[atomic nuclei]] ([Riska 93](Skyrmion#Riska93), [Battye-Manton-Sutcliffe 10](Skyrmion#BattyeMantonSutcliffe10), [Manton 16](Skyrmion#Manton16), [Naya-Sutcliffe 18](Skyrmion#NayaSutcliffe18)), at least for small [[atomic number]].
For instance, various [[experiment|experimentally]] observed resonances of the [[carbon]] [[nucleus]] are modeled well by a Skyrmion with [[atomic number]] 6 and hence baryon number 12 ([Lau-Manton 14](Skyrmion#LauMaonton14)):
\begin{center}
<img src="https://ncatlab.org/nlab/files/SkyrmionB12.jpg" width="200">
\end{center}
> graphics form [Lau-Manton 14](Skyrmion#LauMaonton14)
More generally, the [[Skyrmion]]-model of [[atomic nuclei]] gives good matches with [[experiment]] if not just the [[pi meson]] but also the [[rho meson]]-background is included ([Naya-Sutcliffe 18](Skyrmion#NayaSutcliffe18)):
\begin{center}
<img src="https://ncatlab.org/nlab/files/SkyrmionsWithRho.jpg" width="800">
\end{center}
> graphics form [Naya-Sutcliffe 18](Skyrmion#NayaSutcliffe18)
<br/>
#### WSS-type model for 2d QCD
{#WSSTypeModelFor2dQCD}
There is a direct analogue for [[2d QCD]] of the [above](#WittenSakaiSugimotoModel) [[WSS model]] for 4d [[QCD]]
([Gao-Xu-Zeng 06](#GaoXuZeng06), [Yee-Zahed 11](#YeeZahed11)).
The corresponding [[intersecting D-brane model]] is much as that for 4d QCD [above](#WSSBraneConfiguration), just with
1. [[colour charge|color]] [[D2-branes]] instead of [[D4-branes]];
1. [[baryon]]$\,$ [[D6-branes]] instead of [[D4-branes]];
1. [[meson]]$\,$ [[field (physics)|fields]] given by 3d [[Chern-Simons theory]] instead of [[5d Chern-Simons theory]]:
<center>
<img src="https://ncatlab.org/nlab/files/WSSBraneConfigurationEngineering2dQCD.jpg" width="740"/>
</center>
#### Type0B/$YM_4$-correspondence
{#Type0StringCorrespondence}
Instead of starting with [[M5-branes]] in [[supergravity|locally supersymmetric]] [[M-theory]] and then [[spontaneously broken symmetry|spontaneously breaking]] all [[supersymmetry]] by suitable [[KK-compactification]] as in the [Witten-Sakai-Sugimoto model](#WittenSakaiSugimotoModel), one may start with a non-supersymmetric [[bulk field theory|bulk]] [[string theory]] in the first place.
In this vein, it has been argued in [GLMR 00](#GLMR00) that there is holographic duality between [[type 0 string theory]] and non-supersymmetric 4d [[Yang-Mills theory]] (hence potentially something close to [[QCD]]). See also [AAS 19](#AAS19).
### Bottom-up models
{#BottomUpModels}
A popular [[bottom-up model building|bottom-up approach]] of AdS/QCD is known as the _hard-wall model_ ([Erlich-Katz-Son-Stephanov 05](#ErlichKatzSonStephanov05)).
Further refinement to the "soft-wall model" is due to [KKSS 06](#KKSS06) and further to "improved holographic QCD" is due to [Gursoy-Kiritsis-Nitti 07](#GursoyKiritsisNitti07), [Gursoy-Kiritsis 08](#GursoyKiritsis08), see [GKMMN 10](#GKMMN10).
## Comparison with experiment
{#ComparisonWithExperiment}
Comparison of holographic QCD models with [[experiment]] (mostly using bottom-up models).
Computations due to [Katz-Lewandowski-Schwartz 05](#KatzLewandowskiSchwartz05) using the hard-wall model ([Erlich-Katz-Son-Stephanov 05](#ErlichKatzSonStephanov05)) find the following comparison of AdS/QCD predictions to [[QCD]]-[[experiment]]
<center>
<img src="https://ncatlab.org/nlab/files/HardWallModelPredictions.jpg" width="400">
</center>
> graphics from [Erlich 09, section 1.2](#Erlich09)
Computations due to [KKSS 06](#KKSS06), [Gursoy-Kiritsis-Nitti 07](#GursoyKiritsisNitti07), [Gursoy-Kiritsis 08](#GursoyKiritsis08), see [GKMMN 10](#GKMMN10):
<center>
<img src="https://ncatlab.org/nlab/files/GlueballMasses.jpg" width="660">
</center>
> graphics from [GKMMN 10](#GKMMN10)
<center>
<img src="https://ncatlab.org/nlab/files/ImprovedAdSQCD.jpg" width="680">
</center>
> graphics from [GKMMN 10](#GKMMN10)
\linebreak
From [Pomarol-Wulzer 09](#PomarolWulzer09):
<center>
<img src="https://ncatlab.org/nlab/files/PomarolHolographicBaryonMasses.jpg" width="500">
</center>
\linebreak
{#TablesdRocha21} From [da Rocha 21](#daRocha21), for [[vector mesons]]:
for [[upsilon-mesons]]:
\begin{imagefromfile}
"file_name": "daRocha2021-HolographicMesonMasses-Table1a.jpg",
"width": 440,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [da Rocha 21](#daRocha21)"
\end{imagefromfile}
for [[psi-mesons]]:
\begin{imagefromfile}
"file_name": "daRocha2021-HolographicMesonMasses-Table2.jpg",
"width": 440,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [da Rocha 21](#daRocha21)"
\end{imagefromfile}
for [[omega-mesons]]:
\begin{imagefromfile}
"file_name": "daRocha2021-HolographicMesonMasses-Table3.jpg",
"width": 440,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [da Rocha 21](#daRocha21)"
\end{imagefromfile}
for [[phi-mesons]]:
\begin{imagefromfile}
"file_name": "daRocha2021-HolographicMesonMasses-Table4.jpg",
"width": 440,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [da Rocha 21](#daRocha21)"
\end{imagefromfile}
{#PredictionsIncludingFirstHeavyQuarks} Including the first [[heavy quarks]]:
\begin{imagefromfile}
"file_name": "ChenHuang_HolographicQCD_2021_TableII.jpg",
"width": 800,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [Chen & Huang 2021](#ChenHuang21)"
\end{imagefromfile}
\linebreak
From [CLFH22](#CLFH22):
\begin{imagefromfile}
"file_name": "MesonMassSpectraFromCLFH22.jpg",
"width": 800,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [CLFH22](#CLFH22)"
\end{imagefromfile}
\linebreak
These computations shown so far all use just the field theory in the bulk, not yet the stringy modes ([[limit of a sequence|limit]] of vanishing [[string length]] $\sqrt{\alpha'} \to 0$). Incorporating bulk string corrections further improves these results, see [Sonnenschein-Weissman 18](#SonnenscheinWeissman18).
## Embedding into the standard model of particle physics
[Nastase 03, p. 2](#Nastase03):
> An obvious question then is can one lift this [[intersecting D-brane model|D brane construction]] for the [[holographic QCD|holographic dual of QCD]] to a [[standard model of particle physics|Standard Model]] embedding? I study this question in the context of [[intersecting D-brane models|D-brane-world]] [[GUT]] models and find that one needs to have [[TeV-scale string theory]].
## Related concepts
* [[hadron supersymmetry]], [[hadron Kaluza-Klein theory]]
* [[D=5 Yang-Mills theory]]
* [[confinement]]
* [[chiral perturbation theory]]
* [[quark bag model]], [[Cheshire cat principle]]
* [[Skyrmions]]
* [[lattice QCD]]
* [[AdS-CFT in condensed matter physics]]
* [[holographic Schwinger effect]]
* [[holographic entanglement entropy]]
* [[holography as Koszul duality]]
[[!include effective field theories of nuclear physics -- contents]]
## References
[[!include Polyakov gauge-string duality -- references]]
### General
{#ReferencesGeneral}
#### Review and introduction
* {#Aharony02} [[Ofer Aharony]], _The non-AdS/non-CFT correspondence, or three different paths to QCD_, Progress in string, field and particle theory. Springer, Dordrecht, 2003. 3-24 ([arXiv:hep-th/0212193](https://arxiv.org/abs/hep-th/0212193))
* {#Erlich09} [[Joshua Erlich]], *How Well Does AdS/QCD Describe QCD?*, Int. J. Mod. Phys.A **25** (2010) 411-421 [[arXiv:0908.0312](https://arxiv.org/abs/0908.0312), [doi:10.1142/S0217751X10048718](https://doi.org/10.1142/S0217751X10048718)]
* Marco Panero, _QCD thermodynamics in the large-$N$ limit_, 2010 ([[PaneroAdsQCD.pdf:file|pdf]])
* {#KimYi11} Youngman Kim, Deokhyun Yi, _Holography at Work for Nuclear and Hadron Physics_, Advances in High Energy Physics, Volume 2011, Article ID 259025, 62 pages ([arXiv:1107.0155](https://arxiv.org/abs/1107.0155), [doi:10.1155/2011/259025](http://dx.doi.org/10.1155/2011/259025))
* [[Antoine Van Proeyen]], [[Daniel Freedman]], Chapter VIII of: _Supergravity_, Cambridge University Press (2012) [[doi:10.1017/CBO9781139026833]( https://doi.org/10.1017/CBO9781139026833)]
* [[Koji Hashimoto]], §6.4 in: _D-Brane -- Superstrings and New Perspective of Our World_, Springer (2012) [[doi:10.1007/978-3-642-23574-0](https://link.springer.com/book/10.1007%2F978-3-642-23574-0), [spire:1188897](http://inspirehep.net/record/1188897)]
* M. R. Pahlavani, R. Morad, _Application of AdS/CFT in Nuclear Physics_, Advances in High Energy Physics ([arXiv:1403.2501](https://arxiv.org/abs/1403.2501))
* Jorge Casalderrey-Solana, Hong Liu, [[David Mateos]], Krishna Rajagopal, Urs Achim Wiedemann, _Gauge/string duality, hot QCD and heavy ion collisions_, Cambridge University Press, 2014 ([arXiv:1101.0618](https://arxiv.org/abs/1101.0618))
* {#AHI12} Sinya Aoki, [[Koji Hashimoto]], [[Norihiro Iizuka]], _Matrix Theory for Baryons: An Overview of Holographic QCD for Nuclear Physics_, Reports on Progress in Physics, Volume 76, Number 10 ([arxiv:1203.5386](https://arxiv.org/abs/1203.5386))
* Youngman Kim, Ik Jae Shin, Takuya Tsukioka, _Holographic QCD: Past, Present, and Future_, Progress in Particle and Nuclear Physics **68** (2013) 55-112 [[arXiv:1205.4852](https://arxiv.org/abs/1205.4852)]
* {#Erlich14} [[Joshua Erlich]], *An Introduction to Holographic QCD for Nonspecialists*, Contemporary Physics, **56** 2 (2015) [[arXiv:1407.5002](https://arxiv.org/abs/1407.5002), [doi:10.1080/00107514.2014.942079](https://doi.org/10.1080/00107514.2014.942079)]
* {#Guijosa16} Alberto Guijosa, _QCD, with Strings Attached_, IJMPE Vol. 25, No. 10 (2016) 1630006 ([arXiv:1611.07472](https://arxiv.org/abs/1611.07472))
* [[Mannque Rho]], [[Ismail Zahed]] (eds.) Chapter 4 of: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
* Sophia K Domokos, Robert Bell, Trinh La, Patrick Mazza, *A Pedagogical Introduction to Holographic Hadrons*, published as: *Holographic hadron masses in the language of quantum mechanics*, European Journal of Physics **42** 6 (2021) 065801 [[arXiv:2106.13136](https://arxiv.org/abs/2106.13136), [doi:10.1088/1361-6404/ac1abb](https://iopscience.iop.org/article/10.1088/1361-6404/ac1abb)]
With emphasis on application to the [QCD phase diagram](QCD#PhaseDiagram) and to the description of [[neutron stars]]:
* [[Matti Järvinen]], *Holographic modeling of nuclear matter and neutron stars*, Eur.Phys.J.C. 56 2021 ([arXiv:2110.08281](https://arxiv.org/abs/2110.08281))
* Niko Jokela, *NICER view on holographic QCD* ([arXiv:2111.07940](https://arxiv.org/abs/2111.07940))
See also:
* Wikipedia, _[AdS/QCD correspondence](https://en.wikipedia.org/wiki/AdS/QCD_correspondence)_
Emphasis of the AdS-QCD correspondence at annual String-conferences:
* {#Klebanov21} [[Igor Klebanov]], *Remarks on Color Confinement*, talk at *[Strings 2021](https://www.ictp-saifr.org/strings2021/)* ([pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/06/DiscussionConfinementKlebanov.pdf), [video](https://youtu.be/B9NI2LE7d68?t=262))
* {#Dubovsky21} [[Sergei Dubovsky]], *Comments on (mostly long) QCD strings*, talk at *[Strings 2021](https://www.ictp-saifr.org/strings2021/)* ([pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/06/Sergei.pdf), [video](https://youtu.be/B9NI2LE7d68?t=1183))
* [[Nick Evans]], *Holography for Multi-Representation and Chiral Matter*, talk at *[[Strings 2022]]* [[indico:4940848](https://indico.cern.ch/event/1085701/contributions/4940848), [slides](https://indico.cern.ch/event/1085701/contributions/4940848/attachments/2482358/4261644/Nick%20Evans%20Tuesday-Research-Evans.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=aecca42b-c952-42f0-84c6-5b23230534b7)]
> (emphasis on [[DBI action]]-effects)
* [[Umut Gürsoy]], *Recent developments in gauge-gravity duality applied to quantum many-body systems*, talk at *[[Strings 2022]]* [[indico:4940863](https://indico.cern.ch/event/1085701/contributions/4940863), [pdf](https://indico.cern.ch/event/1085701/contributions/4940863/attachments/2483542/4263813/Slides_Gursoy.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=69bc5a22-b77a-4830-97b8-49c7d6aa1c29) ]
* [[Edward Witten]], *Some Milestones in the Study of Confinement*, talk at *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023) [[web](https://www.ias.edu/video/some-milestones-study-confinement), [YT](https://youtu.be/TvIz-6YOdKs)]
> [40:13](https://youtu.be/TvIz-6YOdKs?t=2413): "Personally, I think [[AdS/QCD|this setup]] really implies that pure [[Yang-Mills theory|$SU(N)$ gauge theory]] is [[gauge-string duality|dual]] to a [[string theory]]. The 'only' problem is that to get the pure gauge theory we need to make a relevant deformation and then take the limit that the deformation parameter is large..."
#### Top-down models
##### Witten-Sakai-Sugimoto model
Precursor developments:
* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))
* [[Joseph Polchinski]], [[Matthew Strassler]], _The String Dual of a Confining Four-Dimensional Gauge Theory_ ([arXiv:hep-th/0003136](https://arxiv.org/abs/hep-th/0003136))
> (discussion of [[confinement]] via [[AdS/CFT]] with a [[Myers effect]] in the bulk)
The top-down Sakai-Sugimoto model is due to
* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Low energy hadron physics in holographic QCD_, Progr. Theor. Phys. 113: 843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))
* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _More on a holographic dual of QCD_, Progr. Theor. Phys. 114: 1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))
along the lines of
* {#KarchKatz02} [[Andreas Karch]], [[Emanuel Katz]], _Adding flavor to AdS/CFT_, JHEP 0206:043, 2002 ([arxiv:hep-th/0205236](https://arxiv.org/abs/hep-th/0205236))
and based on
* {#Witten98} [[Edward Witten]], _Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories_, Adv. Theor. Math. Phys.2:505-532, 1998 ([arXiv:hep-th/9803131](https://arxiv.org/abs/hep-th/9803131))
further developed in
* {#Bartolini17} Lorenzo Bartolini, [[Stefano Bolognesi]], Andrea Proto, _From the Sakai-Sugimoto Model to the Generalized Skyrme Model_, Phys. Rev. D 97, 014024 2018 ([arXiv:1711.03873](https://arxiv.org/abs/1711.03873))
reviewed in:
* Piljin Yi, _Topics in D4-D8 holographic QCD_, 2009 ([[YiD4D8HolographicQCD.pdf:file]])
* [[Shigeki Sugimoto]], _Holographic QCD -- Status and perspectives_, 2012 ([[Sugimoto12.pdf:file]])
* {#Rebhan14} [[Anton Rebhan]], _The Witten-Sakai-Sugimoto model: A brief review and some recent results_, 3rd International Conference on New Frontiers in Physics, Kolymbari, Crete, 2014 ([arXiv:1410.8858](https://arxiv.org/abs/1410.8858))
* {#RhoZahed16} [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
* Si-wen Li, Xiao-tong Zhang, *The D4/D8 model and holographic QCD* [[arXiv:2304.10826](https://arxiv.org/abs/2304.10826)]
See also:
* Vikas Yadav, *String/$\mathcal{M}$-theory Dual of Large-$N$ Thermal QCD-Like Theories at Intermediate Gauge/'t Hooft Coupling and Holographic Phenomenology* ([arXiv:2111.12655](https://arxiv.org/abs/2111.12655))
More on [[D4-D8 brane bound states]]:
* {#Nastase03} [[Horatiu Nastase]], Sections 2, 3 of: _On Dp-Dp+4 systems, QCD dual and phenomenology_ ([arXiv:hep-th/0305069](https://arxiv.org/abs/hep-th/0305069))
The Witten-Sakai-Sugimoto model with [[orthogonal group|orthogonal]] [[gauge groups]] realized by [[D4-D8 brane bound states]] at [[O-planes]]:
* {#ImotoSakaiSugimoto09} Toshiya Imoto, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _$O(N)$ and $USp(N)$ QCD from String Theory_, Prog.Theor.Phys.122:1433-1453, 2010 ([arXiv:0907.2968](https://arxiv.org/abs/0907.2968))
* Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, _5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry_, JHEP 1210 (2012) 142 ([arXiv:1206.6781](https://arxiv.org/abs/1206.6781))
Analogous discussion for [[flavour (particle physics)|flavour]] [[D6-branes]]:
* [[Jérôme Gaillard]], _On $G$-structures in gauge/string duality_, 2011 ([cronfa:42569](https://cronfa.swan.ac.uk/Record/cronfa42569) [spire:1340775](http://inspirehep.net/record/1340775), [[GaillardGStructure.pdf:file]])
(with emphasis of [[G-structures]])
The analogoue of the WSS model for [[2d QCD]]:
* {#GaoXuZeng06} Yi-hong Gao, Weishui Xu, Ding-fang Zeng, _NGN, $QCD_2$ and chiral phase transition from string theory_, Nucl.Phys. B400:181-210, 1993 ([arXiv:hep-th/0605138](https://arxiv.org/abs/hep-th/0605138))
Specifically concerning the 3d [[Chern-Simons theory]] on the [[D8-branes]]:
* {#YeeZahed11} Ho-Ung Yee, Ismail Zahed, _Holographic two dimensional QCD and Chern-Simons term_, JHEP 1107:033, 2011 ([arXiv:1103.6286](https://arxiv.org/abs/1103.6286))
and its relation to [[baryons]]:
* {#SuganumaNakagawaMatsumoto16} Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, _1+1 Large $N_c$ QCD and its Holographic Dual $\sim$ Soliton Picture of Baryons in Single-Flavor World_, JPS Conf. Proc. 13, 020013 (2017) ([arXiv:1610.02074](https://arxiv.org/abs/1610.02074))
On [[jet bundle]] [[cohomology]] in the Sakai-Sugimoto model:
* Ekkehart Winterroth, *Variational cohomology and Chern-Simons gauge theories in higher dimensions* ([arXiv:2103.03037](https://arxiv.org/abs/2103.03037))
##### Further models
Variant with [[D4-brane|D4]] [[flavor branes]]:
* [[Mark Van Raamsdonk]], Kevin Whyte, _Baryon charge from embedding topology and a continuous meson spectrum in a new holographic gauge theory_, JHEP 1005:073, 2010 ([arXiv:0912.0752](https://arxiv.org/abs/0912.0752))
* Shigenori Seki, _Intersecting D4-branes Model of Holographic QCD and Tachyon Condensation_, JHEP 1007:091, 2010 ([arXiv:1003.2971](https://arxiv.org/abs/1003.2971))
See also:
* [[Nick Evans]], [[Jack Mitchell]], *Domain Wall AdS/QCD*, Phys. Rev. D **104** (2021) 094018 [[arXiv:2108.12152](https://arxiv.org/abs/2108.12152), [doi:10.1103/PhysRevD.104.094018](https://doi.org/10.1103/PhysRevD.104.094018)]
* [[Nick Evans]], [[Jack Mitchell]], *Thermal Transitions in Domain Wall AdS/QCD* [[arXiv:2207.10374](https://arxiv.org/abs/2207.10374)]
#### Bottom-up models
##### Hard- and soft-wall model
The bottom-up hard-wall model is due to
* {#ErlichKatzSonStephanov05} Joshua Erlich, [[Emanuel Katz]], Dam T. Son, Mikhail A. Stephanov, _QCD and a Holographic Model of Hadrons_, Phys.Rev.Lett.95:261602, 2005 ([arXiv:hep-ph/0501128](https://arxiv.org/abs/hep-ph/0501128))
while the soft-wall refinement is due to
* {#KKSS06} [[Andreas Karch]], [[Emanuel Katz]], Dam T. Son, Mikhail A. Stephanov, _Linear Confinement and AdS/QCD_, Phys.Rev.D74:015005, 2006 ([arXiv:hep-ph/0602229](https://arxiv.org/abs/hep-ph/0602229))
reviewed in
* Sergey Afonin, Timofey Solomko, *Motivations for the Soft Wall holographic approach to strong interactions* [[arXiv:2209.09042](https://arxiv.org/abs/2209.09042)]
see also
* Alfredo Vega, Paulina Cabrera, _Family of dilatons and metrics for AdS/QCD models_, Phys. Rev. D 93, 114026 (2016) ([arXiv:1601.05999](https://arxiv.org/abs/1601.05999))
* Alfonso Ballon-Bayona, Luis A. H. Mamani, _Nonlinear realisation of chiral symmetry breaking in holographic soft wall models_ ([arXiv:2002.00075](https://arxiv.org/abs/2002.00075))
and the version _improved holographic QCD_ is due to
* {#GursoyKiritsis08} Umut Gursoy, [[Elias Kiritsis]], _Exploring improved holographic theories for QCD: Part I_, JHEP 0802:032, 2008 ([arXiv:0707.1324](https://arxiv.org/abs/0707.1324))
* {#GursoyKiritsisNitti07} Umut Gursoy, [[Elias Kiritsis]], Francesco Nitti, _Exploring improved holographic theories for QCD: Part II_, JHEP 0802:019, 2008 ([arXiv:0707.1349](https://arxiv.org/abs/0707.1349))
reviewed in
* {#GKMMN10} Umut Gürsoy, [[Elias Kiritsis]], Liuba Mazzanti, Georgios Michalogiorgakis, Francesco Nitti, _Improved Holographic QCD_, Lect.Notes Phys.828:79-146,2011 ([arXiv:1006.5461](https://arxiv.org/abs/1006.5461))
More developments on improved holographic QCD:
* Takaaki Ishii, [[Matti Järvinen]], Govert Nijs, _Cool baryon and quark matter in holographic QCD_ ([arXiv:1903.06169](https://arxiv.org/abs/1903.06169))
The extreme form of bottom-up holographic model building is explored in
* {#HashimotoEtAl18} [[Koji Hashimoto]], Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, _Deep Learning and Holographic QCD_, Phys. Rev. D 98, 106014 (2018) ([arXiv:1809.10536](https://arxiv.org/abs/1809.10536))
where an appropriate [[bulk]] geometry is computer-generated from specified boundary behaviour.
More on this:
* Tetsuya Akutagawa, Koji Hashimoto, Takayuki Sumimoto, _Deep Learning and AdS/QCD_ ([arXiv:2005.02636](https://arxiv.org/abs/2005.02636))
On the other hand, an embedding of the hard-wall model into [[type IIB string theory]] is discussed in:
* [[Akash Singh]], [[K. P. Yogendran]], *Phases of a 10-D Holographic hard wall model* [[arXiv:2208.09387](https://arxiv.org/abs/2208.09387)]
##### Holographic light-front QCD
The holographic formulation of [[light cone gauge|light cone quantized]] [[QCD]] as [[holographic light front QCD]]:
Original articles:
* [[Stanley Brodsky]], [[Guy de Teramond]], _Light-Front Hadron Dynamics and AdS/CFT Correspondence_, Phys. Lett. B582:211-221, 2004 ([arXiv:hep-th/0310227](https://arxiv.org/abs/hep-th/0310227))
* [[Guy de Teramond]], [[Stanley Brodsky]], _Light-Front Holography: A First Approximation to QCD_, Phys. Rev. Lett. 102:081601, 2009 ([arXiv:0809.4899](https://arxiv.org/abs/0809.4899))
Review in:
* [[Stanley Brodsky]], [[Guy de Teramond]], [[Hans Günter Dosch]], [[Joshua Erlich]],
_Light-Front Holographic QCD and Emerging Confinement_, Physics Reports Volume 584, 8 July 2015, Pages 1-105 ([arXiv:1407.8131](https://arxiv.org/abs/1407.8131))
* {#ZhouDosch18} Liping Zou, [[Hans Günter Dosch]], _A very Practical Guide to Light Front Holographic QCD_, ([arXiv:1801.00607](https://arxiv.org/abs/1801.00607))
* [[Stanley Brodsky]], _Color Confinement and Supersymmetric Properties of Hadron Physics from Light-Front Holography_, International Conference on Beauty, Charm and Hyperon Hadrons (BEACH 2018) 17–23 June 2018, Peniche, Portugal ([arXiv:1912.12578](https://arxiv.org/abs/1912.12578))
* Ruben Sandapen, _An overview of light-front holography_ ([arXiv:2001.03479](https://arxiv.org/abs/2001.03479))
* [[Stanley Brodsky]], *Supersymmetric and Other Novel Features of Hadron Physics from Light-Front Holography*, Proceedings of the 24th Workshop, "What Comes Beyond the Standard Models" ([arXiv:2112.02453](https://arxiv.org/abs/2112.02453))
See also
* Harun Omer, _Embedding LFHQCD in Worldsheet String Theory_ ([arXiv:1909.12866](https://arxiv.org/abs/1909.12866))
Application to [[B-meson]] physics:
* [[Mohammad Ahmady]], _Holographic light-front QCD in B meson phenomenology_ ([arXiv:2001.00266](https://arxiv.org/abs/2001.00266))
##### Relation to hadron supersymmetry
Discussion of [[hadron supersymmetry]] via [[light cone gauge|light cone]] [[supersymmetric quantum mechanics]] in [[holographic light front QCD]]:
* {#dTDB14} [[Guy de Teramond]], [[Hans Günter Dosch]], [[Stanley Brodsky]], _Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding_, Phys. Rev. D 91, 045040 (2015) ([arXiv:1411.5243](https://arxiv.org/abs/1411.5243))
* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum_, Phys. Rev. D 92, 074010 (2015) ([arXiv:1504.05112](https://arxiv.org/abs/1504.05112))
* {#BTDL16} [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Cédric Lorcé, _Meson/Baryon/Tetraquark Supersymmetry from Superconformal Algebra and Light-Front Holography_, International Journal of Modern Physics AVol. 31, No. 19, 1630029 (2016) ([arXiv:1606.04638](https://arxiv.org/abs/1606.04638))
* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II_, Phys. Rev. D 95, 034016 (2017) ([arXiv:1612.02370](https://arxiv.org/abs/1612.02370))
* {#NielsenBrodsky18} [[Marina Nielsen]], [[Stanley Brodsky]], _Hadronic Superpartners from Superconformal and Supersymmetric Algebra_, Phys. Rev. D 97, 114001 (2018) ([arXiv:1802.09652](https://arxiv.org/abs/1802.09652))
* [[Marina Nielsen]], [[Stanley Brodsky]], Guy F. de Téramond, [[Hans Günter Dosch]], Fernando S. Navarra, Liping Zou, _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))
#### String- and M-theory corrections
{#StringAndMTheoryCorrection}
Generally on [[perturbative string theory]]-corrections (for small [['t Hooft coupling]] $\lambda = g_{YM}^2 N$) and/or [[M-theory]]-corrections ([[large N limit|small N]]) to the [[supergravity]]-approximation of the [[AdS/CFT correspondence]], i.e. the [[small N corrections]] to the correspondence:
On the general need for [[M-theory]] at small $N_c$ in gauge/gravity duality:
* [[Leonard Susskind]], _Another Conjecture about M(atrix) Theory_ ([arXiv:hep-th/9704080](https://arxiv.org/abs/hep-th/9704080))
* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], Section 6 of: _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))
Discussion of [[large N limit|small N]] effects in [[M-theory]] [[AdS4/CFT3]] and using the [[conformal bootstrap]]:
* Nathan B. Agmon, Shai M. Chester, Silviu S. Pufu, _Solving M-theory with the Conformal Bootstrap_, JHEP 06 (2018) 159 ([arXiv:1711.07343](https://arxiv.org/abs/1711.07343))
\linebreak
Specifically on [[small N corrections]] in [[holographic QCD]]:
* {#Basso08} B. Basso, _Cusp anomalous dimension in planar maximally supersymmetric Yang-Mills theory_, Continuous Advances in QCD 2008, pp. 317-328 (2008) ([spire:858223](http://inspirehep.net/record/858223), [doi:10.1142/9789812838667_0027](https://doi.org/10.1142/9789812838667_0027))
> "The result $[$(29)$]$ coincides exactly with the recent two-loop stringy correction computed in [Alday-Maldacena 07](https://arxiv.org/abs/0708.0672), providing a striking confirmation of the AdS/CFT correspondence."
* H. Dorn, H.-J. Otto, _On Wilson loops and $Q\bar Q$-potentials from the AdS/CFT relation at $T\geq 0$_, In: [[Anna Ceresole]], C. Kounnas , [[Dieter Lüst]], [[Stefan Theisen]] (eds.) _Quantum Aspects of Gauge Theories, Supersymmetry and Unification_, Lecture Notes in Physics, vol 525. Springer 2007 ([arXiv:hep-th/9812109](https://arxiv.org/abs/hep-th/9812109),
[doi:10.1007/BFb0104268](https://doi.org/10.1007/BFb0104268))
* Masayasu Harada, Shinya Matsuzaki, and Koichi Yamawaki, _Implications of holographic QCD in chiral perturbation theory with hidden local symmetry_, Phys. Rev. D 74, 076004 (2006) ([doi:10.1103/PhysRevD.74.076004](https://doi.org/10.1103/PhysRevD.74.076004))
(with an eye towards [[hidden local symmetry]])
* Csaba Csaki, [[Matthew Reece]], John Terning, _The AdS/QCD Correspondence: Still Undelivered_, JHEP 0905:067, 2009 ([arXiv:0811.3001](https://arxiv.org/abs/0811.3001))
* Salvatore Baldino, [[Stefano Bolognesi]], Sven Bjarke Gudnason, Deniz Koksal, _A Solitonic Approach to Holographic Nuclear Physics_, Phys. Rev. D 96, 034008 (2017) ([arXiv:1703.08695](https://arxiv.org/abs/1703.08695))
* Vikas Yadav, Aalok Misra, *On M-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to $\mathcal{O}(R^4)$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries* ([arXiv:2004.07259](https://arxiv.org/abs/2004.07259))
[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]
### Hadron physics
Application to [[confinement|confined]] [[hadron]]-physics:
Review:
* Henrique Boschi-Filho, _Hadrons in AdS/QCD models_, Journal of Physics: Conference Series, Volume 706, Section 4 2008 ([doi:10.1088/1742-6596/706/4/042008](http://iopscience.iop.org/article/10.1088/1742-6596/706/4/042008))
* Kanabu Nawa, Hideo Suganuma, Toru Kojo, _Baryons in Holographic QCD_, Phys.Rev.D75:086003, 2007 ([arXiv:hep-th/0612187](https://arxiv.org/abs/hep-th/0612187))
* Deog Ki Hong, Mannque Rho, Ho-Ung Yee, [[Piljin Yi]], _Chiral Dynamics of Baryons from String Theory_, Phys. Rev. D76:061901, 2007 ([arXiv:hep-th/0701276](https://arxiv.org/abs/hep-th/0701276))
* Deog Ki Hong, _Baryons in holographic QCD_, talk at _[From Strings to Things 2008](http://www.int.washington.edu/PROGRAMS/08-1.html)_ ([pdf](http://www.int.washington.edu/talks/WorkShops/int_08_1/People/Hong_D/Hong.pdf))
* [[Johanna Erdmenger]], [[Nick Evans]], [[Ingo Kirsch]], [[Ed Threlfall]], _Mesons in Gauge/Gravity Duals - A Review_, Eur. Phys. J. A **35** (2008) 81-133 [[arXiv:0711.4467](https://arxiv.org/abs/0711.4467), [doi:10.1140/epja/i2007-10540-1](https://doi.org/10.1140/epja/i2007-10540-1)]
* [[Stanley Brodsky]], _Hadron Spectroscopy and Dynamics from Light-Front Holography and Superconformal Algebra_ ([arXiv:1802.08552](https://arxiv.org/abs/1802.08552))
* [[Koji Hashimoto]], [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Holographic Baryons : Static Properties and Form Factors from Gauge/String Duality_, Prog. Theor. Phys.120:1093-1137, 2008 ([arXiv:0806.3122](https://arxiv.org/abs/0806.3122))
* Alex Pomarol, Andrea Wulzer, _Baryon Physics in Holographic QCD_, Nucl. Phys. B809:347-361, 2009 ([arXiv:0807.0316](https://arxiv.org/abs/0807.0316))
* Thomas Gutsche, Valery E. Lyubovitskij, Ivan Schmidt, Alfredo Vega, _Nuclear physics in soft-wall AdS/QCD: Deuteron electromagnetic form factors_, Phys. Rev. D 91, 114001 (2015) ([arXiv:1501.02738](https://arxiv.org/abs/1501.02738))
* {#PomarolWulzer09} Alex Pomarol, Andrea Wulzer, _Baryon physics in a five-dimensional model of hadrons_ ([arXiv:0904.2272](https://arxiv.org/abs/0904.2272)), Chapter 18 in: [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
* Marco Claudio Traini, _Generalized Parton Distributions: confining potential effects within AdS/QCD_, Eur. Phys. J. C (2017) 77:246 ([arXiv:1608.08410](https://arxiv.org/abs/1608.08410))
* {#CaiLi17} Wenhe Cai, Si-wen Li, _Holographic three flavor baryon in the Witten-Sakai-Sugimoto model with the D0-D4 background_, Eur. Phys. J. C (2018) 78: 446 ([arXiv:1712.06304](https://arxiv.org/abs/1712.06304))
* Valery E. Lyubovitskij, Ivan Schmidt, _Gluon parton densities in soft-wall AdS/QCD_ ([arXiv:2012.01334](https://arxiv.org/abs/2012.01334))
* Lorenzo Bartolini, Sven Bjarke Gudnason, *Symmetry energy in holographic QCD* [[arXiv:2209.14309](https://arxiv.org/abs/2209.14309)]
See also:
* {#CLFH22} Ruixiang Chen, Danning Li, Kazem Bitaghsir Fadafan, Mei Huang, *The hadron spectra and pion form factor in dynamical holographic QCD model with anomalous 5D mass of scalar field* [[arXiv:2212.10363](https://arxiv.org/abs/2212.10363)]
#### Baryons as instantons
[[baryons]] as [[instantons]]:
* {#KatzLewandowskiSchwartz05} Emanuel Katz, Adam Lewandowski, Matthew D. Schwartz, Phys. Rev. D74:086004, 2006 ([arXiv:hep-ph/0510388](https://arxiv.org/abs/hep-ph/0510388))
* {#HSSY07} Hiroyuki Hata, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Shinichiro Yamato, _Baryons from instantons in holographic QCD_, Prog.Theor.Phys.117:1157, 2007 ([arXiv:hep-th/0701280](https://arxiv.org/abs/hep-th/0701280))
* Hiroyuki Hata, Masaki Murata, _Baryons and the Chern-Simons term in holographic QCD with three flavors_ ([arXiv:0710.2579](https://arxiv.org/abs/0710.2579))
* Salvatore Baldino, [[Stefano Bolognesi]], Sven Bjarke Gudnason, Deniz Koksal, _A Solitonic Approach to Holographic Nuclear Physics_, Phys. Rev. D 96, 034008 (2017) ([arXiv:1703.08695](https://arxiv.org/abs/1703.08695))
* Chandan Mondal, Dipankar Chakrabarti, Xingbo Zhao, _Deuteron transverse densities in holographic QCD_, Eur. Phys. J. A 53, 106 (2017) ([arXiv:1705.05808](https://arxiv.org/abs/1705.05808))
* Stanley J. Brodsky, _Color Confinement, Hadron Dynamics, and Hadron Spectroscopy from Light-Front Holography and Superconformal Algebra_ ([arXiv:1709.01191](https://arxiv.org/abs/1709.01191))
* Alfredo Vega, M. A. Martin Contreras, _Melting of scalar hadrons in an AdS/QCD model modified by a thermal dilaton_ ([arXiv:1808.09096](https://arxiv.org/abs/1808.09096))
* Meng Lv, Danning Li, Song He, _Pion condensation in a soft-wall AdS/QCD model_ ([arXiv:1811.03828](https://arxiv.org/abs/1811.03828))
* Kazem Bitaghsir Fadafan, Farideh Kazemian, Andreas Schmitt, _Towards a holographic quark-hadron continuity_ ([arXiv:1811.08698](https://arxiv.org/abs/1811.08698))
* {#SonnenscheinWeissman18} [[Jacob Sonnenschein]], Dorin Weissman, _Excited mesons, baryons, glueballs and tetraquarks: Predictions of the Holography Inspired Stringy Hadron model_, ([arXiv:1812.01619](https://arxiv.org/abs/1812.01619))
* Kazem Bitaghsir Fadafan, Farideh Kazemian, Andreas Schmitt, _Towards a holographic quark-hadron continuity_ ([arXiv:1811.08698](https://arxiv.org/abs/1811.08698))
* {#AbdolmalekiBoroun18} M. Abdolmaleki, G.R. Boroun, _The Survey of Proton Structure Function with the AdS/QCD Correspondence_ Phys.Part.Nucl.Lett. 15 (2018) no.6, 581-584 ([doi:10.1134/S154747711806002X](https://doi.org/10.1134/S154747711806002X))
* Si-wen Li, Hao-qian Li, Sen-kai Luo, *Corrections to the instanton configuration as baryon in holographic QCD* [[arXiv:2209.12521](https://arxiv.org/abs/2209.12521)]
On relation to [[type 0 string theory]]:
* {#GLMR00} Roberto Grena, Simone Lelli, Michele Maggiore, Anna Rissone, _Confinement, asymptotic freedom and renormalons in type 0 string duals_, JHEP 0007 (2000) 005 ([arXiv:hep-th/0005213](https://arxiv.org/abs/hep-th/0005213))
* {#AAS19} Mohammad Akhond, Adi Armoni, Stefano Speziali, _Phases of $U(N_c)$ $QCD_3$ from Type 0 Strings and Seiberg Duality_ ([arxiv:1908.04324](https://arxiv.org/abs/1908.04324))
See also
* S. S. Afonin, _AdS/QCD without Kaluza-Klein modes: Radial spectrum from higher dimensional QCD operators_ ([arXiv:1905.13086](https://arxiv.org/abs/1905.13086))
In relation to the [[open string]] [[tachyon]]:
* M. Järvinen, E. Kiritsis, F. Nitti, E. Préau, *Tachyon-dependent Chern-Simons terms and the V-QCD Baryon* [[arXiv:2209.05868](https://arxiv.org/abs/2209.05868)]
See also:
* Keiichiro Hori, Hideo Suganuma, Hiroki Kanda, *Numerical analysis of a baryon and its dilatation modes in holographic QCD* [[arXiv:2307.16590](https://arxiv.org/abs/2307.16590)]
#### Baryons as wrapped branes
[[baryons]] as [[wrapped brane|wrapped]] [[D4-branes]]:
original articles:
* {#Witten98b} [[Edward Witten]], _Baryons And Branes In Anti de Sitter Space_, JHEP 9807:006, 1998 ([arXiv:hep-th/9805112](https://arxiv.org/abs/hep-th/9805112))
* {#GrossOoguri98} [[David Gross]], [[Hirosi Ooguri]], _Aspects of Large $N$ Gauge Theory Dynamics as Seen by String Theory_, Phys. Rev. D58:106002, 1998 ([arXiv:hep-th/9805129](https://arxiv.org/abs/hep-th/9805129))
* {#BISY98} A. Brandhuber, N. Itzhaki, [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], _Baryons from Supergravity_, JHEP 9807:020, 1998 ([arxiv:hep-th/9806158](https://arxiv.org/abs/hep-th/9806158))
* Yosuke Imamura, _Supersymmetries and BPS Configurations on Anti-de Sitter Space_, Nucl. Phys. B537:184-202, 1999 ([arxiv:hep-th/9807179](https://arxiv.org/abs/hep-th/9807179))
* {#CGS98} [[Curtis Callan]], Alberto Guijosa, Konstantin G. Savvidy, _Baryons and String Creation from the Fivebrane Worldvolume Action_ ([arxiv:hep-th/9810092](https://arxiv.org/abs/hep-th/9810092))
* [[Curtis Callan]], Alberto Guijosa, Konstantin G. Savvidy, Oyvind Tafjord, _Baryons and Flux Tubes in Confining Gauge Theories from Brane Actions_, Nucl. Phys. B555 (1999) 183-200 ([arxiv:hep-th/9902197](https://arxiv.org/abs/hep-th/9902197))
Review:
* {#Yi09} [[Piljin Yi]], _Holographic Baryons_ ([arXiv:0902.4515](https://arxiv.org/abs/0902.4515), [doi:10.1142/9789814280709_0016](https://www.worldscientific.com/doi/abs/10.1142/9789814280709_0016)), Chapter 16 in: [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
* {#Yi11} [[Piljin Yi]], _Precision Holographic Baryons_, AIP Conference Proceedings 1388, 106 (2011) ([arXiv:1103.1684](https://arxiv.org/abs/1103.1684), [doi:10.1063/1.3647358](https://aip.scitation.org/doi/abs/10.1063/1.3647358))
* {#Yi13} [[Piljin Yi]], _Two Approaches to Holographic Baryons/Nuclei_, Few-Body Syst (2013) 54: 77. ([doi:10.1007/s00601-012-0373-7](https://doi.org/10.1007/s00601-012-0373-7))
#### Baryons as Skyrmions
{#ReferencesBaryonsSkyrmions}
[[baryons]] as [[Skyrmions]]:
Review:
* {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmion and String theory_, chapter 15 in [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))
* {#RhoEtAl16} [[Mannque Rho]] et al., Introduction to _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710), [pdf](https://www.worldscientific.com/doi/suppl/10.1142/9710/suppl_file/9710_intro.pdf))
Original articles
* Kanabu Nawa, Hideo Suganuma, Toru Kojo, _Brane-induced Skyrmions: Baryons in Holographic QCD_, Prog.Theor.Phys.Suppl.168:231-236, 2007 ([arXiv:hep-th/0701007](https://arxiv.org/abs/hep-th/0701007))
* Hovhannes R. Grigoryan, _Baryon as skyrmion-like soliton from the
holographic dual model of QCD_, talk at _[From Strings to Things 2008](http://www.int.washington.edu/PROGRAMS/08-1.html)_ ([pdf](https://www.jlab.org/div_dept/theory/talks/2008/grigoryan08_INT.pdf))
* [[Paul Sutcliffe]], _Skyrmions, instantons and holography_, JHEP 1008:019, 2010 ([arXiv:1003.0023](https://arxiv.org/abs/1003.0023))
* [[Stefano Bolognesi]], [[Paul Sutcliffe]], _The Sakai-Sugimoto soliton_, JHEP 1401:078, 2014 ([arXiv:1309.1396](https://arxiv.org/abs/1309.1396))
* [[Paul Sutcliffe]], _Holographic Skyrmions_, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 ([spire:1383608](http://inspirehep.net/record/1383608))
#### Nucleons
nucleon [[form factors]] via [[holographic QCD]]:
* Kiminad A. Mamo, [[Ismail Zahed]], _Nucleon mass radii and distribution: Holographic QCD, Lattice QCD and GlueX data_ ([arXiv:2103.03186](https://arxiv.org/abs/2103.03186))
* [[Roldão da Rocha]], *Information entropy of nuclear electromagnetic transitions in AdS/QCD* [[arXiv:2208.07191](https://arxiv.org/abs/2208.07191)]
> The derived parameters are shown
to corroborate experimental data with great accuracy
via a [[nuclear matrix model]]:
* [[Koji Hashimoto]], [[Norihiro Iizuka]], [[Piljin Yi]], _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))
* Si-wen Li, Tuo Jia, _Matrix model and Holographic Baryons in the D0-D4 background_, Phys. Rev. D 92, 046007 (2015) ([arXiv:1506.00068](https://arxiv.org/abs/1506.00068))
* [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))
* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))
nuclear binding energy
* Salvatore Baldino, Lorenzo Bartolini, [[Stefano Bolognesi]], Sven Bjarke Gudnason, _Holographic Nuclear Physics with Massive Quarks_ ([arXiv:2102.00680](https://arxiv.org/abs/2102.00680))
nuclear binding energy via the [[nuclear matrix model]]:
* [[Koji Hashimoto]], [[Yoshinori Matsuo]], _Nuclear binding energy in holographic QCD_ ([arXiv:2103.03563](https://arxiv.org/abs/2103.03563))
#### Pentaquarks
[[pentaquarks]]:
* Kazuo Ghoroku, Akihiro Nakamura, Tomoki Taminato, Fumihiko Toyoda, _Holographic Penta and Hepta Quark State in Confining Gauge Theories_,
JHEP 1008:007,2010 ([arxiv:1003.3698](https://arxiv.org/abs/1003.3698))
#### Parton distribution functions
* Matteo Rinaldi, _Double parton correlations in mesons within AdS/QCD soft-wall models: a first comparison with lattice data_ ([arXiv:2003.09400](https://arxiv.org/abs/2003.09400))
[[!include heavy flavor hadrodynamics via holographic QCD -- references]]
### Flux string breaking
* Oleg Andreev, _String Breaking, Baryons, Medium, and Gauge/String Duality_ ([arXiv:2003.09880](https://arxiv.org/abs/2003.09880))
### Glueball physics
* {#Suzuki01} Kenji Suzuki, _D0-D4 system and $QCD_{3+1}$_, Phys.Rev. D63 (2001) 084011 ([arXiv:hep-th/0001057](https://arxiv.org/abs/hep-th/0001057))
* S.S. Afonin, A.D. Katanaeva, _Glueballs and deconfinement temperature in AdS/QCD_ ([arXiv:1809.07730](https://arxiv.org/abs/1809.07730))
* S. S. Afonin, A. D. Katanaeva, E. V. Prokhvatilov, M. I. Vyazovsky, _Deconfinement temperature in AdS/QCD from the spectrum of scalar glueballs_ ([arXiv:2001.07990](https://arxiv.org/abs/2001.07990))
* Cornélio Rodrigues Filho, _Glueballs in the Klebanov-Strassler Theory: Pseudoscalars vs Scalars_ ([arXiv:2011.12689](https://arxiv.org/abs/2011.12689))
[[!include holographic Schwinger effect -- references]]
### Application to vector meson dominance
Derivation of [[vector meson dominance]] via [[holographic QCD]]:
* D.T. Son, M.A. Stephanov, _QCD and dimensional deconstruction_, Phys. Rev. D69 (2004) 065020 ([arXiv:hep-ph/0304182](https://arxiv.org/abs/hep-ph/0304182))
* Sungho Hong, Sukjin Yoon, [[Matthew Strassler]], _On the Couplings of Vector Mesons in AdS/QCD_, JHEP 0604 (2006) 003 ([arXiv:hep-th/0409118](https://arxiv.org/abs/hep-th/0409118))
* Sungho Hong, Sukjin Yoon, [[Matthew Strassler]], _On the Couplings of the Rho Meson in AdS/QCD_ ([cds:816440](https://cds.cern.ch/record/816440), [arXiv:hep-ph/0501197](https://arxiv.org/abs/hep-ph/0501197))
* Leandro Da Rold, Alex Pomarol, _Chiral symmetry breaking from five dimensional spaces_, Nucl. Phys. B721:79-97, 2005 ([arXiv:hep-ph/0501218](https://arxiv.org/abs/hep-ph/0501218))
and specifically in the [[Witten-Sakai-Sugimoto model]]:
* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], p. 18 and Section 5 of: _More on a holographic dual of QCD_, Progr. Theor. Phys. 114: 1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))
### Application to the quark-gluon plasma
Application to the [[quark-gluon plasma]]:
Expositions and reviews include
* Pavel Kovtun, _Quark-Gluon Plasma and String Theory_, RHIC news (2009) ([blog entry](http://www.bnl.gov/rhic/news/091107/story2.asp))
* Makoto Natsuume, _String theory and quark-gluon plasma_ ([arXiv:hep-ph/0701201](http://arxiv.org/abs/hep-ph/0701201))
* [[Steven Gubser]], _Using string theory to study the quark-gluon plasma: progress and perils_ ([arXiv:0907.4808](http://arxiv.org/abs/0907.4808))
* {#BiagazziCotrone12} Francesco Biagazzi, A. Cotrone, _Holography and the quark-gluon plasma_, AIP Conference Proceedings 1492, 307 (2012) ([doi:10.1063/1.4763537]( https://doi.org/10.1063/1.4763537), [slides pdf](http://cp3-origins.dk/content/movies/2013-01-14-bigazzi.pdf))
* {#Brambilla14} Brambilla et al., section 9.2.2 of _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981 ([doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))
Holographic discussion of the [[shear viscosity]] of the quark-gluon plasma goes back to
* {#PolicastroSonStarinets01} [[Giuseppe Policastro]], D.T. Son, A.O. Starinets, _Shear viscosity of strongly coupled $N=4$ supersymmetric Yang-Mills plasma_, Phys. Rev. Lett.87:081601, 2001 ([arXiv:hep-th/0104066](http://arxiv.org/abs/hep-th/0104066))
Other original articles include:
* Brett McInnes, _Holography of the Quark Matter Triple Point_ ([arXiv:0910.4456](http://arxiv.org/abs/0910.4456))
* Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, _Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma_ ([arXiv:1003.1138](http://arxiv.org/abs/1003.1138))
* Mansi Dhuria, Aalok Misra, _Towards MQGP_, JHEP 1311 (2013) 001 ([arXiv:1306.4339](https://arxiv.org/abs/1306.4339))
* Irina Ya. Aref'eva, Kristina Rannu, Pavel Slepov, _Energy Loss in Holographic Anisotropic Model for Heavy Quarks in External Magnetic Field_ ([arXiv:2012.05758](https://arxiv.org/abs/2012.05758))
See also:
* Si-wen Li, Yi-peng Zhang, *Correlation function of fundamental fermion in holographic QCD* [[arXiv:2307.13357](https://arxiv.org/abs/2307.13357)]
### Application to lepton anomalous magnetic moment
Application to [[anomalous magnetic moment]] of the [[muon]]:
* Luigi Cappiello, _What does Holographic QCD predict for anomalous $(g-2)_\mu$?_, 2015 ([pdf](https://agenda.infn.it/getFile.py/access?contribId=19&sessionId=5&resId=0&materialId=paper&confId=9430))
* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the anomalous magnetic moment of the muon_ ([arXiv:1912.01596](https://arxiv.org/abs/1912.01596))
* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the muon $g-2$_ ([arXiv:2012.06514](https://arxiv.org/abs/2012.06514))
* [[Josef Leutgeb]], Jonas Mager, [[Anton Rebhan]], *Holographic QCD and the muon anomalous magnetic moment* ([arXiv:2110.07458](https://arxiv.org/abs/2110.07458))
### Application to the Higgs field
{#ReferencesApplicationToHiggsField}
Application to the [[Higgs field]]:
* {#EspiruKatanaeva18} Domenec Espriu, Alisa Katanaeva, _Composite Higgs Models: a new holographic approach_ ([arXiv:1812.01523](https://arxiv.org/abs/1812.01523))
* [[Johanna Erdmenger]], [[Nick Evans]], [[Werner Porod]], [[Konstantinos Rigatos]], *Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models*, JHEP **58** (2021) [[arXiv:2010.10279](https://arxiv.org/abs/2010.10279), <a href="https://doi.org/10.1007/JHEP02(2021)058">doi:10.1007/JHEP02(2021)058</a>]
### Application to $\theta$-angle axions and strong CP-problem
Realization of [[axions]] and solution of [[strong CP-problem]]:
* Francesco Bigazzi, Alessio Caddeo, Aldo L. Cotrone, Paolo Di Vecchia, Andrea Marzolla, _The Holographic QCD Axion_ ([arXiv:1906.12117](https://arxiv.org/abs/1906.12117))
Discussion of the [[theta angle]] via the the [[graviphoton]] in the [[higher WZW term]] of the [[D4-brane]]:
* Si-wen Li, around (3.1) of _The theta-dependent Yang-Mills theory at finite temperature in a holographic description_ ([arXiv:1907.10277](https://arxiv.org/abs/1907.10277))
Discussion of the Witten-Veneziano mechanism
* [[Josef Leutgeb]], [[Anton Rebhan]], _Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD_ ([arxiv:1909.12352](https://arxiv.org/abs/1909.12352))
See also:
* Si-wen Li, Hao-qian Li, Yi-peng Zhang, *The worldvolume fermion as baryon in holographic QCD with a theta angle* [[arXiv:2402.01197](https://arxiv.org/abs/2402.01197)]
### Application to the QCD trace anomaly
Discussion of the [[QCD trace anomaly]]:
* Jose L. Goity, Roberto C. Trinchero, _Holographic models and the QCD trace anomaly_, Phys. Rev. D 86, 034033 – 2012 ([arXiv:1204.6327](https://arxiv.org/abs/1204.6327))
* Aalok Misra, Charles Gale, _The QCD Trace Anomaly at Strong Coupling from M-Theory_ ([arXiv:1909.04062](https://arxiv.org/abs/1909.04062))
The [[QCD trace anomaly]] affects notably the [[equation of state]] of the [[quark-gluon plasma]], see there at _[References -- Holographic description of quark-gluon plasma](quark-gluon+plasma#ReferencesViaAdSCFT)_
### Application to parton distribution
* Akira Watanabe, Takahiro Sawada, Mei Huang, _Extraction of gluon distributions from structure functions at small x in holographic QCD_ ([arxiv:1910.10008](https://arxiv.org/abs/1910.10008))
> Understanding the nucleon structure is one of the most
important research topics in fundamental science, and tremendous efforts have been done to deepen our knowledge over several decades. $[...]$ Since $[these]$ are highly nonperturbative physical quantities, in principle they are not calculable by the direct use of QCD. Furthermore, although there is available data, this has large errors. These facts cause the huge uncertainties which can be seen in the preceding studies based on the global QCD analysis.
> In this work, we investigate the gluon distribution in nuclei by calculating the structure functions in the framework of holographic QCD, which is constructed based on the AdS/CFT correspondence.
### Application to QCD phases
{#ReferencesColourSuperconductivity}
Application to [[phase of matter|phases]] of [[QCD]]:
* R. Narayanan, H. Neuberger, _A survey of large $N$ continuum phase transitions_, PoSLAT 2007:020, 2007 ([arXiv:0710.0098](https://arxiv.org/abs/0710.0098))
To [[colour superconductivity]]:
* Kazem Bitaghsir Fadafan, Jesus Cruz Rojas, [[Nick Evans]], _A Holographic Description of Colour Superconductivity_, Phys. Rev. D **98** (2018) 066010 [[arXiv:1803.03107](https://arxiv.org/abs/1803.03107), [doi:10.1103/PhysRevD.98.066010](https://doi.org/10.1103/PhysRevD.98.066010)]
to [[confinement]]/[[quark-gluon plasma|deconfinement]] phase transiton:
* {#LYY18} Meng-Wei Li, Yi Yang, Pei-Hung Yuan _Imprints of Early Universe on Gravitational Waves from First-Order Phase Transition in QCD_ ([arXiv:1812.09676](https://arxiv.org/abs/1812.09676))
With [[magnetic fields]]:
* Umut Gursoy, *Holographic QCD and magnetic fields* ([arXiv:2104.02839](https://arxiv.org/abs/2104.02839))
Of relevance in [[neutron stars]]:
* Carlos Hoyos, Niko Jokela, [[Matti Jarvinen]], Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, *Transport in strongly coupled quark matter*, Phys.Rev.Lett. 125 (2020) 241601 ([arXiv:2005.14205](https://arxiv.org/abs/2005.14205))
* Carlos Hoyos, Niko Jokela, [[Matti Järvinen]], Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, *Holographic approach to transport in dense QCD matter*, Phys.Rev.D 105 (2022) 6, 066014 ([arXiv:2109.12122](https://arxiv.org/abs/2109.12122))
* [[Matti Järvinen]], *Holographic modeling of nuclear matter and neutron stars*, Eur.Phys.J.C. 56 2021 ([arXiv:2110.08281](https://arxiv.org/abs/2110.08281))
* Carlos Hoyos, Niko Jokela, Aleksi Vuorinen, *Holographic approach to compact stars and their binary mergers*, Prog.Part.Nucl.Phys. 126 (2022) 103972 ([arXiv:2112.08422] (https://arxiv.org/abs/2112.08422))
See also
* Yosuke Imamura, _Baryon Mass and Phase Transitions in Large N Gauge Theory_, Prog. Theor. Phys. 100 (1998) 1263-1272 ([arxiv:hep-th/9806162](https://arxiv.org/abs/hep-th/9806162))
* Varun Sethi, _A study of phases in two flavour holographic QCD_ ([arXiv:1906.10932](https://arxiv.org/abs/1906.10932))
* {#ABBCN18} Riccardo Argurio, Matteo Bertolini, Francesco Bigazzi, Aldo L. Cotrone, Pierluigi Niro, _QCD domain walls, Chern-Simons theories and holography_, J. High Energ. Phys. (2018) 2018: 90 ([arXiv:1806.08292](https://arxiv.org/abs/1806.08292))
* Alfonso Ballon-Bayona, Jonathan P. Shock, Dimitrios Zoakos, _Magnetic catalysis and the chiral condensate in holographic QCD_ ([arXiv:2005.00500](https://arxiv.org/abs/2005.00500))
* Yi Yang, Pei-Hung Yuan, _QCD Phase Diagram by Holography_ ([arXiv:2011.11941](https://arxiv.org/abs/2011.11941))
* Nicolas Kovensky, Aaron Poole, Andreas Schmitt, *Phases of cold holographic QCD: baryons, pions and rho mesons* [[arXiv:2302.10675](https://arxiv.org/abs/2302.10675)]
### Application to meson physics
Application to [[meson]] physics:
* Daniel Ávila, Leonardo Patiño, _Melting holographic mesons by cooling a magnetized quark gluon plasma_ ([arXiv:2002.02470](https://arxiv.org/abs/2002.02470))
* Xuanmin Cao, Hui Liu, Danning Li, _Pion quasiparticles and QCD phase transitions at finite temperature and isospin density from holography_, Phys. Rev. D 102, 126014 (2020) ([arXiv:2009.00289](https://arxiv.org/abs/2009.00289))
* Xuanmin Cao, Songyu Qiu, Hui Liu, Danning Li, _Thermal properties of light mesons from holography_ ([arXiv:2102.10946](https://arxiv.org/abs/2102.10946))
Application to [[quarkonium]]:
* Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, _Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma_ ([arXiv:1003.1138](http://arxiv.org/abs/1003.1138))
* Rico Zöllner, Burkhard Kampfer, _Holographic vector meson melting in a thermal gravity-dilaton background related to QCD_ ([arXiv:2002.07200](https://arxiv.org/abs/2002.07200))
* Miguel Angel Martin Contreras, Saulo Diles, Alfredo Vega, _Heavy quarkonia spectroscopy at zero and finite temperature in bottom-up AdS/QCD_ ([arXiv:2101.06212](https://arxiv.org/abs/2101.06212))
[[!include application of holographic QCD to B-meson physics -- references]]
### Relation to holographic entanglement entropy
Relating to [[holographic entanglement entropy]]:
* Zhibin Li, Kun Xu, Mei Huang, _The entanglement properties of holographic QCD model with a critical end point_ ([arXiv:2002.08650](https://arxiv.org/abs/2002.08650))
### Application to defects
Application to QCD [[QFT with defects|with defects]]:
* Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, _On Classification of QCD defects via holography_, Phys. Rev. D79:106003, 2009 ([arxiv:0902.1842](https://arxiv.org/abs/0902.1842))
### Application to thermal QCD
Application to [[thermal field theory|thermal]] [[QCD]]:
* Vikas Yadav, Aalok Misra, _Towards Thermal QCD from M theory at Intermediate 't Hooft Coupling and G-Structure Classification of Non-supersymmetric Underlying Geometries_ ([arXiv:2004.07259](https://arxiv.org/abs/2004.07259))
* Alfonso Ballon-Bayona, Luis A. H. Mamani, Alex S. Miranda, Vilson T. Zanchin, _Effective holographic models for QCD: Thermodynamics and viscosity coefficients_ ([arXiv:2103.14188](https://arxiv.org/abs/2103.14188))
[[!redirects AdS-QCD correspondences]]
[[!redirects AdS/QCD correspondence]]
[[!redirects AdS-QCD]]
[[!redirects AdS/QCD]]
[[!redirects AdS-QCD duality]]
[[!redirects AdS-QCD dualities]]
[[!redirects Sakai-Sugimoto model]]
[[!redirects Sakai-Sugimoto models]]
[[!redirects Witten-Sakai-Sugimoto model]]
[[!redirects Witten-Sakai-Sugimoto models]]
[[!redirects Sakai-Sugimoto-Witten model]]
[[!redirects Sakai-Sugimoto-Witten models]]
[[!redirects WSS model]]
[[!redirects WSS models]]
[[!redirects WSS-model]]
[[!redirects WSS-models]]
[[!redirects holographic QCD]]
[[!redirects holographic quantum chromodynamics]]
[[!redirects improved holographic QCD]]
[[!redirects improved holographic quantum chromodynamics]]
[[!redirects Ads/QCD]]
|
AdS2-CFT1 -- references | https://ncatlab.org/nlab/source/AdS2-CFT1+--+references |
[[!include SYK-model and AdS2-CFT1 -- references]]
Discussion of [[small N corrections]] via a [[lattice QFT]]-Ansatz on the AdS side:
* Richard C. Brower, Cameron V. Cogburn, A. Liam Fitzpatrick, Dean Howarth, Chung-I Tan, _Lattice Setup for Quantum Field Theory in $AdS_2$_ ([arXiv:1912.07606](https://arxiv.org/abs/1912.07606))
See also:
* Gregory J. Galloway, Melanie Graf, Eric Ling, _A conformal infinity approach to asymptotically $AdS_2 \times S^{n-1}$ spacetimes_ ([arXiv:2003.00093](https://arxiv.org/abs/2003.00093))
### Random matrix theory in $AdS_2/CFT_1$
On [[Jackiw-Teitelboim gravity]] dual to [[random matrix theory]] (via [[AdS2/CFT1]] and [[topological recursion]]):
* {#AlmheiriPolchinski14} [[Ahmed Almheiri]], [[Joseph Polchinski]], _Models of $AdS_2$ Backreaction and Holography_, J. High Energ. Phys. (2015) 2015: 14. ([arXiv:1402.6334](https://arxiv.org/abs/1402.6334))
* [CGHSS 16](#CGHSS16)
* {#SaadShenkerStanford19} [[Phil Saad]], [[Stephen Shenker]], [[Douglas Stanford]], _JT gravity as a matrix integral_ ([arXiv:1903.11115](https://arxiv.org/abs/1903.11115))
* {#StanfordWitten19} [[Douglas Stanford]], [[Edward Witten]], _JT Gravity and the Ensembles of Random Matrix Theory_ ([arXiv:1907.03363](https://arxiv.org/abs/1907.03363))
### BFSS matrix model in $AdS_2/CFT_1$
On [[AdS2/CFT1]] with the [[BFSS matrix model]] on the CFT side and [[black hole in string theory|black hole-like solutions]] in [[type IIA supergravity]] on the AdS side:
* [[Juan Maldacena]], Alexey Milekhin, _To gauge or not to gauge?_, JHEP 04 (2018) 084 ([arxiv:1802.00428](https://arxiv.org/abs/1802.00428))
and on its analog of [[holographic entanglement entropy]]:
* Tarek Anous, Joanna L. Karczmarek, Eric Mintun, [[Mark Van Raamsdonk]], Benson Way, _Areas and entropies in BFSS/gravity duality_ ([arXiv:1911.11145](https://arxiv.org/abs/1911.11145))
See also
* Takeshi Morita, Hiroki Yoshida, _A Critical Dimension in One-dimensional Large-N Reduced Models_ ([arXiv:2001.02109](https://arxiv.org/abs/2001.02109))
### Flat space limit
The [[SYK model]] in [[flat space holography]]:
* Hamid Afshar, Hernan Gonzalez, [[Daniel Grumiller]], Dmitri Vassilevich, *Flat space holography and complex SYK*, Phys. Rev. D 101, 086024 ([arXiv:1911.05739](https://arxiv.org/abs/1911.05739), [doi:10.1103/PhysRevD.101.086024](http://dx.doi.org/10.1103/PhysRevD.101.086024))
[[!include D1-D3 intersections in AdS2-CFT1 -- references]]
[[!include weight systems on chord diagrams in physics]]
|
AdS3-CFT2 and CS-WZW correspondence | https://ncatlab.org/nlab/source/AdS3-CFT2+and+CS-WZW+correspondence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
One incarnation of the [[holographic principle]] in [[quantum field theory]] is the correspondence between 3d $G$-[[Chern-Simons theory]] as the [[bulk field theory]] and the 2d [[Wess-Zumino-Witten model]] on a suitable [[Lie group]] $G$ as the [[boundary field theory]]. This case stands out in that it was known and understood already before the [[holographic principle]] was formulated as such, motivated from [[bulk field theories]] of [[gravity]]. Notably the CS/WZW correspondence is an actual [[theorem]] instead of just a vague [[conjecture]], as for much of the [[AdS-CFT correspondence]].
Indeed, the natural equivalence between the [[space of quantum states]] of [[Chern-Simons theory]] on a [[surface]] $\Sigma$ and the space of [[conformal blocks]] of the [[WZW model]] on $\Sigma$ was understood in the seminal article ([Witten 89](#Witten89)) and subsequently discussed in much detail, see also at [CS-theory -- References](Chern-Simons%20theory#References). The explicit holographic correspondence between the [[wavefunctions]] of Chern-Simons theory and the [[correlators]] of the WZW model is reviewed for instance in ([Gawędzki 99, around p. 30](#Gawedzki99)). For the case of abelian gauge group and with an eye towards generalization to [[self-dual higher gauge theory]] a review is in ([Witten 96, section 2](#Witten96)).
(This correspondence is captured [[functor|functorially]] by the notion of the _[[modular functor]]_ of the 2d theory, see there for more.)
For instance the [[FRS formalism]] _constructs_ all [[rational conformal field theories]] as full [[FQFTs]] holographically from the [[Reshetikhin-Turaev construction]] of the 3d Chern-Simons theory and fully classifies them this way.
Later the [[AdS-CFT correspondence]] came to be understood as a canonical or default implementation of the [[holographic principle]]. Here the [[bulk field theory]] is a theory of [[3d quantum gravity]] which is very much like traditional [[Chern-Simons theory]] but may crucially differ from it, see at _[[Chern-Simons gravity]]_ the [comments on the non-perturbative regime](Chern-Simons%20gravity#ProblemsInTheNonPerturbativeRegime). Instead some variant of CS3/WTW2 appears as one "sector" inside AdS3/CFT2, this is discussed in ([Gukov-Martinec-Moore-Strominger 04](#GukovMartinecMooreStrominger04)).
But notice that also plain [[Chern-Simons theory]] is a [[string theory]], but of [[topological strings]]. For more on this see at _[[TCFT]]_ the section _[Worldsheet and effective background theories](TCFT#ActionFunctionals)_.
A general argument that in sectors of the [[AdS-CFT correspondence]] the [[conformal blocks]] on the CFT-side are given just by the [[higher dimensional Chern-Simons theory]]-sector inside the dual [[gravity]] theory is in ([Witten98](#Witten98)). This applies notably to the duality between [[7-dimensional Chern-Simons theory]] and the conformal blocks in the [[6d (2,0)-superconformal QFT]] on the [[M5-brane]].
## Related concepts
* [[p-adic AdS/CFT correspondence]]
* [[3d quantum gravity]]
* [[BTZ black hole]]
* [[quantization of Chern-Simons theory]]/[[quantization of loop groups]]
* [[super 1-brane in 3d]]
* [[TT deformation]]
## References
### CS/WZW
The original article on the CS/WZW correspondence is
* {#Witten89} [[Edward Witten]], _Quantum Field Theory and the Jones Polynomial_, Commun. Math. Phys. **121** 3 (1989) 351399 [[euclid:.cmp/1104178138](http://projecteuclid.org/euclid.cmp/1104178138), [doi:10.1007/BF01217730](https://doi.org/10.1007/BF01217730), MR0990772]
More details worked out:
* [[Daniel C. Cabra]], [[Gerardo L. Rossini]], *Explicit connection between conformal field theory and 2+1 Chern-Simons theory*, Mod. Phys. Lett. A **12** (1997) 1687-1697 [[arXiv:hep-th/9506054](https://arxiv.org/abs/hep-th/9506054), [doi:10.1142/S0217732397001722](https://doi.org/10.1142/S0217732397001722)]
> (with motivation from [[Laughlin wavefunctions]] for [[anyons]] in [[condensed matter theory]])
Reviews:
* {#Gawedzki99} [[Krzysztof Gawędzki]], *Conformal field theory: a case study*, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) *Conformal Field Theory -- New Non-perturbative Methods In String And Field Theory*, CRC Press (2000) [[arXiv:hep-th/9904145](https://arxiv.org/abs/hep-th/9904145), [doi:10.1201/9780429502873](https://doi.org/10.1201/9780429502873)]
* {#Witten96} [[Edward Witten]], section 2 of _Five-Brane Effective Action In M-Theory_ J. Geom. Phys. 22: 103-133, 1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
The relation of this $CS_3/WZW_2$-duality to the [[AdS-CFT correspondence]] is discussed in
* {#GukovMartinecMooreStrominger04} [[Sergei Gukov]], [[Emil Martinec]], [[Gregory Moore]], [[Andrew Strominger]], _Chern-Simons Gauge Theory and the $AdS_3/CFT_2$ Correspondence_, in: [[Mikhail Shifman]] et al. (eds.) _From fields to strings_, vol. 2, 1606-1647, 2004 ([arXiv:hep-th/0403225](https://arxiv.org/abs/hep-th/0403225))
* Kristan Jensen, _Chiral anomalies and AdS/CMT in two dimensions_, JHEP 1101:109,2011 ([arXiv:1012.4831](https://arxiv.org/abs/1012.4831))
* [[Per Kraus]], [[Finn Larsen]], _Partition functions and elliptic genera from supergravity_, JHEP 0701:002, 2007 ([arXiv:hep-th/0607138](https://arxiv.org/abs/hep-th/0607138))
* [[Per Kraus]], _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_, Lect. Notes Phys. 755: 193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074))
* Ville Keranen, _Chern-Simons interactions in AdS3 and the current conformal block_ ([arXiv:1403.6881](https://arxiv.org/abs/1403.6881))
A general argument about the relation between AdS/CFT duality and [[schreiber:infinity-Chern-Simons theory]] is in
* {#Witten98} [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ JHEP 9812:012,1998 ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))
An argument (via [[Chern-Simons gravity]], but see the caveats there) that [[3d quantum gravity]] with negative [[cosmological constant]] has as [[boundary field theory]] 2d [[Liouville theory]] is due to
* {#CoussaertHenneauxvanDriel95} O. Coussaert, [[Marc Henneaux]], P. van Driel, _The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant_, Class. Quant. Grav. 12 (1995) 2961-2966 ([arXiv:gr-qc/9506019](http://arxiv.org/abs/gr-qc/9506019))
Discussion of the [[Ising model]] [[2d CFT]] as a boundary theory to a 3d [[TQFT]] based on the [[Turaev-Viro model]], and the phenomenon of [[Kramers-Wannier duality]], is discussed in
* {#FreedTeleman18} [[Daniel Freed]], [[Constantin Teleman]], _Topological dualities in the Ising model_ ([arXiv:1806.00008](https://arxiv.org/abs/1806.00008))
Via [[factorization algebras]]:
* {#GwilliamRabinovichWilliams2022} [[Owen Gwilliam]], [[Eugene Rabinovich]], [[Brian R. Williams]], *Factorization algebras and abelian CS/WZW-type correspondences*, Pure and Applied Mathematics Quarterly **18** 4 (2022) 1485–1553 [[arXiv:2001.07888](https://arxiv.org/abs/2001.07888), [doi:10.4310/PAMQ.2022.v18.n4.a7](https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a7)]
Formulation in [[homotopical AQFT]]:
* [[Marco Benini]], [[Alastair Grant-Stuart]], [[Alexander Schenkel]], *The linear CS/WZW bulk/boundary system in AQFT*, Annales Henri Poincaré (2023) [[arXiv:2302.06990](https://arxiv.org/abs/2302.06990)]
[[!include 3d gravity and Chern-Simons theory -- references]]
### $AdS_3$/$CFT_2$
{#ReferencesAdS3CFT2}
Discussion of [[AdS/CFT correspondence]] for [[3d gravity]]/[[2d CFT]]:
* [[Andrea Prinsloo]], [[Vidas Regelskis]], [[Alessandro Torrielli]], _Integrable open spin-chains in AdS3/CFT2_ ([arXiv:1505.06767](http://arxiv.org/abs/1505.06767))
An exact correspondence of the symmetric [[orbifold]] [[CFT]] of [[Liouville theory]] with a string theory on $AdS_3$ is claimed in:
* {#EberhardtGaberdiel19a} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _String theory on $AdS_3$ and the symmetric orbifold of Liouville theory_ ([arXiv:1903.00421](https://arxiv.org/abs/1903.00421))
* {#EberhardtGaberdiel19b} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Strings on $AdS_3 \times S^3 \times S^3 \times S^1$_ ([arXiv:1904.01585](https://arxiv.org/abs/1904.01585))
* {#EberhardtGaberdielGopakumar19} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _Deriving the $AdS_3/CFT_2$ Correspondence_ ([arXiv:1911.00378](https://arxiv.org/abs/1911.00378))
* Andrea Dei, [[Lorenz Eberhardt]], _Correlators of the symmetric product orbifold_ ([arXiv:1911.08485](https://arxiv.org/abs/1911.08485))
based on
* Shouvik Datta, [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$_ JHEP 1801 (2018) 146 ([arXiv:1709.06393](https://arxiv.org/abs/1709.06393))
See also
* [[Lorenz Eberhardt]], _$AdS_3/CFT_2$ at higher genus_ ([arXiv:2002.11729](https://arxiv.org/abs/2002.11729))
* [[Lorenz Eberhardt]], *A perturbative CFT dual for pure NS-NS $AdS_3$ strings*, J. Phys. A: Math. Theor. **55** 064001 [[arXiv:2110.07535](https://arxiv.org/abs/2110.07535), [arXiv:10.1088/1751-8121/ac47b2](https://doi.org/10.1088/1751-8121/ac47b2)]
* Kiarash Naderi, *DDF operators in the Hybrid Formalism* [[arXiv:2208.01617](https://arxiv.org/abs/2208.01617)]
Relation of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]:
* [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036))
In the context of [[holography as Koszul duality]]:
* [[Kevin Costello]], [[Natalie Paquette]], _Twisted Supergravity and Koszul Duality: A case study in $AdS_3$_ ([arXiv:2001.02177](https://arxiv.org/abs/2001.02177))
Generalization to [[boundary field theory]]:
* Sanjit Shashi, _Quotient-AdS/BCFT: Holographic Boundary $CFT_2$ on $AdS_3$ Quotients_ ([arXiv:2005.10244](https://arxiv.org/abs/2005.10244))
* [[Tadashi Takayanagi]], Takahiro Uetoko, _Chern-Simons Gravity Dual of BCFT_ ([arXiv:2011.02513](https://arxiv.org/abs/2011.02513))
See also:
* Stefano Speziali, _Spin 2 fluctuations in 1/4 BPS AdS3/CFT2_ ([arxiv:1910.14390](https://arxiv.org/abs/1910.14390))
* Bruno Balthazar, Amit Giveon, David Kutasov, Emil J. Martinec, *Asymptotically Free AdS3/CFT2* ([arXiv:2109.00065](https://arxiv.org/abs/2109.00065))
Relating to [[random matrix theory]]:
* Gabriele Di Ubaldo, [[Eric Perlmutter]], *$AdS_3/RMT_2$ Duality* [[arXiv:2307.03707](https://arxiv.org/abs/2307.03707)]
[[!include AdS3-CFT2 on D1-D5 branes -- references]]
[[!include AdS3-CFT2 on D2-D4-D6-D8 branes -- references]]
[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]
[[!redirects CS-WZW correspondence]]
[[!redirects CS/WZW correspondence]]
[[!redirects AdS3-CFT2]]
[[!redirects AdS3/CFT2]]
[[!redirects AdS3/CFT2 duality]]
[[!redirects AdS3-CFT2 duality]]
[[!redirects AdS3/CFT2 dualities]]
[[!redirects AdS3-CFT2 dualities]]
|
AdS3-CFT2 on D1-D5 branes -- references | https://ncatlab.org/nlab/source/AdS3-CFT2+on+D1-D5+branes+--+references | [[!redirects AdS3/CFT2 on D1/D5 branes -- references]]
### $AdS_3/CFT_2$ on D1/D5 branes
On [[AdS3-CFT2]] for [[D1/D5 brane bound states]] and [[black hole entropy]] [[black holes in string theory|in string theory]]:
* {#Maldacena97a} [[Juan Maldacena]], Section 4 of: _The Large N limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys. 2:231, 1998 ([hep-th/9711200](http://arxiv.org/abs/hep-th/9711200))
* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], Section 5 of: _Large $N$ Field Theories, String Theory and Gravity_, Phys. Rept. 323:183-386, 2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))
* Gautam Mandal, _A review of the D1/D5 system and five dimensional black hole from supergravity and brane viewpoint_ ([arXiv:hep-th/0002184](https://arxiv.org/abs/hep-th/0002184))
* [[Robbert Dijkgraaf]], [[Juan Maldacena]], [[Gregory Moore]], [[Erik Verlinde]], _A Black Hole Farey Tail_ ([arXiv:hep-th/0005003](https://arxiv.org/abs/hep-th/0005003), [spire:526744](http://inspirehep.net/record/526744))
* E. Gava, A.B. Hammou, J.F. Morales, K.S.Narain, _AdS/CFT correspondence and D1/D5 systems in theories with 16 supercharges_, JHEP 0103:035, 2001 ([arXiv:hep-th/0102043](https://arxiv.org/abs/hep-th/0102043))
* [[Per Kraus]], Section 4 of: _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_, Lect. Notes Phys. 755: 193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074)) |
AdS3-CFT2 on D2-D4-D6-D8 branes -- references | https://ncatlab.org/nlab/source/AdS3-CFT2+on+D2-D4-D6-D8+branes+--+references | [[!redirects AdS3/CFT2 on D2/D4-D6/D8 branes -- references]]
### $AdS_3/CFT_2$ on D2/D4-D6/D8 branes
On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] as [[defect field theory]] "inside" [[AdS7-CFT6]]:
* {#DibitettoPetri17} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arxiv:1707.06154](https://arxiv.org/abs/1707.06154))
* [[Nicolò Petri]], section 6.5 of: _Supersymmetric objects in gauged supergravities_ ([arxiv:1802.04733](https://arxiv.org/abs/1802.04733))
* {#Petri18} [[Nicolò Petri]], _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))
* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))
* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))
* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))
* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))
* Kostas Filippas, _Non-integrability on $AdS_3$ supergravity_ ([arxiv:1910.12981](https://arxiv.org/abs/1910.12981))
See also
* Andrea Legramandi, [[Niall Macpherson]], _$AdS_3$ solutions with $\mathcal{N}=(3,0)$ from $S^3 \timesS^3$ fibrations_, ([arXiv:1912.10509](https://arxiv.org/abs/1912.10509)
|
advanced and retarded causal propagators | https://ncatlab.org/nlab/source/advanced+and+retarded+causal+propagators |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What are called _advanced_ and _retarded causal Green functions_ $\Delta_{A/R}$ are [[Green functions]] for [[hyperbolic differential operators]] on manifolds with [[causal structure]] (e.g. [[spacetimes]]) whose [[support]] is in the [[future cone]] or [[past cone]], respectively, of the source excitation. The corresponding [[integral kernels]] hence say how a [[delta distribution|point excitation]] _propagates_ into the future or past, respectively, via the given [[differential equation]], and therefore these are also called the advanced/future _propagators_.
If both advanced and retarded Green functions exist for a differential operator as well as for its [[formally adjoint differential operator|formal adjoint]], then the differental operator is called a _[[Green hyperbolic differential operator]]_. The archetypical examples are, on [[globally hyperbolic spacetimes]]:
1. [[normally hyperbolic differential operators]] such as the [[wave operator]] and the [[Klein-Gordon operator]];
1. [[Dirac operators]] on [[spinor bundles]] whose square is a normally hyperbolic differential operator as above.
The advanced/retarded [[integral kernel]]
$$
\Delta_{A/R}
\in \mathcal{D}'(X \times X)
$$
is such that
1. $(x,y) \in supp(\Delta_R)$ precisely if $x$ is in the [[causal future]] of $y$;
1. $(x,y) \in supp(\Delta_A)$ precisely if $x$ is in the [[causal past]] of $y$.
Written as [[generalized functions]] these satisfy
$$
\Delta_A(x,y) = \Delta_R(y,x)
\,.
$$
This implies in particular that
1. the _[[causal propagator]]_, which is the difference of the two
$$
\Delta_S \coloneqq \Delta_R - \Delta_A
$$
is skew-symmetric in its arguments (reflecting the fact that this is the [[integral kernel]] for the [[Peierls-Poisson bracket]] for the [[free field|free]] [[scalar field]] on the given spacetime);
1. the _[[Dirac propagator]]_, which is the sum of the two
$$
\Delta_D \coloneqq \Delta_R + \Delta_A
$$
is symmetric in its arguments, reflecting the fact that this is the integral kernel for [[time-ordered products]] away from the [[diagonal]].
## Definition
+-- {: .num_defn #CompactlySourceCausalSupport}
###### Definition
**(compactly sourced causal support)
Given a [[vector bundle]] $E \overset{}{\to} \Sigma$ over a manifold $\Sigma$ with [[conal causal structure|causal structure]]
Write $\Gamma_{\Sigma}(-)$ for [[space of sections|spaces of smooth sections]], and write
$$
\array{
\Gamma_{cp}(-) & \text{compact support}
\\
\Gamma_{\Sigma,\pm cp}(-) & \text{compactly sourced future/past support}
\\
\Gamma_{\Sigma,scp}(-) & \text{spacelike compact support}
\\
\Gamma_{\Sigma,(f/p)cp}(-) & \text{future/past compact support}
\\
\Gamma_{\Sigma,tcp}(-) & \text{timelike compact support}
}
$$
for the [[linear subspaces]] on those smooth sections whose [[support]] is
1. ($cp$) inside a [[compact subset]]
1. ($\pm cp$) inside the [[closed future cone]]/[[closed past cone]], respectively, of a [[compact subset]],
1. ($scp$) inside the [[closed causal cone]] of a [[compact subset]], which equivalently means that the [[intersection]] with every ([[spacelike]]) [[Cauchy surface]] is compact ([Sanders 13, theorem 2.2](#Sanders12)),
1. ($fcp$) inside the past of a Cauchy surface ([Sanders 13, def. 3.2](#Sanders12)),
1. ($pcp$) inside the future of a Cauchy surface ([Sanders 13, def. 3.2](#Sanders12)),
1. ($tcp$) inside the future of one Cauchy surface and the past of another ([Sanders 13, def. 3.2](#Sanders12))
=--
([Bär 14, section 1](#Baer14), [Khavkine 14, def. 2.1](#Khavkine14))
+-- {: .num_defn #AdvancedAndRetardedGreenFunctions}
###### Definition
**([[advanced and retarded Green functions]], [[causal Green function]] and [[propagators]])**
Let $\Sigma$ be a [[smooth manifold]] with [[causal structure]], let $E \to \Sigma$ be a [[smooth vector bundle]] and let
$P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$ be a [[differential operator]] on its [[space of smooth sections]].
Then a [[linear map]]
$$
\mathrm{G}_{P,\pm}
\;\colon\;
\Gamma_{\Sigma, cp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma, \pm cp}(E)
$$
from spaces of sections of [[compact support]] to spaces of sections of causally sourced future/past support (def. \ref{CompactlySourceCausalSupport}) is called an _[[advanced or retarded Green function]]_ for $P$, respectively, if
1. for all $\Phi \in \Gamma_{\Sigma,cp}(E_1)$ we have
$$
\label{AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator}
G_{P,\pm} \circ P(\Phi) = \Phi
$$
and
$$
\label{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator}
P \circ G_{P,\pm}(\Phi) = \Phi
$$
1. the [[support]] of $G_{P.\pm}(\Phi)$ is in the [[closed future cone]] or [[closed past cone]] of the support of $\Phi$, respectively.
If the advanced/retarded Green functions $G_{P\pm}$ exists, then the difference
$$
\mathrm{G}_P \coloneqq \mathrm{G}_{P,+} - \mathrm{G}_{P,-}
$$
is called the _[[causal Green function]]_.
If there are [[integral kernel]], hence [[distributions in two variables]]
$$
\Delta_{P,\pm} \in \Gamma'\Sigma( \tilde E^\ast \boxtimes_\Sigma E )
$$
such that these Green functions are given by the corresponding [[integral transform]], in that
(in [[generalized function]]-notation)
$$
(G_{P,\pm} \Phi)(x)
\;=\;
\underset{y \in \Sigma}{\int}
\Delta_{P, \pm}(x,y) \cdot \Phi(y)
$$
then these integral kernels are called the advanced/retarded _[[propagators]]_; similarly then their difference
$$
\label{CausalPropagator}
\Delta_{P,S} \coloneqq \Delta_{P,+} - \Delta_{P,-}
$$
is the corresponding _[[causal propagator]]_.
=--
(e.g. [Bär 14, def. 3.2, cor. 3.10](#Baer14}))
## Properties
### Existence and uniqueness
+-- {: .num_prop #AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}
###### Proposition
**([[advanced and retarded Green functions]] of [[Green hyperbolic differential operator]] are unique)**
The [[advanced and retarded Green functions]] (def. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}) of a [[Green hyperbolic differential operator]] are unique.
=--
([Bär 14, cor. 3.12](#Baer14})
### Continuity
{#Continuity}
+-- {: .num_defn #TVSStructureOnSpacesOfSmoothSections}
###### Definition
**([[Fréchet space|Fréchet]] [[topological vector space]] on [[spaces of smooth sections]] of a [[smooth vector bundle]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]]. On its [[real vector space]] $\Gamma_\Sigma(E)$ [[space of sections|of smooth sections]] consider the [[seminorms]] indexed by a [[compact subset]] $K \subset \Sigma$ and a [[natural number]] $N \in \mathbb{N}$ and given by
$$
\array{
\Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}&
[0,\infty)
\\
\Phi &\mapsto& \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,,
}
$$
where on the right we have the [[absolute values]] of the [[covariant derivatives]] of $\Phi$ for any fixed choice of [[connection on a bundle|connection]] on $E$ and [[norm]] on the [[tensor product of vector bundles]] $(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E $.
This makes $\Gamma_\Sigma(E)$ a [[Fréchet space|Fréchet]] [[topological vector space]].
For $K \subset \Sigma$ any [[closed subset]] then the sub-space of sections
$$
\Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)
$$
of sections whose [[support]] is inside $K$ becomes a [[Fréchet space|Fréchet]] [[topological vector spaces]] with the induced [[subspace topology]], which makes these be [[closed subspaces]].
=--
([Bär 14, 2.1, 2.2](#Baer14))
+-- {: .num_defn #DistributionalSections}
###### Definition
**([[distribution|distributional]] [[sections]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over a [[smooth manifold]] with [[causal structure]].
The [[vector space|vector]] [[spaces of smooth sections]] with restricted support from def. \ref{CompactlySourceCausalSupport} structures of [[topological vector spaces]] via def. \ref{TVSStructureOnSpacesOfSmoothSections}. We denote the topological [[dual spaces]] by
$$
\Gamma'_{\Sigma}(\tilde{E}^*) \coloneqq (\Gamma_{\Sigma,cp}(E))^*
$$
etc.
This is the space of _distributional sections_ of the bundle $\tilde{E}^*$.
With this notations, smooth compactly supported sections of the same bundle, regarded as the [[non-singular distributions]], constitute a [[dense subset]]
$$
\Gamma_{\Sigma,cp}(\tilde{E}^*)
\underset{\text{dense}}{\hookrightarrow}
\Gamma'_{\Sigma}(\tilde{E}^*)
\,.
$$
Imposing the same restrictions to the [[supports of distributions]] as in def. \ref{CompactlySourceCausalSupport}, we have the following subspaces of distributional sections:
$$
\Gamma'_{\Sigma,cp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,\pm cp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,scp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,fcp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,pcp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,tcp}(\tilde E^\ast)
\subset
\Gamma'_{\Sigma}(\tilde E^\ast) .
$$
=--
([Sanders 13](#Sanders12), [Bär 14](#Baer14))
+-- {: .num_prop #GreenFunctionsAreContinuous}
###### Proposition
**([[causal Green functions]] of [[Green hyperbolic differential operators]] are [[continuous linear maps]])**
Given a [[Green hyperbolic differential operator]] $P$ (def. \ref{GreenHyperbolicDifferentialOperator}), the advanced, retarded and causal Green functions of $P$ (def. \ref{AdvancedAndRetardedGreenFunctions}) are [[continuous linear maps]] with respect to the [[topological vector space]] structure from def. \ref{TVSStructureOnSpacesOfSmoothSections} and also have a unique continuous extension to the spaces of sections with .larger support (def. \ref{CompactlySourceCausalSupport}) as follows:
$$
\begin{aligned}
\mathrm{G}_{P,+}
&\;\colon\;
\Gamma_{\Sigma, pcp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma, pcp}(E) ,
\\
\mathrm{G}_{P,-}
&\;\colon\;
\Gamma_{\Sigma, fcp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma, fcp}(E) ,
\\
\mathrm{G}_{P}
&\;\colon\;
\Gamma_{\Sigma, tcp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma}(E) ,
\end{aligned}
$$
such that we still have the relation
$$
\mathrm{G}_P = \mathrm{G}_{P,+} - \mathrm{G}_{P,-}
$$
and
$$
P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id
$$
and
$$
supp \mathrm{G}_{P,\pm}(\tilde{\alpha}^*) \subseteq J^\pm(supp \tilde{\alpha}^*)
\,.
$$
=--
([Bär 14, thm. 3.8, cor. 3.11](#Baer14))
## Examples
### For Klein-Gordon operator on Minkowski spacetime
{#ForKleinGordonOperatorOnMinkowskiSpacetime}
On [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ consider the [[Klein-Gordon operator]]
$$
\eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} \Phi - \left( \tfrac{m c}{\hbar} \right)^2 \Phi \;=\; 0 \,.
$$
Its [[Fourier transform]] is
$$
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2
\;=\;
(k_0)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2
\,.
$$
The [[dispersion relation]] of this equation we write
$$
\label{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime}
\omega(\vec k)
\;\coloneqq\;
+ c \sqrt{ {\vert \vec k \vert}^2 + \left( \tfrac{m c}{\hbar}\right)^2 }
\,,
$$
where on the right we choose the [[non-negative real number|non-negative]] [[square root]].
$\,$
We now discuss
1. _[Advanced and regarded propagators](#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime)_
1. _[Causal propagator](#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)_
1. _[Wightman propagator](#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime)_
1. _[Feynman propagator](#FeynmanPropagator)_
1. _[Singular support and Wave front sets](#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime)_
$\,$
**[[advanced and retarded propagators]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]]**
{#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime}
+-- {: .num_prop #AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}
###### Proposition
**(mode expansion of [[advanced and retarded propagators]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])**
The [[advanced and retarded Green functions]] $G_\pm$ of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] are given by [[integral kernels]] ("[[propagators]]")
$$
\Delta_\pm \in \mathcal{D}'(\mathbb{R}^{p,1}\times \mathbb{R}^{p,1})
$$
by (in [[generalized function]]-notation)
$$
G_\pm(\Phi)
\;=\;
\underset{\mathbb{R}^{p,1}}{\int}
\Delta_{\pm}(x,y) \Phi(y) \, dvol(y)
$$
where the [[advanced and retarded propagators]] $\Delta_{\pm}(x,y)$ have the following equivalent expressions:
$$
\label{ModeExpansionForMinkowskiAdvancedRetardedPropagator}
\begin{aligned}
\Delta_\pm(x-y)
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^{p}}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\end{aligned}
$$
Here $\omega(\vec k)$ denotes the [[dispersion relation]] (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) of the [[Klein-Gordon equation]].
=--
+-- {: .proof}
###### Proof
The [[Klein-Gordon operator]] is a [[Green hyperbolic differential operator]] ([this example](Green+hyperbolic+partial+differential+equation#GreenHyperbolicKleinGordonOperator)) therefore its advanced and retarded Green functions exist uniquely (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}).
Moreover, prop. \ref{GreenFunctionsAreContinuous} says that they are [[continuous linear functionals]] with respect to the [[topological vector space]] [[structures]] on [[spaces of smooth sections]] (def. \ref{TVSStructureOnSpacesOfSmoothSections}). In the case of the [[Klein-Gordon operator]] this just means that
$$
G_{\pm}
\;\colon\;
C^\infty_{cp}(\mathbb{R}^{p,1})
\longrightarrow
C^\infty_{\pm cp}(\mathbb{R}^{p,1})
$$
are [[continuous linear functionals]] in the standard sense of [[distributions]]. Therefore the
[[Schwartz kernel theorem]] implies the existence of [[integral kernels]] being [[distributions in two variables]]
$$
\Delta_{\pm} \in \mathcal{D}(\mathbb{R}^{p,1} \times \mathbb{R}^{p,1})
$$
such that, in the notation of [[generalized functions]],
$$
(G_\pm \alpha)(x)
\;=\;
\underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \alpha(y) \, dvol(y)
\,.
$$
These integral kernels are the advanced/retarded "[[propagators]]". We now compute
these [[integral kernels]] by making an Ansatz and showing that it has the defining properties,
which identifies them by the uniqueness statement of prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.
We make use of the fact that the [[Klein-Gordon equation]] is [[invariant]] under the defnining [[action]]
of the [[Poincaré group]] on [[Minkowski spacetime]], which is a [[semidirect product group]] of the [[translation group]]
and the [[Lorentz group]].
Since the [[Klein-Gordon operator]] is invariant, in particular, under [[translations]] in $\mathbb{R}^{p,1}$ it is clear that the propagators, as a [[distribution in two variables]], depend only on the difference of its two arguments
$$
\label{TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime}
\Delta_{\pm}(x,y) = \Delta_{\pm}(x-y)
\,.
$$
Since moreover the [[Klein-Gordon operator]] is [[formally adjoint differential operator|formally self-adjoint]] ([this prop.](Klein-Gordon+equation#FormallySelfAdjointKleinGordonOperator)) this implies that for $P$ the Klein the equation (eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator)
$$
P \circ G_\pm = id
$$
is equivalent to the equation (eq:AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator)
$$
G_\pm \circ P = id
\,.
$$
Therefore it is sufficient to solve for the first of these two equation, subject to
the defining support conditions. In terms of the [[propagator]] [[integral kernels]] this means that we have to solve the [[distribution|distributional]] equation
$$
\label{KleinGordonEquationOnAdvacedRetardedPropagator}
\left(
\eta^{\mu \nu}
\frac{\partial}{\partial x^\mu}
\frac{\partial}{\partial x^\nu}
-
\left( \tfrac{m c}{\hbar} \right)^2
\right)
\Delta_\pm(x-y)
\;=\;
\delta(x-y)
$$
subject to the condition that the [[support of a distribution|distributional support]] is
$$
supp\left( \Delta_{\pm}(x-y) \right)
\subset
\left\{
{\vert x-y\vert^2_\eta}\lt 0
\;\,,\;
\pm(x^0 - y^ 0) \gt 0
\right\}
\,.
$$
We make the _Ansatz_ that we assume that $\Delta_{\pm}$, as a distribution in a single variable $x-y$, is a [[tempered distribution]]
$$
\Delta_\pm \in \mathcal{S}'(\mathbb{R}^{p,1})
\,,
$$
hence amenable to [[Fourier transform of distributions]]. If we do find a solution this way, it is guaranteed to be the unique solution by prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.
By [this prop.](Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces)
the [[Fourier transform of distributions|distributional Fourier transform]] of equation (eq:KleinGordonEquationOnAdvacedRetardedPropagator) is
$$
\begin{aligned}
\label{FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator}
\left(
- \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2
\right)
\widehat{\Delta_{\pm}}(k)
& =
\widehat{\delta}(k)
\\
& =
1
\end{aligned}
\,,
$$
where in the second line we used the [[Fourier transform of distributions|Fourier transform]] of the [[delta distribution]]
from [this example](Dirac+distribution#FourierTransformOfDeltaDistribution).
Notice that this implies that the [[Fourier transform]] of the [[causal propagator]]
$$
\Delta_S \coloneqq \Delta_+ - \Delta_-
$$
satisfies the homogeneous equation:
$$
\label{FourierVersionOfPDEForKleinGordonCausalPropagator}
\left(
- \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2
\right)
\widehat{\Delta_S}(k)
\;=\;
0
\,,
$$
Hence we are now reduced to finding solutions $\widehat{\Delta_\pm} \in \mathcal{S}'(\mathbb{R}^{p,1})$ to (eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator) such that their [[Fourier inversion theorem|Fourier inverse]] $\Delta_\pm$ has the required [[support of a distribution|support]] properties.
We discuss this by a variant of the [[Cauchy principal value]]:
Suppose the following [[limit of a sequence|limit]] of [[non-singular distributions]] in the [[variable]] $k \in \mathbb{R}^{p,1}$
exists in the space of [[distributions]]
$$
\label{LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator}
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\frac{1}{ (k_0 \mp i \epsilon)^2 - {\vert \vec k\vert^2} - \left( \tfrac{m c}{\hbar} \right)^2 }
\;\in\;
\mathcal{D}'(\mathbb{R}^{p,1})
$$
meaning that for each [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ the [[limit of a sequence|limit]] in $\mathbb{C}$
$$
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\underset{\mathbb{R}^{p,1}}{\int} \frac{b(k)}{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 }
d^{p+1}k
\;\in\;
\mathbb{C}
$$
exists. Then this limit is clearly a solution to the distributional equation (eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator)
because on those bump functions $b(k)$ which happen to be products with $\left(-\eta^{\mu \nu}k_\mu k-\nu - \left( \tfrac{m c}{\hbar}\right)^2\right)$
we clearly have
$$
\begin{aligned}
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\underset{\mathbb{R}^{p,1}}{\int}
\frac{
\left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) b(k)
}{
(k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2
}
d^{p+1}k
& =
\underset{\mathbb{R}^{p,1}}{\int}
\underset{= 1}{
\underbrace{
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\frac{
\left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) }{
(k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 }
}
}
b(k)\, d^{p+1}k
\\
& =
\langle 1, b\rangle
\,.
\end{aligned}
$$
Moreover, if the limiting distribution (eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator)
exists, then it is clearly a [[tempered distribution]], hence we may apply [[Fourier inversion theorem|Fourier inversion]]
to obtain [[Green functions]]
$$
\label{AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets}
\Delta_{\pm}(x,y)
\;\coloneqq\;
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{1}{(2\pi)^{p+1}}
\underset{\mathbb{R}^{p,1}}{\int}
\frac{e^{i k_\mu (x-y)^\mu}}{
(k_0 \mp i \epsilon )^2 - {\vert \vec k\vert}^2 - \left(\tfrac{m c}{\hbar}\right)^2
}
d k_0 d^p \vec k
\,.
$$
To see that this is the correct answer, we need to check the defining support property.
Finally, by the [[Fourier inversion theorem]], to show that the [[limit of a sequence|limit]] (eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator) indeed exists it is sufficient to show that the
limit in (eq:AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets) exists.
We compute as follows
$$
\label{TheSupportOfTheCandidateAdvancedRetardedPropagatorIsinTheFutureOrPastRespectively}
\begin{aligned}
\Delta_\pm(x-y)
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2
}
\, d k_0 \, d^p \vec k
\\
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i \epsilon)^2 - \left(\omega(\vec k)/c\right)^2
}
\, d k_0 \, d^p \vec k
\\
&=
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
\left(
(k_0 \mp i\epsilon) - \omega(\vec k)/c
\right)
\left(
(k_0 \mp i \epsilon) + \omega(\vec k)/c
\right)
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^{p}}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\end{aligned}
$$
where $\omega(\vec k)$ denotes the [[dispersion relation]] (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) of the [[Klein-Gordon equation]]. The last step is simply the application of [[Euler's formula]] $\sin(\alpha) = \tfrac{1}{2 i }\left( e^{i \alpha} - e^{- i \alpha}\right)$.
Here the key step is the application of [[Cauchy's integral formula]] in the fourth step. We spell this out now for $\Delta_+$, the discussion for $\Delta_-$ is the same, just with the appropriate signs reversed.
1. If $(x^0 - y^0) \gt 0$ thn the expression $e^{ik_0 (x^0 - y^0)}$ decays with _[[positive number|positive]] [[imaginary part]]_ of $k_0$, so that we may expand the [[integration]] [[domain]] into the [[upper half plane]] as
$$
\begin{aligned}
\int_{-\infty}^\infty d k_0
& = \phantom{+}
\int_{-\infty}^0 d k_0 + \int_{0}^{+ i \infty} d k_0
\\
& = + \int_{+i \infty}^0 d k_0 + \int_0^\infty d k_0
\,;
\end{aligned}
$$
Conversely, if $(x^0 - y^0) \lt 0$ then we may analogously expand into the [[lower half plane]].
1. This integration domain may then further be completed to two [[contour integrations]]. For the expansion into the [[upper half plane]] these encircle counter-clockwise the [[poles]] at $\pm \omega(\vec k)+ i\epsilon \in \mathbb{C}$, while for expansion into the [[lower half plane]] no poles are being encircled.
<img src="https://ncatlab.org/nlab/files/ContourForAdvancedPropagator.png" height="280">
1. Apply [[Cauchy's integral formula]] to find in the case $(x^0 - y^0)\gt 0$ the sum of the [[residues]] at these two [[poles]] times $2\pi i$, zero in the other case. (For the retarded propagator we get $- 2 \pi i$ times the residues, because now the contours encircling non-trivial poles go clockwise).
1. The result is now non-singular at $\epsilon = 0$ and therefore the [[limit of a sequence|limit]] $\epsilon \to 0$ is now computed by evaluating at $\epsilon = 0$.
This computation shows a) that the limiting distribution indeed exists, and b) that the [[support of a distribution|support]]
of $\Delta_+$ is in the future, and that of $\Delta_-$ is in the past.
Hence it only remains to see now that the support of $\Delta_\pm$ is inside the [[causal cone]].
But this follows from the previous argument, by using that the [[Klein-Gordon equation]] is invariant under
[[Lorentz transformations]]: This implies that the support is in fact in the [[future]] of _every_ spacelike
slice through the origin in $\mathbb{R}^{p,1}$, hence in the [[closed future cone]] of the origin.
=--
+-- {: .num_cor #CausalPropagatorIsSkewSymmetric}
###### Corollary
**([[causal propagator]] is skew-symmetric)**
Under reversal of arguments the [[advanced and retarded causal propagators]] from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} are related by
$$
\Delta_{\pm}(y-x) = \Delta_\mp(x-y)
\,.
$$
It follows that the [[causal propagator]] $\Delta \coloneqq \Delta_+ - \Delta_-$ is skew-symmetric in its arguments:
$$
\Delta_S(x-y) = - \Delta_S(y-x)
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} we have with (eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator)
$$
\begin{aligned}
\Delta_\pm(y-x)
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x -\vec y)}
-
e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\Delta_\mp(x-y)
\end{aligned}
$$
Here in the second step we applied [[change of integration variables]] $\vec k \mapsto - \vec k$ (which introduces _no_ sign because in addition to $d \vec k \mapsto - d \vec k$ the integration domain reverses [[orientation]]).
=--
$\,$
**[[causal propagator]]**
{#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
+-- {: .num_prop #ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}
###### Proposition
**(mode expansion of [[causal propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[causal propagator]] (eq:CausalPropagator) for the [[Klein-Gordon equation]] for [[mass]] $m$ on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ is given, in [[generalized function]] notation, by
$$
\label{CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime}
\begin{aligned}
\Delta_S(x,y)
& =
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\,,
\end{aligned}
$$
where in the second line we used [[Euler's formula]] $sin(\alpha)= \tfrac{1}{2i}\left( e^{i \alpha} - e^{-i \alpha} \right)$.
In particular this shows that the [[causal propagator]] is [[real part|real]], in that it is equal to its [[complex conjugation|complex conjugate]]
$$
\label{CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal}
\left(\Delta_S(x,y)\right)^\ast = \Delta_S(x,y)
\,.
$$
=--
+-- {: .proof}
###### Proof
By definition and using the expression from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} for the [[advanced and retarded causal propagators]] we have
$$
\begin{aligned}
\Delta_S(x,y)
& \coloneqq
\Delta_+(x,y) - \Delta_-(x,y)
\\
& =
\left\{
\array{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
& \vert & \text{if} \, + (x^0 - y^0) \gt 0
\\
\frac{(-1) (-1) i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
& \vert & \text{if} \, - (x^0 - y^0) \gt 0
}
\right.
\\
& =
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\end{aligned}
$$
For the reality, notice from the last line that
$$
\begin{aligned}
\left(\Delta_S(x,y)\right)^\ast
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{-i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{+i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\\
& =
\Delta_S(x,y)
\,,
\end{aligned}
$$
where in the last step we used the [[change of integration variables]] $\vec k \mapsto - \vec k$
(whih introduces no sign, since on top of $d \vec k \mapsto - d \vec k$ the [[orientation]] of the integration [[domain]] changes).
=--
We consider a couple of equivalent expressions for the causal propagator which are useful for computations:
+-- {: .num_prop #CausalPropagatorForKleinGordonOnMinkowskiAsContourIntegral}
###### Proposition
**([[causal propagator]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]] as a [[contour integral]])**
The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]], given as a [[contour integral]] along a curve $C(\vec k)$ going counter-clockwise around the two [[poles]] at $k_0 = \pm \omega(\vec k)/c$:
$$
\Delta_S(x,y)
\;=\;
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{e^{i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2g }
\,d k_0
\,d^{p} k
\,.
$$
=--
<img src="https://ncatlab.org/nlab/files/ContourForCausalPropagator.png" height="160">
> graphics grabbed from [Kocic 16](#Kocic16)
+-- {: .proof}
###### Proof
By [[Cauchy's integral formula]] we compute as follows:
$$
\begin{aligned}
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{e^{i k_\mu (x^\mu - y^\mu)}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
\,d k_0
\,d^{p} k
& =
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{
e^{i k_0 x^0} e^{ i \vec k \cdot (\vec x - \vec y)}
}{
k_0^2 - \omega(\vec k)^2/c^2
}
\,d k_0
\,d^p \vec k
\\
& =
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
( k_0 + \omega(\vec k)/c )
( k_0 - \omega(\vec k)/c )
}
\,d k_0
\,d^p \vec k
\\
& =
(2\pi)^{-(p+1)}
2\pi i
\int
\left(
\frac{
e^{i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
}
{
2 \omega(\vec k)/c
}
-
\frac{
e^{ - i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
}{
2 \omega(\vec k)/c
}
\right)
\,d^p \vec k
\\
& =
i
(2\pi)^{-p}
\int
\frac{1}{\omega(\vec k)/c}
sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\,d^p \vec k
\,.
\end{aligned}
$$
The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}
=--
+-- {: .num_prop #CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator}
###### Proposition
**([[causal propagator]] as [[Fourier transform]] of [[delta distribution]] on the [[Fourier transform|Fourier transformed]] [[Klein-Gordon operator]])**
The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]]:
$$
\Delta_S(x,y)
\;=\;
i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k
\,,
$$
where the [[integrand]] is the product of the [[sign function]] of $k_0$ with the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] and a [[plane wave]] factor.
=--
+-- {: .proof}
###### Proof
By decomposing the integral over $k_0$ into its negative and its positive half, and applying the [[change of integration variables]] $k_0 = \pm\sqrt{h}$ we get
$$
\begin{aligned}
i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k
& =
+ i (2\pi)^{-p} \int \int_0^\infty \delta\left( -k_0^2 + \vec k^2 + \left( \tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0) + i \vec k \cdot (\vec x - \vec y)} d k_0 \, d^p \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \int_{-\infty}^0 \delta\left( -k_0^2 + \vec k^2 + \left(\tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0)+ i \vec k \cdot (\vec x - \vec y) } d k_0 \, d^{p} \vec k
\\
& =
+i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( -h + \omega(\vec k)^2/c^2 \right) e^{ + i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( - h + \omega(\vec k)^2/c^2 \right) e^{ - i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k
\\
& =
+i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x} d^{p} \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ - i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x } d^{p} \vec k
\\
& = -(2 \pi)^{-p} \int \frac{1}{\omega(\vec k)/c}
sin\left( \omega(\vec k)(x-y)^0/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\end{aligned}
$$
The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}.
=--
$\,$
**[[Wightman propagator]]**
{#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime}
Prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} exhibits the [[causal propagator]] of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] as the difference of a contribution for [[positive real number|positive]] temporal [[angular frequency]] $k_0 \propto \omega(\vec k)$ (hence positive [[energy]] $\hbar \omega(\vec k)$ and a contribution of negative temporal [[angular frequency]].
The [[positive real number|positive]] [[frequency]] contribution to the [[causal propagator]] is called the _[[Wightman propagator]]_ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below), also known as the the _[[vacuum state]] [[2-point function]] of the [[free field|free]] [[real scalar field]] on [[Minkowski spacetime]]_. Notice that the temporal component
of the [[wave vector]] is proportional to the _negative_ [[angular frequency]]
$$
k_0 = -\omega/c
$$
(see at _[[plane wave]]_), therefore the appearance of the [[step function]] $\Theta(-k_0)$ in (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) below:
+-- {: .num_defn #StandardHadamardDistributionOnMinkowskiSpacetime}
###### Definition
**([[Wightman propagator]] or [[vacuum state]] [[2-point function]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])**
The _[[Wightman propagator]]_ for the [[Klein-Gordon operator]] at [[mass]] $m$ on [[Minkowski spacetime]] is the [[tempered distribution|tempered]] [[distribution in two variables]] $\Delta_H \in \mathcal{S}'(\mathbb{R}^{p,1})$ which as a [[generalized function]] is given by the expression
$$
\label{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime}
\begin{aligned}
\Delta_H(x,y)
& \coloneqq
\frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k
\\
& =
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\,,
\end{aligned}
$$
Here in the first line we have in the [[integrand]] the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] times a [[plane wave]] and times the [[step function]] $\Theta$ of the temporal component of the [[wave vector]]. In the second line we used the [[change of integration variables]] $k_0 = \sqrt{h}$, then the definition of the [[delta distribution]] and the fact that $\omega(\vec k)$ is by definition the [[non-negative real number|non-negative]] solution to the Klein-Gordon [[dispersion relation]].
=--
(e.g. [Khavkine-Moretti 14, equation (38) and section 3.4](Hadamard+distribution#KhavineMoretti14))
+-- {: .num_prop #ContourIntegralForStandardHadamardPropagatorOnMinkowskiSpacetime}
###### Proposition
**([[contour integral]] representation of the [[Wightman propagator]] for the [[Klein-Gordon operator]] on [[Minkowski spacetime]])
The [[Wightman propagator]] from
def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} is
equivalently given by the [[contour integral]]
$$
\label{StandardHadamardPropagatorOnMinkowskiSpacetimeInTermsOfContourIntegral}
\Delta_H(x,y)
\;=\;
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{e^{-i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
d k_0
d^{p} k
\,,
$$
where the [[Jordan curve]] $C_+(\vec k) \subset \mathbb{C}$ runs counter-clockwise, enclosing the point $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$, but not enclosing the point $- \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$.
<img src="https://ncatlab.org/nlab/files/ContourForHadamardPropagator.png" height="200">
> graphics grabbed from [Kocic 16](#Kocic16)
=--
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{e^{ - i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
d k_0
d^{p} k
& =
-i(2\pi)^{-(p+1)}
\int
\oint_{C_+(\vec k)}
\frac{
e^{ -i k_0 x^0} e^{i \vec k \cdot (\vec x - \vec y)}
}{
k_0^2 - \omega(\vec k)^2/c^2
}
d k_0
d^p \vec k
\\
& =
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{
e^{ - i k_0 (x^0-y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
( k_0 - \omega_\epsilon(\vec k) )
( k_0 + \omega_\epsilon(\vec k) )
}
d k_0
d^p \vec k
\\
& =
(2\pi)^{-p}
\int
\frac{1}{2 \omega(\vec k)}
e^{-i \omega(\vec k) (x^0-y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
d^p \vec k
\,.
\end{aligned}
$$
The last step is application of [[Cauchy's integral formula]], which says that the [[contour integral]] picks up the [[residue]] of the [[pole]] of the [[integrand]] at $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$. The last line is $\Delta_H(x,y)$, by definition \ref{StandardHadamardDistributionOnMinkowskiSpacetime}.
=--
+-- {: .num_prop #SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**(skew-symmetric part of [[Wightman propagator]] is the [[causal propagator]])**
The [[Wightman propagator]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) is of the form
$$
\label{DeompositionOfHadamardPropagatorOnMinkowkski}
\begin{aligned}
\Delta_H
& =
\tfrac{i}{2} \Delta_S
+
H
\\
& =
\tfrac{i}{2}
\left(
\Delta_+ - \Delta_-
\right)
+
H
\end{aligned}
\,,
$$
where
1. $\Delta_S$ is the [[causal propagator]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}),
which is real (eq:CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal) and skew-symmetric (prop. \ref{CausalPropagatorIsSkewSymmetric})
$$
(\Delta_S(x,y))^\ast = \Delta_S(x,y)
\phantom{AA}
\,,
\phantom{AA}
\Delta_S(y,x) = - \Delta_S(x,y)
$$
1. $H$ is real and symmetric
$$
(H(x,y))^\ast = H(x,y)
\phantom{AA}
\,,
\phantom{AA}
H(y,x) = H(x,y)
$$
=--
+-- {: .proof}
###### Proof
By applying [[Euler's formula]] to (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) we obtain
$$
\label{SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime}
\begin{aligned}
\Delta_H(x,y)
& =
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\\
& =
\tfrac{i}{2}
\underset{= \Delta_S(x,y)}{
\underbrace{
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\;+\;
\underset{
\coloneqq H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\end{aligned}
$$
On the left this identifies the [[causal propagator]] by (eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime), prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}.
The second summand changes, both under complex conjugation as well as under $(x-y) \mapsto (y-x)$,
via [[change of integration variables]] $\vec k \mapsto - \vec k$ (because the [[cosine]] is an even function).
This does not change the integral, and hence $H$ is symmetric.
=--
$\,$
**[[Feynman propagator]]**
{#FeynmanPropagator}
We have seen that the [[positive real number|positive]] [[frequency]] component of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) is the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) given, according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}, by (eq:DeompositionOfHadamardPropagatorOnMinkowkski)
$$
\begin{aligned}
\Delta_H
& =
\tfrac{i}{2} \Delta_S
+
H
\\
& =
\tfrac{i}{2}
\left(
\Delta_+ - \Delta_-
\right)
+
H
\end{aligned}
\,.
$$
There is an evident variant of this combination, which will be of interest:
+-- {: .num_defn #FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Definition
**([[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The _[[Feynman propagator]]_ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[linear combination]]
$$
\Delta_F
\coloneqq
\tfrac{i}{2}
\left(
\Delta_+ + \Delta_-
\right)
+
H
$$
where the first term is proportional to the sum of the [[advanced and retarded propagators]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) and the second is the symmetric part of the [[Wightman propagator]] according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}.
Similarly the _[[anti-Feynman propagator]]_ is
$$
\Delta_{\overline{F}}
\coloneqq
\tfrac{i}{2}
\left(
\Delta_+ + \Delta_-
\right)
-
H
\,.
$$
=--
+-- {: .num_prop #ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**(mode expansion for [[Feynman propagator]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is given by the following equivalent expressions
$$
\begin{aligned}
\Delta_F(x,y)
& =
\left\{
\array{
\frac{1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)c}
e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0
\\
\Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
Similarly the [[anti-Feynman propagator]] is equivalently given by
$$
\begin{aligned}
\Delta_{\overline{F}}(x,y)
& =
\left\{
\array{
\frac{-}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{-}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
-\Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0
\\
-\Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
=--
+-- {: .proof}
###### Proof
By the mode expansion of $\Delta_{\pm}$ from (eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator) and the mode expansion of $H$ from (eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime) we have
$$
\begin{aligned}
\Delta_F(x,y)
& =
\left\{
\array{
\underset{
= \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0
}{
\underbrace{
\frac{- i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
+
\underset{
= H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \gt 0
\\
\underset{
= 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0
}{
\underbrace{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
+
\underset{ = H(x,y) }{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\frac{1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0
\\
\Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
where in the second line we used [[Euler's formula]]. The last line follows by comparison with (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) and using that the integral over $\vec k$
is invariant under $\vec k \mapsto - \vec k$.
The computation for $\Delta_{\overline{F}}$ is the same, only now with a minus sign in front of the [[cosine]]:
$$
\begin{aligned}
\Delta_{\overline{F}}(x,y)
& =
\left\{
\array{
\underset{
= \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0
}{
\underbrace{
\frac{- i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
-
\underset{
= H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \gt 0
\\
\underset{
= 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0
}{
\underbrace{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
-
\underset{ = H(x,y) }{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\frac{-1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{-1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{-1i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
- \Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0
\\
- \Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
=--
As before for the [[causal propagator]], there are equivalent reformulations of the [[Feynman propagator]], which are useful for computations:
+-- {: .num_prop #FeynmanPropagatorAsACauchyPrincipalvalue}
###### Proposition
**([[Feynman propagator]] as a [[Cauchy principal value]])**
The [[Feynman propagator]] and [[anti-Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is equivalently given by the following expressions, respectively:
$$
\begin{aligned}
\left.
\array{
\Delta_F(x,y)
\\
\Delta_{\overline{F}}(x,y)
}
\right\}
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
\end{aligned}
$$
where we have a [[limit of a sequence|limit]] of [[distributions]] as for the [[Cauchy principal value]] ([this prop](Cauchy+principal+vlue#CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta)).
=--
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
(k_0)^2
-
\underset{
\coloneqq \omega_{\pm\epsilon}(\vec k)^2/c^2
}{\underbrace{ \left( \omega(\vec k)^2/c^2 \pm i \epsilon \right) }}
}
\, d k_0 \, d^p \vec k
\\
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
\left(
k_0 - \omega_{\pm \epsilon}(\vec k)/c
\right)
\left(
k_0 + \omega_{\pm \epsilon}(\vec k)/c
\right)
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{\pm i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{\mp 1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{\mp i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\Delta_F(x,y)
\\
\Delta_{\overline{F}}(x,y)
}
\right.
\end{aligned}
$$
Here
1. In the first step we introduced the [[complex number|complex]] [[square root]] $\omega_{\pm \epsilon}(\vec k)$. For this to be compatible with the choice of _non-negative_ square root for $\epsilon = 0$ in (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) we need to choose that complex square root whose [[complex phase]] is one half that of $\omega(\vec k)^2 - i \epsilon$ (instead of that plus [[π]]). This means that $\omega_{+ \epsilon}(\vec k)$ is in the _[[upper half plane]]_ and $\omega_-(\vec k)$ is in the [[lower half plane]].
1. In the third step we observe that
1. for $(x^0 - y^0) \gt 0$ the [[integrand]] decays for [[positive real number|positive]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[upper half plane]];
1. for $(x^0 - y^0) \lt 0$ the integrand decays for [[negative real number|negative]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[lower half plane]]
and then apply [[Cauchy's integral formula]] which picks out $2\pi i$ times the [[residue]] a these poles.
<img src="https://ncatlab.org/nlab/files/ContourForFeynmanPropagator.png" height="300">
Notice that when completing to a contour in the [[lower half plane]] we pick up a minus signs from the fact that now the contour runs clockwise.
1. In the fourth step we used prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}.
=--
$\,$
$\,$
**[[singular support]] and [[wave front sets]]**
{#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime}
We now discuss the [[singular support]] and the [[wave front sets]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]].
+-- {: .num_prop #SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proposition
**([[singular support]] of the [[causal propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])**
The [[singular support]] of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]],
regarded via [[translation]] [[invariant|invariance]] as a [[generalized function]] in a single variable (eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime)
is the [[light cone]] of the origin:
$$
supp_{sing}(\Delta_S)
\;=\;
\left\{
x \in \mathbb{R}^{p,1}
\,\vert\,
{\vert x\vert}^2_\eta = 0
\right\}
\,.
$$
=--
+-- {: .proof #ProofThatSingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proof
By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]];
and by basic properties of the Fourier transform this is the [[convolution of distributions]] of the separate
Fourier transforms:
$$
\begin{aligned}
\Delta_S(x)
& \propto
\widehat{
\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 )
}
\\
&\propto
\widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)}
\star
\widehat{sgn( k_0 )}
\end{aligned}
$$
By ([Gel'fand-Shilov 66, III 2.11 (7), p 294](#GelfandShilov66)), see [this prop.](Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell), the [[singular support]]
of the first convolution factor is the [[light cone]].
The second factor is
$$
\begin{aligned}
\widehat{sgn(k_0)}
& \propto
\left(2\widehat{\Theta(k_0)} - \widehat{1}\right) \delta(\vec k)
\\
& \propto
\left(2\tfrac{1}{i x^0 + 0^+} - \delta(x^0)\right) \delta(\vec k)
\end{aligned}
$$
(by [this example](Dirac+distribution#FourierTransformOfDeltaDistribution) and [this example](Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution)) and hence the [[wave front set]] of the second
factor is
$$
WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\}
$$
(by [this example](wavefront+set#WaveFrontOfDeltaDistribution) and [this example](Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation)).
With this the statement follows, via a [[partition of unity]], from [this prop.](convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions).
For illustration we now make this general argument more explicit in the special case of [[spacetime]] [[dimension]]
$$
p + 1 = 3 + 1
$$
by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]],
the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]].
We follow ([Scharf 95 (2.3.18)](causal+perturbation+theory#Scharf95)).
Consider the formula for the [[causal propagator]] in terms of the mode expansion (eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime). Since the [[integrand]] here depends on the [[wave vector]] $\vec k$ only via its [[norm]] ${\vert \vec k\vert}$ and the [[angle]] $\theta$ it makes with the given [[spacetime]] [[vector]] via
$$
\vec k \cdot (\vec x - \vec y)
\;=\;
{\vert \vec k\vert} \, {\vert \vec x\vert} \, \cos(\theta)
$$
we may express the [[integration]] in terms of [[polar coordinates]] as follws:
$$
\begin{aligned}
\Delta_S(x - y)
& =
\frac{-1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
\\
& =
\frac{- vol_{S^{p-2}}}{(2\pi)^p}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\underset{ \theta \in [0,\pi] }{\int}
\frac{ 1 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) }
{\vert \vec k\vert} ({\vert \vec k\vert} \sin(\theta))^{p-2}
\,
d \theta
\wedge
d {\vert \vec k\vert}
\end{aligned}
$$
In the special case of [[spacetime]] [[dimension]] $p + 1 = 3 + 1$ this becomes
$$
\label{StepsInComputingCausalPropagatorIn3plus1Dimension}
\begin{aligned}
\Delta_S(x - y)
& =
\frac{- 2\pi}{(2\pi)^{3}}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ {\vert \vec k \vert}^2 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\underset{
=
\tfrac{1}{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} }
\left(
e^{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}}
-
e^{-i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}}
\right)
}{
\underbrace{
\underset{ \cos(\theta) \in [-1,1] }{\int}
e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) }
d \cos(\theta)
}
}
\wedge
d {\vert \vec k \vert}
\\
& =
\frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ {\vert \vec k \vert} }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\sin\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right)
\, d {\vert \vec k\vert}
\\
& =
\frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y \vert } }
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ 1 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\cos\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right)
\, d {\vert \vec k\vert}
\\
& =
\frac{- 1}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y \vert } }
\underset{ \kappa \in \mathbb{R} }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left( \omega(\kappa) (x^0 - y^0) /c \right)
\cos\left( \kappa\, {\vert \vec x - \vec y\vert} \right)
\, d \kappa
\\
& =
\frac{- 1}{2(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y} \vert }
\left(
\underset{\coloneqq I_+}{
\underbrace{
\underset{ \kappa \in \mathbb{R} }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left(
\omega(\kappa) (x^0 - y^0) /c
+
\kappa\, {\vert \vec x - \vec y\vert}
\right)
d\kappa
}
}
+
\underset{ \coloneqq I_- }{
\underbrace{
\underset{ \kappa \in \mathbb{R} }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left(
\omega(\kappa) (x^0 - y^0) /c
-
\kappa\, {\vert \vec x - \vec y\vert}
\right)
\,
d \kappa
}
}
\right)
\,.
\end{aligned}
$$
Here in the second but last step we renamed $\kappa \coloneqq {\vert \vec k\vert}$ and
doubled the integration domain for convenience,
and in the last step we used the [[trigonometric identity]] $\sin(\alpha) \cos(\beta)\;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)$.
In order to further evaluate this, we parameterize the remaining components $(\omega/c, \kappa)$ of the [[wave vector]]
by the dual [[rapidity]] $z$, via
$$
\left(\cosh(z)\right)^2 - \left( \sinh(z)\right)^2 = 1
$$
as
$$
\omega(\kappa)/c
\;=\;
\left(
\tfrac{m c}{\hbar}
\right)
\cosh(z)
\phantom{AA}
\,,
\phantom{AA}
\kappa
\;=\;
\left(
\tfrac{m c}{\hbar}
\right)
\sinh(z)
\,,
$$
which makes use of the fact that $\omega(\kappa)$ is non-negative, by construction.
This [[change of integration variables]] makes the integrals under the braces above become
$$
\label{TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski}
I_\pm
\;=\;
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\left(
(x^0 - y^0) \cosh(z)
\pm
{\vert \vec x - \vec y\vert}
\sinh(z)
\right)
\right)
\, d z
\,.
$$
Next we similarly parameterize the vector $x-y$ by its [[rapidity]] $\tau$. That parameterization depends on whether
$x-y$ is spacelike or not, and if not, whether it is future or past directed.
First, if $x-y$ is [[spacelike]] in that ${\vert x-y\vert}^2_\eta \gt 0$
then we may parameterize as
$$
(x^0 - y^0)
=
\sqrt{{\vert x-y\vert}^2_\eta} \sinh(\tau)
\phantom{AA}
\,,
\phantom{AA}
{\vert \vec x - \vec y\vert}
=
\sqrt{ {\vert x-y\vert}^2_\eta} \cosh(\tau)
$$
which yields
$$
\begin{aligned}
I_{\pm}
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\sinh(\tau) \cosh(z)
\pm
\cosh(\tau)
\sinh(z)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta}
\left(
\sinh\left( \tau \pm z\right)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\sinh\left( z \right)
\right)
\right)
\, d z
\\
& = 0
\,,
\end{aligned}
$$
where in the last line we observe that the integrand is a skew-symmetric function of $z$.
Second, if $x-y$ is [[timelike]] with $(x^0 - y^0) \gt 0$ then we may parameterize as
$$
(x^0 - y^0)
=
\sqrt{ -{\vert x-y\vert}^2_\eta} \cosh(\tau)
\phantom{AA}
\,,
\phantom{AA}
{\vert \vec x - \vec y\vert}
=
\sqrt{ -{\vert x - y\vert}^2_\eta }
\sinh(\tau)
$$
which yields
$$
\label{IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski}
\begin{aligned}
I_\pm
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\left(
(x^0 - y^0) \cosh(z)
\pm
{\vert \vec x - \vec y\vert}
\sinh(z)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\sqrt{ - {\vert x-y\vert}^2_\eta }
\tfrac{m c}{\hbar}
\left(
\cosh(\tau)\cosh(z)
\pm
\cosh(\tau)
\sinh(z)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\sqrt{ - {\vert x-y\vert}^2_\eta }
\tfrac{m c}{\hbar}
\left(
\cosh(z \pm \tau)
\right)
\right)
\, d z
\\
& =
\pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right)
\end{aligned}
\,.
$$
Here in the last line we identified the integral representation of the [[Bessel function]] $J_0$ of order 0 (see [here](Bessel+function#eq:J0AsIntSinOfxCoshtdt)). The important point here is that this is a smooth function.
Similarly, if $x-y$ is [[timelike]] with $(x^0 - y^0) \lt 0$ then the same argument yields
$$
I_\pm = - \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right)
$$
In conclusion, the general form of $I_\pm$ is
$$
I_\pm
=
\pi
sgn(x^0 - y^0)
\Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right)
\,.
$$
Therefore we end up with
$$
\label{FinalResultOfComputationOf3Plus1dCausalPropagator}
\begin{aligned}
\Delta_S(x,y)
& =
\frac{1}{4 \pi {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y\vert}}
sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left(
\sqrt{ -{\vert x-y\vert}^2_\eta }
\tfrac{m c}{\hbar}
\right)
\\
& =
\frac{-1}{2 \pi }
\frac{d}{d (-{\vert x-y\vert}^2_\eta)}
sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left(
\sqrt{-{\vert x-y \vert}^2_\eta} \tfrac{m c}{\hbar}
\right)
\\
& =
-\frac{1}{2 \pi }
\frac{d}{d (- \vert x-y\vert^2_{\eta})}
sgn(x^0) \Theta\left( - {\vert x - y\vert}^2_\eta \right)
J_0\left(
\tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\\
& =
\frac{-1}{2\pi}
sgn(x^0)
\left(
\delta\left(
-{\vert x-y\vert}^2_\eta
\right)
\;-\;
\Theta\left( -{\vert x-y\vert}^2_\eta \right)
\frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) }
J_0\left(
\tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\right)
\end{aligned}
$$
=--
+-- {: .num_prop #SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proposition
**([[singular support]] of the [[Wightman propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])**
The [[singular support]] of the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]]
in a single variable, is the [[light cone]] of the origin:
$$
supp_{sing}(\Delta_H)
=
\left\{
x \in \mathbb{R}^{p,1}
\;\vert\;
{\vert x\vert}^2_\eta = 0
\right\}
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]];
and by basic properties of the Fourier transform this is the [[convolution of distributions]] of the separate
Fourier transforms:
$$
\begin{aligned}
\Delta_S(x)
& \propto
\widehat{
\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 )
}
\\
&\propto
\widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)}
\star
\widehat{sgn( k_0 )}
\end{aligned}
$$
By ([Gel'fand-Shilov 66, III 2.11 (7), p 294](#GelfandShilov66)), see [this prop.](Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell), the [[singular support]]
of the first convolution factor is the [[light cone]].
The second factor is
$$
\widehat{\Theta(k_0)}
\propto
\tfrac{1}{i x^0 + 0^+} \delta(\vec k)
$$
(by [this example](Dirac+distribution#FourierTransformOfDeltaDistribution) and [this example](Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution)) and hence the [[wave front set]] of the second
factor is
$$
WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\}
$$
(by [this example](wavefront+set#WaveFrontOfDeltaDistribution) and [this example](Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation)).
With this the statement follows, via a [[partition of unity]], from [this prop.](convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions).
For illustration, we now make this general statement fully explicit in the special case of [[spacetime]] [[dimension]]
$$
p + 1 = 3 + 1
$$
by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]],
the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]].
We follow ([Scharf 95 (2.3.36)](causal+perturbation+theory#Scharf95)).
By (eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime) we have
$$
\begin{aligned}
\Delta_H(x,y)
& =
\tfrac{i}{2}
\underset{= \Delta_S(x,y)}{
\underbrace{
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\;+\;
\underset{
\coloneqq H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\end{aligned}
$$
The first summand, proportional to the [[causal propagator]], which we computed as (eq:FinalResultOfComputationOf3Plus1dCausalPropagator)
in prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} to be
$$
\tfrac{i}{2}\Delta_S(x,y)
\;=\;
\frac{-i}{4\pi}
sgn(x^0)
\left(
\delta\left(
-{\vert x-y\vert}^2_\eta
\right)
\;-\;
\Theta\left( -{\vert x-y\vert}^2_\eta \right)
\frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) }
J_0\left(
\tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\right)
\,.
$$
The second term is computed in a directly analogous fashion: The integrals $I_\pm$ from
(eq:TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski) are now
$$
I_\pm
\coloneqq
\int_{-\infty}^\infty
\cos\left(
\tfrac{m c}{\hbar}
\left(
(x^0 - y^0) \cosh(z)
\pm
{\vert \vec x - \vec y\vert}
\sinh(z)
\right)
\right)
\, d z
$$
Parameterizing by [[rapidity]], as in the proof of prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}, one finds that for [[timelike]] $x-y$ this is
$$
\begin{aligned}
I_\pm
& =
\int_{-\infty}^\infty
\cos\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\cosh\left( z \right)
\right)
\right)
\, d z
\\
& =
- \pi N_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
\end{aligned}
$$
while for [[spacelike]] $x-y$ it is
$$
\begin{aligned}
I_\pm
& =
\int_{-\infty}^\infty
\cos\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\sinh\left( z \right)
\right)
\right)
\, d z
\\
& =
2 K_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
\,,
\end{aligned}
$$
where we identified the integral representations of the [[Neumann function]] $N_0$ (see [here](Bessel+function#N0AsIntSinOfxCoshtdt))
and of the [[modified Bessel function]] $K_0$ (see [here](Bessel+function#eq:K0AsIntSinOfxCoshtdt)).
As for the [[Bessel function]] $J_0$ in (eq:IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski) the key point is that these are [[smooth functions]]. Hence we conclude that
$$
H(x,y)
\;\propto\;
\frac{d}{d \left( {\vert x-y\vert}^2_\eta \right)}
\left(
-\Theta\left( -{\vert x-y\vert}^2_\eta \right)
N_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
+
\Theta\left( {\vert x-y\vert}^2_\eta \right)
\tfrac{2}{\pi}
K_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
\right)
\,.
$$
This expression has singularities on the [[light cone]] due to the [[step functions]].
In fact the expression being differentiated is continuous at the light cone
([Scharf 95 (2.3.34)](#Scharf95)), so that the singularity on the light cone is not a [[delta distribution]]
singularity from the derivative of the step functions. Accordingly it does not cancel the singularity
of $\tfrac{i}{2}\Delta_S(x,y)$ as above, and hence the singular support of $\Delta_H$ is still the whole
light cone.
=--
+-- {: .num_prop #SingularSupportOfFeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**([[singular support]] of [[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[singular support]] of the [[Feynman propagator]] $\Delta_H$ and of the [[anti-Feynman propagator]] $\Delta_{\overline{F}}$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]]
in a single variable, is the [[light cone]] of the origin:
$$
\left.
\array{
supp_{sing}(\Delta_F)
\\
supp_{sing}(\Delta_{\overline{F}})
}
\right\}
=
\left\{
x \in \mathbb{R}^{p,1}
\;\vert\;
{\vert x\vert}^2_\eta = 0
\right\}
\,.
$$
=--
(e.g [DeWitt 03 (27.85)](Feynman+propagator#DeWitt03))
+-- {: .proof}
###### Proof
By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the Feynman propagator is equivalently
the [[Cauchy principal value]] of the inverse of the Fourier transformed Klein-Gordon operator:
$$
\Delta_F
\;\propto\;
\widehat{
\frac{1}{-k_\mu k^\mu - \left(\tfrac{m c}{\hbar}\right)^2 + i 0^+}
}
\,.
$$
With this the statement follows immediately from the result
([Gel'fand-Shilov 66, III 2.8 (8) and (9), p 289](#GelfandShilov66)), see [this prop.](Cauchy+principal+value#FourierTransformOfPrincipalValueOfPowerOfQuadraticForm).
=--
+-- {: .num_prop #WaveFronSetsForKGPropagatorsOnMinkowski}
###### Proposition
**([[wave front sets]] of [[propagators]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[wave front set]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded, via [[translation]] [[invariant|invariance]], as [[distributions]] in a single variable, are as follows:
* the [[causal propagator]] $\Delta_S$ (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the lightcone:
$$
WF(\Delta_S) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \, k \neq 0 \right\}
$$
<center>
<img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60"> <br/> - <br/> <img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60">
</center>
* the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $k^0 \gt 0$:
$$
WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; k^0 \gt 0 \right\}
$$
<center>
<img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60">
</center>
* the [[Feynman propagator]] $\Delta_S$ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0$
$$
WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \;
\left(
\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0
\right)
\right\}
$$
<center>
<img src="https://ncatlab.org/nlab/files/FeynmanPropagator.png" width="60">
</center>
=--
([Radzikowski 96, (16)](Hadamard+distribution#Radzikowski96))
+-- {: .proof}
###### Proof
First regarding the causal propagator:
By prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} the [[singular support]] of $\Delta_S$ is the [[light cone]].
Since the causal propagator is a solution to the homogeneous Klein-Gordon equation, the
[[propagation of singularities theorem]] says that also all [[wave vectors]] in the wave front set are lightlike.
Hence it just remains to show that all non-vanishing lightlike wave vectors based on the lightcone in spacetime
indeed do appear in the wave front set.
To that end, let $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ be a [[bump function]] whose [[compact support]] includes the origin.
For $a \in \mathbb{R}^{p,1}$ a point on the light cone, we need to determine the decay property of the Fourier transform of $x \mapsto b(x-a)\Delta_S(x)$. This is the [[convolution of distributions]] of $\hat b(k)e^{i k_\mu a^\mu}$ with $\widehat \Delta_S(k)$. By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} we have
$$
\widehat \Delta_{S}(k)
\;\propto\;
\delta\left( -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \right)
sgn(k_0)
\,.
$$
This means that the convolution product is the smearing of the mass shell by $\widehat b(k)e^{i k_\u a^\mu}$.
Since the mass shell asymptotes to the light cone, and since $e^{i k_\mu a^\mu} = 1$ for $k$ on the light cone (given that $a$ is on the light cone), this implies the claim.
Now for the [[Wightman propagator]]:
By def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} its Fourier transform is of the form
$$
\widehat \Delta_H(k)
\;\propto\;
\delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 )
$$
Moreover, its [[singular support]] is also the light cone (prop. \ref{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}).
Therefore now same argument as before says that the wave front set consists of wave vectors $k$ on the light cone, but now due to the [[step function]] factor $\Theta(-k_0)$ it must satisfy $0 \leq - k_0 = k^0$.
Finally regarding the [[Feynman propagator]]:
By prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime} the Feynman propagator
coincides with the positive frequency Wightman propagator for $x^0 \gt 0$ and with the "negative frequency Hadamard operator"
for $x^0 \lt 0$. Therefore the form of $WF(\Delta_F)$ now follows directly with that of $WF(\Delta_H)$ above.
=--
### For Dirac operator on Minkowski spacetime
{#ExampleForDiracOperatorOnMinkowskiSpacetime}
Finally we observe that the [[propagators]] for the [[Dirac field]] on [[Minkowski spacetime]]
follow immediately from the propagators for the [[scalar field]]:
+-- {: .num_prop #DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}
###### Proposition
**([[advanced and retarded propagator]] for [[Dirac equation]] on [[Minkowski spacetime]])**
Consider the [[Dirac operator]] on [[Minkowski spacetime]], which in [[Feynman slash notation]] reads
$$
\begin{aligned}
D
& \coloneqq
-i {\partial\!\!\!/\,}
+ \tfrac{m c}{\hbar}
\\
& =
-i \gamma^\mu \frac{\partial}{\partial x^\mu} + \tfrac{m c}{\hbar}
\end{aligned}
\,.
$$
Its [[advanced and retarded propagators]] (def. \ref{AdvancedAndRetardedGreenFunctions})
are the [[derivatives of distributions]] of the advanced and retarded propagators $\Delta_\pm$ for the [[Klein-Gordon equation]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) by ${\partial\!\!\!/\,} + m$:
$$
\Delta_{D, \pm}
\;=\;
\left(
-i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
\right)
\Delta_{\pm}
\,.
$$
Hence the same is true for the [[causal propagator]]:
$$
\Delta_{D, S}
\;=\;
\left(
-i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
\right)
\Delta_{S}
\,.
$$
=--
+-- {: .proof}
###### Proof
Applying a [[differential operator]] does not change the [[support]] of a [[smooth function]], hence also not the [[support of a distribution]]. Therefore the uniqueness of the advanced and retarded propagators (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique})
together with the translation-invariance and the anti-[[formally self-adjoint differential operator|formally self-adjointness]] of the [[Dirac operator]] (as for the [[Klein-Gordon operator]] (eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime)
implies that it is sufficent to check that applying the [[Dirac operator]] to the $\Delta_{D, \pm}$
yields the [[delta distribution]]. This follows since the Dirac operator squares to the Klein-Gordon operator:
$$
\begin{aligned}
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)
\Delta_{D, \pm}
& =
\underset{ = \Box - \left(\tfrac{m c}{\hbar}\right)^2}{
\underbrace{
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)
\left(
-i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
\right)
}
}
\Delta_{\pm}
\\
& =
\delta
\end{aligned}
\,.
$$
=--
Similarly we obtain the other [[propagators]] for the [[Dirac field]] from those of the [[real scalar field]]:
+-- {: .num_defn #HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}
###### Definition
**([[Wightman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])**
The _[[Wightman propagator]]_ for the [[Dirac operator]] on [[Minkowski spacetime]] is the [[positive real number|positive]] [[frequency]] part of the [[causal propagator]] (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}),
hence the [[derivative of distributions]] of the Wightman propagator for the Klein-Gordon field (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) by the [[Dirac operator]]:
$$
\begin{aligned}
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)\Delta_{H}(x,y)
& =
\frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) ( {k\!\!\!/\,} + \tfrac{m c}{\hbar}) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k
\\
& =
\frac{1}{(2\pi)^p}
\int \frac{ \gamma^0 \omega(\vec k)/c + \vec \gamma \cdot \vec k + \tfrac{m c}{\hbar} }{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\,.
\end{aligned}
$$
Here we used the expression (eq:StandardHadamardDistributionOnMinkowskiSpacetime) for the Wightman propagator of the Klein-Gordon equation.
=--
+-- {: .num_defn #FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}
###### Definition
**([[Feynman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])**
The _[[Feynman propagator]]_ for the [[Dirac operator]] on [[Minkowski spacetime]] is the linear combination
$$
\Delta_{D, F}
\;\coloneqq\;
\Delta_{D,H}
+
i \Delta_{D, -}
$$
of the [[Wightman propagator]] (def. \ref{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}) and the retarded propagator (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}). By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}
this means that it is the
[[derivative of distributions]] of the [[Feynman propagator]] of the [[Klein-Gordon equation]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) by the [[Dirac operator]]
$$
\begin{aligned}
\Delta_{D, F}
& =
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)\Delta_{F}(x,y)
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{-i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
\left( {k\!\!\!/\,} + \tfrac{m c}{\hbar} \right) e^{i k_\mu (x^\mu - y^\mu)}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
}
\, d k_0 \, d^p \vec k
\,.
\end{aligned}
$$
=--
## Related concepts
[[!include propagators - table]]
$\,$
* [[Green hyperbolic differential operator]]
## References
General discussion includes
* {#Baer14} [[Christian Bär]], _Green-hyperbolic operators on globally hyperbolic spacetimes_, Communications in Mathematical Physics 333, 1585-1615 (2014) ([doi](http://dx.doi.org/10.1007/s00220-014-2097-7), [arXiv:1310.0738](https://arxiv.org/abs/1310.0738))
* {#Khavkine14} [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv:1402.1282](https://arxiv.org/abs/1402.1282))
based on
* {#Sanders12} [[Ko Sanders]], _A note on spacelike and timelike compactness_, Classical and Quantum Gravity 30, 115014 (2012) ([doi](http://dx.doi.org/10.1088/0264-9381/30/11/115014), [arXiv:1211.2469](https://arxiv.org/abs/1211.2469))
Textbook discussion for [[free fields]] in [[Minkowski spacetime]] is in
* {#Scharf95} [[Günter Scharf]], section 2.3 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Springer 1995
* {#Scharf01} [[Günter Scharf]], section 1 of _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001
An overview of the [[Green functions]] of the [[Klein-Gordon operator]], hence of the [[Feynman propagator]], [[advanced propagator]], [[retarded propagator]], [[causal propagator]] etc. is given in
* {#Kocic16} [[Mikica Kocic]], _Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions_, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])
Discussion on general [[globally hyperbolic spacetimes]] includes
* F. Friedlander, _The Wave Equation on a Curved Space-Time_, Cambridge: Cambridge University Press, 1975
* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))
* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_, Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009
Review in the context of [[perturbative algebraic quantum field theory]] includes
* [[Katarzyna Rejzner]], sections 4.1 and 6.2.3 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([pdf](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))
[[!redirects retarded and advanced causal propagators]]
[[!redirects advanced and retarded propagators]]
[[!redirects retarded and advanced propagators]]
[[!redirects advanced and retarded propagator]]
[[!redirects retarded and advanced propagator]]
[[!redirects advanced propagator]]
[[!redirects advanced propagators]]
[[!redirects retarded propagator]]
[[!redirects retarded propagators]]
[[!redirects advanced causal propagator]]
[[!redirects advanced causal propagators]]
[[!redirects retarded causal propagator]]
[[!redirects retarded causal propagators]]
[[!redirects advanced Green function]]
[[!redirects advanced Green functions]]
[[!redirects retarded Green function]]
[[!redirects retarded Green functions]]
[[!redirects retarded and advanced Green functions]]
[[!redirects advanced and retarded Green functions]]
[[!redirects advanced Green's function]]
[[!redirects advanced Green's functions]]
[[!redirects retarded Green's function]]
[[!redirects retarded Green's functions]]
[[!redirects advanced or retarded Green function]]
[[!redirects advanced or retarded Green functions]]
[[!redirects retarded and advanced Green's functions]]
[[!redirects advanced and retarded Green's functions]]
|
Ady Stern | https://ncatlab.org/nlab/source/Ady+Stern |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Ady_Stern)
* [Institute page](https://webhome.weizmann.ac.il/home/stern/)
## Selected writings
On [[anyons]] in the [[quantum Hall effect]]:
* [[Ady Stern]], _Anyons and the quantum Hall effect -- A pedagogical review_, Annals of Physics Volume 323, Issue 1, January 2008, Pages 204-249 ([doi:10.1016/j.aop.2007.10.008](https://doi.org/10.1016/j.aop.2007.10.008), [arXiv:0711.4697](https://arxiv.org/abs/0711.4697))
On [[anyons]] in application to [[topological quantum computation]]:
* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. 80, 1083 (2008) ([arXiv:0707.1888] (http://arxiv.org/abs/0707.1889))
* [[Ady Stern]], Netanel H. Lindner, *Topological Quantum Computation -- From Basic Concepts to First Experiments*, Science 08 Mar 2013: Vol. 339, Issue 6124, pp. 1179-1184 ([doi:10.1126/science.1231473](https://science.sciencemag.org/content/339/6124/1179.full.pdf+html))
category: people
|
Adámek's fixed point theorem | https://ncatlab.org/nlab/source/Ad%C3%A1mek%27s+fixed+point+theorem | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
# Idea
_Adámek's fixed point theorem_ gives a method to construct an [[initial algebra of an endofunctor]] by "iterating" the functor.
## Construction
This theorem is attributed to [[Jiří Adámek]] but note that the same construction for set-functors already appeared in ([Pohlova,1973](#Pohlova))).
+-- {: .num_theorem #AdameksTheorem }
######Theorem (Adámek)
Let $C$ be a category with an [[initial object]] $0$ and [[transfinite composition]] of length $\omega$, hence [[colimits]] of sequences $\omega \to C$ (where $\omega$ is the first infinite [[ordinal]]), and suppose $F: C \to C$ preserves colimits of C$\omega$-chains. Then the colimit $\gamma$ of the chain
$$0 \overset{i}{\to} F(0) \overset{F(i)}{\to} \ldots \to F^{(n)}(0) \overset{F^{(n)}(i)}{\to} F^{(n+1)}(0) \to \ldots$$
carries a structure of initial $F$-algebra.
=--
+-- {: .proof}
###### Proof
The $F$-algebra structure $F\gamma \to \gamma$ is inverse to the canonical map $\gamma \to F\gamma$ out of the colimit (which is invertible by the hypothesis on $F$). The proof of initiality may be extracted by dualizing the corresponding proof given at [[terminal coalgebra]].
=--
This approach can be generalized to the [[transfinite construction of free algebras]].
## Related entries
* [[fixed point]]
* [[Kleene's fixed point theorem]] is precisely the [[de-categorification]] of this theorem to [[posets]]/[[preorders]].
* [[Knaster-Tarski's fixed point theorem]]
* [[Lawvere's fixed point theorem]]
* [[Lefschetz fixed point theorem]]
* [[Brouwer's fixed point theorem]]
* [[Atiyah-Bott fixed point formula]]
## References
* {#Pohlova} Věra Pohlová. "On sums in generalized algebraic categories." Czechoslovak Mathematical Journal **23**.2 (1973) 235-251 [[eudml:12718](http://eudml.org/doc/12718)]
* [[Jiří Adámek]], *Free algebras and automata realizations in the language of categories*, Commentationes Mathematicae Universitatis Carolinae **15**.4 (1974) 589-602 [[eudml:16649](https://eudml.org/doc/16649)]
* [[Jiří Adámek]], Věra Trnková, *Automata and algebras in categories* **37** Springer (1990) [[ISBN:9780792300106](https://link.springer.com/book/9780792300106)]
[[!redirects Adamek's fixed point theorem]]
[[!redirects Adamek's theorem]]
[[!redirects Adámek's theorem]] |
Aeysha Khalique | https://ncatlab.org/nlab/source/Aeysha+Khalique |
* [institute page](https://sns.nust.edu.pk/faculty/aeysha-khalique/)
* [GoogleScholar page](https://scholar.google.ca/citations?user=mKbWAvcAAAAJ&hl=en)
## Selected writings
On practical realization of [[quantum teleportation]]:
* Syed Tahir Amin, [[Aeysha Khalique]], *Practical Quantum Teleportation of an Unknown Quantum State*, Can. J. Phys. **95** 5 (2017) 498 [[arXiv:1508.01141](https://arxiv.org/abs/1508.01141), [doi:10.1139/cjp-2016-0758](https://doi.org/10.1139/cjp-2016-0758)]
category: people |
affiliated operator | https://ncatlab.org/nlab/source/affiliated+operator | ## Motivation
Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on $L^2(\mathbb{R}^n)$ which is proportional to the momentum operator in quantum mechanics.
## Idea
The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of "approximable" operators by means of operator algebra itself. One way to do this is to define the __affiliated elements__ of $C^\ast$-algebra, or the operators affiliated with the $C^\ast$-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the [[spectral theorem|spectral decomposition]] will supply the approximation.
## Details
Let $T$ be a normal unbounded operator. Thanks to the [[spectral theorem]] for unbounded operators, $T$ has a [[spectral measure]] $P_{\lambda}$. Every single spectral projection is bounded, of course, so we may look for von Neumann algebras that contain them. Since a von Neumann algebra may be characterised as the algebra of all operators that commute with some set of unitary operators, we give the following definition:
* Definition (affiliation): A closed operator $T$ is **affiliated** with a von Neumann algebra $\mathcal{M}$, written as $T \eta \mathcal{M}$, if every unitary operator $U$ in the commutant of $\mathcal{M}$ transforms $D_T$ to itself, and we have $U^*TU = T$.
We mention some interesting theorems using this concept.
* Theorem (Kadison-Ringrose 5.6.18): An operator is normal iff it is affiliated with an abelian von Neumann algebra. If A is normal, there is a smallest von Neumann algebra that A is affiliated with, this algebra is abelian.
* Definition: Given a normal operator A, the smallest (and necessarily abelian) von Neumann algebra that A is associated with is called the **von Neumann algebra generated by A**.
## Literature
* [[S. L. Woronowicz]], K. Napiórkowski, _Operator theory in $C^\ast$-framework_, Reports on Mathematical Physics __31__, Issue 3 (1992), 353-371,
<a href="http://dx.doi.org/10.1016/0034-4877(92)90025-V">doi</a>, [pdf](http:www.fuw.edu.pl/~slworono/PDF-y/OP.pdf)
* [[S. L. Woronowicz]], _Unbounded elements affiliated with $C^\ast$-algebras and non-compact quantum groups_, Commun. Math. Phys. __136__, 399--432 (1991) [euclid](http://projecteuclid.org/euclid.cmp/1104202358) [MR1096123](http://www.ams.org/mathscinet-getitem?mr=1096123); _$C^\ast$-algebras generated by unbounded elements_, [pdf](http://www.fuw.edu.pl/~slworono/PDF-y/GENER.pdf)
* wikipedia [affiliated operator](http://en.wikipedia.org/wiki/Affiliated_operator)
[[!redirects affiliated element]]
[[!redirects affiliated elements]] |
affine algebra | https://ncatlab.org/nlab/source/affine+algebra | Given a field $k$ an __affine algebra__ is any finitely generated [[noetherian ring|noetherian]] associative commutative unital $k$-algebra without [[nilpotent elements]]. This term is limited to a small community of algebraists and may bring confusion if quoted outside of that context. The name is from the fact that the affine algebras are the coordinate rings of [[affine varieties]]; equivalently, an affine $k$-variety is precisely the [[maximal spectrum]] of an affine $k$-algebra.
Some people drop the condition of commutativity and talk about noncommutative affine algebras. This is even less standard as in the noncommutative context there is a further hesitation to drop noetherianess as well.
Yet another confusion may arise when people informally say affine algebra for an [[affine Lie algebra]]; of course, the latter, full, term is preferred (or "affine Kac-Moody algebra").
[[!redirects affine algebras]]
|
affine connection | https://ncatlab.org/nlab/source/affine+connection |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
An _affine connection_ $\nabla$ on a [[smooth manifold]] $M$ is a [[connection on a bundle|connection]] on the [[frame bundle]] $F M$ of $M$, i.e., the [[principal bundle]] of [[frame of a vector space|frames]] in the [[tangent bundle]] $T M$.
The components of the local [[Lie-algebra valued 1-form]] of an affine connection are called [[Christoffel symbols]].
## Related concepts
* [[connection on a bundle]]
* [[parallel transport]], [[holonomy]]
* [[principal connection]]
* [[affine connection]], [[Levi-Civita connection]], [[Cartan connection]], [[symplectic connection]]
* [[connection on a 2-bundle]]
* [[connection on an infinity-bundle]]
* [[higher parallel transport]]
## References
A coordinate-free treatment first appeared in
* [[Harley Flanders]], _Development of an extended exterior differential calculus_. Transactions of the American Mathematical Society 75:2 (1953), 311–311. [doi](https://doi.org/10.1090/s0002-9947-1953-0057005-8).
* wikipedia [affine connection](http://en.wikipedia.org/wiki/Affine_connection)
* eom [affine connection](http://eom.springer.de/a/a010950.htm)
* А.П. Норден, _Пространства аффинной связности_, 1976, [djvu](http://gen.lib.rus.ec/get?nametype=orig&md5=05EC4513F05BA7E7F05195880858AA29)
[[!redirects affine connection]]
[[!redirects affine connections]]
|
affine Grassmannian | https://ncatlab.org/nlab/source/affine+Grassmannian |
#Contents#
* table of contents
{:toc}
## Idea
According to [wikipedia](https://en.wikipedia.org/wiki/Affine_Grassmannian),
> the affine Grassmannian of an algebraic group $G$ over a field $k$ is an [[ind-scheme]] which can be thought of as a flag variety for the loop group $G(k((t)))$
The affine Grassmannian is ind-representable.
Affine Grassmannian of $SL_n$ admits embedding into [[Sato Grassmanian]].
## Definitions
### The $\GL_{n}$ case
\begin{definition}(Definition 1.1.1 of [#Zhu2016](#Zhu2016))
Let $k$ be a field and let $R$ be a $k$-algebra. An _$R$-family of lattices in $k((t))^{n}$_ is a finitely generated projective submodule $\Lambda$ of $R((t))^{n}$ such that $\Lambda \otimes_{R[[t]]}R((t))=R((t))^{n}$.
\end{definition}
\begin{definition}(Definition 1.1.2 of [#Zhu2016](#Zhu2016))
The _affine Grassmannian $\Gr_{\GL_{n}}$ for $\GL_{n}$_ is the presheaf that assigns to every $k$-algebra $R$ the set of $R$-families of lattices in $k((t))^{n}$.
\end{definition}
### The general case
\begin{definition}(Section 1.2 of [#Zhu2016](#Zhu2016))
Let $k$ be a field and let $G$ be an affine $k$-group. If $R$ is a $k$-algebra, let $D_{R}=\Spec k[[t]]\widehat{\times}\Spec R$ and let $D^{*}_{R}=\Spec k((t))\widehat{\times} \Spec R$. Let $\mathcal{E}^{0}$ be the trivial $G$-torsor over $D_{R}$.
The _affine Grassmannian $\Gr_{G}$ of $G$_ is the presheaf that assigns to every $k$-algebra $R$ the set of pairs $(\mathcal{E},\beta)$, where $\mathcal{E}$ is a $G$-torsor on $D_{R}$ and $\beta:\mathcal{E}\vert_{D^{*}_{R}}\xrightarrow{\simeq}\mathcal{E}^{0}\vert_{D^{*}_{R}}$ is a trivialization.
\end{definition}
## Related
* [[geometric Langlands correspondence]], [[Ran space]]
* [[G-torsor]]
* [[Grassmannian]], [[flag variety]]
## References
* Evgeny Feigin, Michael Finkelberg, Markus Reineke, _Degenerate affine Grassmannians and loop quivers_, [http://arxiv.org/abs/1410.0777](http://arxiv.org/abs/1410.0777)
* {#Zhu2016} Xinwen Zhu, _An introduction to affine Grassmannians and the geometric Satake equivalence_, [arXiv:1603.05593](http://arxiv.org/abs/1603.05593v2).
* [[Bhargav Bhatt]], [[Peter Scholze]], _Projectivity of the Witt vector affine Grassmannian_, Invent. math. 209, 329-423 (2017) [doi](https://doi.org/10.1007/s00222-016-0710-4) [arXiv:1507.06490](https://arxiv.org/abs/1507.06490)
* Alexander Schmitt, _Affine flag manifolds and principal bundles_, Trends in Mathematics, Springer 2010
category: algebra, algebraic geometry |
affine group | https://ncatlab.org/nlab/source/affine+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Given an [[affine space]] underlying a [[vector space]] $V$, then its _affine group_ is its group of [[automorphisms]], i.e. the [[semidirect product group]]
$$
Aff(V) = V \rtimes GL(V)
$$
of the [[translation group]] of $V$ and the [[general linear group]] of $V$.
## Related concepts
* [[affine symplectic group]]
* [[Galilei group]]
* [[Poincare group]]
[[!redirects affine groups]]
[[!redirects affine transformation]]
[[!redirects affine transformations]]
|
affine group scheme | https://ncatlab.org/nlab/source/affine+group+scheme | > Distinguish from the notion of a group of affine transformations (of an affine space) or the (general) affine group.
## Definition
An __affine group scheme__ is a [[group object]] (hence a [[group scheme]]) [[internalization|internal]] to the [[category]] of [[affine schemes]].
## Related notions
If $k$ is a [[commutative ring]], the category of affine group $k$-schemes is [[opposite category|opposite]] to the category of commutative [[Hopf algebra|Hopf $k$-algebras]].
By an __affine algebraic group__, one additionally assumes that the underlying scheme is over an [[algebraic variety]] (in particular reducible) and over a field. The narrower notion then agrees with the notion of a [[linear algebraic group]].
[Waterhouse](#Waterhouse)12.1 The Lie algebra of the affine group $k$-scheme $X$ is the Lie algebra of left-invariant $k$-linear derivations $T:\mathcal{O}(X)\to\mathcal{O}(X)$ of the Hopf algebra of regular functions of $X$, which are also morphisms of right comodules: $\Delta\circ T = (id\otimes T)\circ\Delta:\mathcal{O}(X)\to\mathcal{O}(X)\otimes\mathcal{O}(X)$
(one says also that $T$ is left-invariant).
This Lie algebra is also isomorphic to the Lie algebra of $k$-linear $k$-valued derivations of $\mathcal{O}(X)$.
## Examples
* The *[[additive group]] scheme* (viewed as a group valued functor on the opposite of the category of affine schemes) assigns to a commutative ring its additive group.
## Literature
Comprehensive textbooks include
* [[James Milne|James S. Milne]], _Basic theory of affine group schemes_, [pdf](https://www.jmilne.org/math/CourseNotes/AGS.pdf)
* {#Waterhouse} William C. Waterhouse, _Introduction to affine group schemes_, Graduate texts in mathematics 66, 1979
category: algebra, algebraic geometry
[[!redirects affine group schemes]]
[[!redirects affine algebraic group]]
[[!redirects affine algebraic groups]]
|
affine Lie algebra | https://ncatlab.org/nlab/source/affine+Lie+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
__Affine Lie algebras__ (sometimes: _[[current algebras]]_) are the most important class of [[Kac-Moody Lie algebras]]. They should be viewed as tangent Lie algebras to the [[loop groups]], with a correction term which is sometimes related to quantization/quantum anomaly.
These affine Lie algebras appear in [[quantum field theory]] as the [[current algebras]] in the [[WZW model]] as well as in its "chiral halfs", as such for instance in the [[heterotic string]] [[2d CFT]].
## Related concepts
* [[Kac-Weyl character]]
* [[quantum affine algebra]]
* [[Borcherds algebra]]
* [[double affine Hecke algebra]]
* [[su(2)-anyon]]
## References
Textbook account
* [[Victor G. Kac]], *Infinite Dimensional Lie Algebras*, Progress in Mathematics **44** Springer 1983 [[doi:10.1007/978-1-4757-1382-4](https://link.springer.com/book/10.1007/978-1-4757-1382-4)], Cambridge University Press (1990) [[doi:10.1017/CBO9780511626234](https://doi.org/10.1017/CBO9780511626234)]
Lecture notes:
* [[Minoru Wakimoto ]], *Lectures on Infinite-Dimensional Lie Algebra*, World Scientific (2001) [[doi:10.1142/4269](https://doi.org/10.1142/4269)]
* David Hernandez, _An introduction to affine Kac-Moody algebras_ (2006) ([pdf](http://www.ctqm.au.dk/events/2006/October/Week42/Masterclassnotes.pdf))
* {#Gordon08} [[Iain Gordon]], _Infinite-dimensional Lie algebras_ (2008/9) ([pdf](http://www.maths.ed.ac.uk/~igordon/LA1.pdf)
* {#Wassermann11} [[Antony Wassermann]], _Kac-Moody and Virasoro algebras_, course notes (2011) ([pdf](https://www.dpmms.cam.ac.uk/~ajw/course11.pdf))
The standard textbook on [[loop groups]] is
* A. Pressley, [[Graeme Segal]], _Loop groups_, Oxford Univ. Press 1988
The relation to [[quantum physics|quantum]] [[physics]] ([[WZW model]]) is highlighted in:
* S. Kass, R. V. Moody, J. Patera, _Affine Lie Algebras, Weight Multiplicities, and Branching Rules _
* Louise Dolan, _The Beacon of Kac-Moody symmetry for physics_, Notices of the AMS 1995 ([pdf](http://www.ams.org/notices/199512/dolan.pdf))
and specifically a review in the context of the [[Witten genus]] is in
* [[Kefeng Liu]], section 2.2 of _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 ([EUCLID](http://projecteuclid.org/euclid.jdg/1214456221), [pdf](http://www.math.ucla.edu/~liu/Research/loja.pdf))
The famous quote by Kac is from:
* [[Victor Kac]], _The idea of locality_ ([q-alg/9709008](http://arxiv.org/abs/q-alg/9709008))
> It is a well kept secret that the theory of Kac-Moody algebras has been a disaster. True, it is a generalization of a very important object, the simple finite-dimensional Lie algebras, but a generalization too straightforward to expect anything interesting from it. True, it is remarkable how far one can go with all these ei's, fi's and hi's. Practically all, even most difficult results of finite-dimensional theory, such as the theory of characters, Schubert calculus and cohomology theory, have been extended to the general set-up of Kac-Moody algebras. But the answer to the most important question is missing: what are these algebras good for? Even the most sophisticated results, like the connections to the theory of quivers, seem to be just scratching the surface.
> However, there are two notable exceptions. The best known one is, of course, the theory of affine Kac-Moody algebras. This part of the Kac-Moody theory has deeply penetrated many branches of mathematics and physics. The most important single reason for this success is undoubtedly the isomorphism of affine algebras and central extensions of loop algebras, often called current algebras. The second notable exception is provided by [[Borcherds' algebras]] which are roughly speaking the spaces of physical states of certain chiral algebras.
Relation to [[modular forms]]:
* [[Victor G. Kac]], [[Dale H. Peterson]], *Affine Lie algebras and Hecke modular forms*, Bull. Amer. Math. Soc. (N.S.) **3** 3 (1980) 1057-1061 [[bams:1183547694](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-3/issue-3/Affine-Lie-algebras-and-Hecke-modular-forms/bams/1183547694.full)]
* [[Victor G. Kac]], [[Dale H. Peterson]], *Infinite-dimensional Lie algebras, theta functions and modular forms*, Advances in Mathematics **53** 2 (1984) 125-264 [<a href="https://doi.org/10.1016/0001-8708(84)90032-X">doi:10.1016/0001-8708(84)90032-X</a>]
review:
* [[Ian G. MacDonald]], *Affine Lie algebras and modular forms*, Séminaire Bourbaki: vol. 1980/81, exposés 561-578, Séminaire Bourbaki, no. 23 (1981), Exposé no. 577 [[numdam:SB_1980-1981__23__258_0](http://www.numdam.org/book-part/SB_1980-1981__23__258_0/)]
* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular and conformal invariance constraints in representation theory of affine algebras*, Advances in Mathematics **70** 2 (1988) 156-236 [<a href="https://doi.org/10.1016/0001-8708(88)90055-2">doi:10.1016/0001-8708(88)90055-2</a>, [spire:275458](https://inspirehep.net/literature/275458)]
On non-integrable but "admissible" [[irreps]] of [[affine Lie algebras]]:
* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*, PNAS **85** 14 (1988) 4956-4960 [[doi:10.1073/pnas.85.14.4956](https://doi.org/10.1073/pnas.85.14.4956)]
* [[Victor G. Kac]], [[Minoru Wakimoto]], *Classification of modular invariant representations of affine algebras*, p. 138-177 in V. G. Kač (ed.): *Infinite dimensional Lie algebras and groups*, Advanced series in Mathematical physics **7**, World Scientific (1989) [[pdf](https://math.mit.edu/~kac/not-easily-available/admissible.pdf), [cds:268092](https://cds.cern.ch/record/268092)]
For the special case $\mathfrak{g} = $ [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]] the formula for the "admissible" weights is made explicit in
* [[Boris Feigin]], [[Fyodor Malikov]], *Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.) *Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics **202** , AMS (1997) [[arXiv:q-alg/9511011](https://arxiv.org/abs/q-alg/9511011), [ams:conm-202](https://bookstore.ams.org/conm-202)]
[[!redirects affine Lie algebras]]
[[!redirects affine Kac-Moody Lie algebra]]
[[!redirects affine Kac-Moody Lie algebras]]
|
affine line | https://ncatlab.org/nlab/source/affine+line |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For every [[Lawvere theory]] $T$ containing the theory of [[abelian group]]s [[Isbell duality|Isbell dual]] [[sheaf topos]] over formal duals of $T$-algebras contains a canonical [[line object]] $\mathbb{A}^1$.
For $T$ the theory of commutative [[ring]]s this is called the _affine line_ .
## Definition
### Affine line
Let $k$ be a [[ring]], and $T$ the [[Lawvere theory]] of [[associative algebra]]s over $k$, such that the category of [[algebras over a Lawvere theory]] $T Alg = Alg_k$ is the [[category]] of $k$-algebras.
+-- {: .num_defn}
###### Definition
The canonical $T$-[[line object]] is the **affine line**
$$
\mathbb{A}_k := Spec(F_T(*)) = Spec (k[t])
\,.
$$
=--
Here the [[free functor|free]] $T$-algebra on a single generator $F_T(*)$ is the [[polynomial algebra]] $k[t] \in Alg_k$ on a single generator $* = t$ and $Spec k[t]$ may be regarded as the corresponding object in the [[opposite category]] $Aff_k := Alg_k^{op}$ of [[affine scheme]]s over $Spec k$.
### Multiplicative group
{#MultiplicativeGroup}
The [[multiplicative group]] object in $Ring^{op}$ corresponding to the affine line -- usually just called the **multiplicative group** -- is the [[group scheme]] denoted $\mathbb{G}_m$
* whose underlying affine scheme is
$$
(\mathbb{A}^1 - \{0\}) := Spec \left(k[t,t^{-1}]\right)
\,,
$$
where $k[t,t^{-1}]$ is the [[localization]] of the ring $k[t]$ at the element $t = (t-0)$.
* whose multiplication operation
$$
\cdot \mathbb{G}_m \times \mathbb{G}_m \to \mathbb{G}_m
$$
is the morphism in $Ring^{op}$ corresponding to the morphism in [[Ring]]
$$
k[t_1,t_1^{-1}] \otimes_k k[t_2, t_2^{-1}] \leftarrow k[t,t^{-1}]
$$
given by $t \mapsto t_1 \cdot t_2$;
* whose unit map $Spec k \to Spec k[t,t^{-1}]$ is given by
$$
t \mapsto 1
$$
* and whose inversion map $Spec k[t,t^{-1}] \to Spec[t,t^{-1}]$ is given by
$$
t \mapsto t^{-1}
\,.
$$
Therefore for $R$ any [[ring]] a morphism
$$
Spec R \longrightarrow \mathbb{G}_m
$$
is equivalently a ring homomorphism
$$
R \leftarrow k[t,t^{-1}]
$$
which is equivalently a choice of multiplicatively invertible element in $R$. Therefore
$$
Hom(Spec R , \mathbb{G}_m) \simeq R^\times = GL_1(R)
$$
is the [[group of units]] of $R$.
### Additive group
{#AdditiveGroup}
The [[additive group]] in $Ring^{op}$ corresponding to the affine line -- usually just called the **additive group** -- is the [[group scheme]] denoted $\mathbb{G}_a$
* whose underlying object is $\mathbb{A}^1$ itself;
* whose addition operation $\mathbb{G}_a \times \mathbb{G}_a \to \mathbb{G}_a$ is dually the ring homomorphism
$$
k[t_1] \otimes_k k[t_2] \leftarrow k[t]
$$
given by
$$
t \mapsto t_1 + t_2
\,;
$$
* whose unit map is given by
$$
t \mapsto 0
\,;
$$
* whose inversion map is given by
$$
t \mapsto -t
\,.
$$
### Group of roots of unity
The group of $n$th [[roots of unity]] is
$$
\mu_n = Spec(k[t](t^n -1))
\,.
$$
This sits inside the [[multiplicative group]] via the [[Kummer sequence]]
$$
\mu_n \longrightarrow \mathbb{G}_m \stackrel{(-)^n}{\longrightarrow}\mathbb{G}_m
\,.
$$
## Properties
{#Properties}
### Grading
{#Grading}
+-- {: .num_prop}
###### Proposition
Let $R$ be a commutative $k$-algebra. There is a [[natural isomorphism]] between
* $\mathbb{Z}$-[[graded algebra|gradings]] on $R$;
* $\mathbb{G}_m$-[[action]]s on $Spec R$.
=--
+-- {: .proof}
###### Proof
For the first direction, let $R$ be a $\mathbb{Z}$-[[graded algebra|graded commutative algebra]]. Then $X = Spec R$ comes with a $\mathbb{G}$-action given as follows: the action morphism
$$
\rho : X \times \mathbb{G}_m \to X
$$
is dually the ring homomorphism
$$
R \otimes_k \mathbb{Z}[t,t^{-1}] \leftarrow R
$$
defined on homogeneous elements $r$ of degree $n$ by
$$
r \mapsto r \cdot t^n
\,.
$$
The action property
$$
\array{
X \times \mathbb{G}_m \times \mathbb{G}_m &\stackrel{Id \times \cdot}{\to}& X \times \mathbb{G}
\\
{}^{\mathllap{\rho} \times Id}\downarrow && \downarrow^{\mathrlap{\rho}}
\\
X \times \mathbb{G}_m &\stackrel{\rho}{\to}& X
}
$$
is equivalently the equation
$$
r (t_1)^n \cdot (t_2)^n = r (t_1 \cdot t_2)^n
$$
for all $n \in \mathbb{Z}$. Similarly the [[unitality]] of the action is the equation
$$
(1)^n = 1
\,.
$$
Conversely, given an action of $\mathbb{G}_m$ on $Spec R$ we have some morphism
$$
R[t,t^{-1}] \leftarrow R
$$
that sends
$$
r \mapsto \sum_{n \in \mathbb{Z}} r_n t^n
\,.
$$
By the action property we have that
$$
\sum_n r_n (t_1 t_2)^n = \sum_{n,k} (r_n)_k t_1^n t_2^k
\,.
$$
Hence
$$
(r_n)_k =
\left\{
\array{
r_n & if \; n = k
\\
0 & otherwise
}
\right.
$$
and so the morphism gives a decomposition of $R$ into pieces labeled by $\mathbb{Z}$.
One sees that these two constructions are [[inverse]] to each other.
=--
### Étale homotopy type
+-- {: .num_example}
###### Example
For $k$ a [[field]] of [[characteristic]] 0, then the affine line $\mathbb{A}^1_k$ has a [[contractible homotopy type|contractible]] [[étale homotopy type]] . This is no longer the case in [[positive number|positive]] [[characteristic]].
=--
([HSS 13, section 1](#HSS13))
### Internal formulation
{#InternalFormulation}
+-- {: .num_prop}
###### Proposition
Let $X$ be a [[scheme]] and $Sh(Sch/X)$ the [[Zariski site|big Zariski topos]] associated to $X$. Denote by $\mathbb{A}^1$ (the [[affine line]]) the [[ring object]] $T \mapsto \Gamma(T,\mathcal{O}_T)$, i.e. the functor represented by the $X$-scheme $\mathbb{A}^1_X \coloneqq X \times Spec(\mathbb{Z}[t])$. Then:
* $\mathbb{A}^1$ is [[internal logic|internally]] a [[local ring]].
* $\mathbb{A}^1$ is internally a [[field]] in the sense that any nonzero element is invertible.
* Internally, any [[function]] $f : \mathbb{A}^1 \to \mathbb{A}^1$ is a [[polynomial]] function, i.e. of the form $f(x) = \sum_i a_i x^i$ for some coefficients $a_i : \mathbb{A}^1$. More precisely,
$$ Sh(Sch/X) \models \forall f : [\mathbb{A}^1,\mathbb{A}^1]. \bigvee_{n \in \mathbb{N}} \exists a_0,\ldots,a_n : \mathbb{A}^1. \forall x : \mathbb{A}^1. f(x) = \sum_i a_i x^i. $$
Furthermore, these coefficients are uniquely determined.
=--
+-- {: .proof}
###### Proof
Since the internal logic is local, we can assume that $X = Spec(R)$ is affine. The interpretations of the asserted statements using the [[Zariski site#KripkeJoyal|Kripke?Joyal semantics]] are:
* Let $S$ be an $R$-algebra and $f, g \in S$ be elements such that $f + g = 1$. Then there exists a partition $1 = \sum_i s_i \in S$ such that in the localized rings $S[s_i^{-1}]$, $f$ or $g$ is invertible.
* Let $S$ be an $R$-algebra and $f \in S$ an element. Assume that any $S$-algebra $T$ in which $f$ is zero is trivial (fulfills $1 = 0 \in T$). Then $f$ is invertible in $S$.
* Let $S$ be an $R$-algebra and $f \in [\mathbb{A}^1,\mathbb{A}^1](S) = S[T]$ be an element. Then there exists a partition $1 = \sum_i s_i \in S$ such that in the localized rings $S[s_i^{-1}]$, $f$ is a polynomial with coefficients in $S[s_i^{-1}]$.
For the first statement, simply choose $s_1 \coloneqq f$, $s_2 \coloneqq g$.
For the second statement, consider the $S$-algebra $T \coloneqq S/(f)$.
The third statement is immediate, localization is not even necessary.
=--
+-- {: .num_remark}
###### Remark
Since the big Zariski topos is [[cocomplete category|cocomplete]] (being a [[Grothendieck topos]]), one can also get rid of the external [[disjunction]] and refer to the object $\mathbb{A}^1[X]$ of internal polynomials: The canonical ring homomorphism $\mathbb{A}^1[X] \to [\mathbb{A}^1,\mathbb{A}^1]$ (given by evaluation) is an [[isomorphism]].
=--
See also at _[[synthetic differential geometry applied to algebraic geometry]]_.
## Examples
### Projective space
The [[diagonal]] [[action]] of the multiplicative group on the [[product]] $\mathbb{A}^n := \prod_{i = 1 \cdots n} \mathbb{A}^1$ for $n \in \mathbb{N}$
$$
\mathbb{A}^n \times \mathbb{G}_m \to \mathbb{A}^n
$$
is dually the morphism
$$
k[t, t_1, \cdots, t_n] \leftarrow k[t_1, \cdots, t_n]
$$
given by
$$
t_i \mapsto t \cdot t_i
\,.
$$
This makes $k[t,\{t_i\}]$ the free [[graded algebra]] over $k$ on $n$ generators $t_i$ in degree 1. This is $\mathbb{N} \subset \mathbb{Z}$-graded. What is genuinely $\mathbb{Z}$-graded is
$$
\mathcal{O} (\mathbb{A}^n - \{0\}) \simeq k[t_1, t_1^{-1}, \cdots, t_n, t_n^{-1}]
\,.
$$
The quotient by the multiplicative group action
$$
\mathbb{A} P^n_k := (\mathbb{A}^{n+1} - \{0\})/\mathbb{G}_m
$$
is the [[projective space]] over $k$ of [[dimension]] $n$.
### $\mathbb{A}^1$-homotopy theory
In [[A1-homotopy theory|A^1 homotopy theory]] one considers the [[reflective sub-(∞,1)-category|reflective]] [[localization of an (∞,1)-category|localization]]
$$
Sh_\infty(C)_{\mathbb{A}^1}
\stackrel{\leftarrow}{\hookrightarrow}
Sh_\infty(C)
$$
of the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] over a [[site]] $C$ such as the [[Nisnevich site]], at the morphisms of the form
$$
p_1 : X \times \mathbb{A}^1 \to X
$$
that contract away cartesian factors of the affine line.
## Related concepts
* [[analytic affine line]]
* [[spectral affine line]]
* [[Tate sphere]]
## References
Discussion of [[étale homotopy type]] is in
* {#HSS13} Armin Holschbach, Johannes Schmidt, Jakob Stix, _Étale contractible varieties in positive characteristic_ ([arXiv:1310.2784](http://arxiv.org/abs/1310.2784))
[[!redirects affine lines]] |
affine localization | https://ncatlab.org/nlab/source/affine+localization | A **[[localization functor]]** $Q^* : A\to B$ is **affine** if it has a right adjoint $Q_*$ (which is automatically fully faithful, i.e. the localized category if reflective), and which itself is having its own right adjoint $Q^!$. Therefore $Q^*\dashv Q_* \dashv Q^!$.
For example, every left [[Ore localization]] of [[ring]]s $i: R\to S^{-1}R$, induces a [[flat functor|flat]] affine localization functor $M\mapsto S^{-1}M = S^{-1}R\otimes_R M$.
A functor is an affine localization if it is an inverse image part of an [[affine morphism]] with *full* direct image functor. Here affine morphism means an adjoint triple like above with *conservative* direct image functor; the conservativeness of an exact additive functor among abelian categories is equivalent to its faithfulness. While the notion of affine localization is fixed, the notion of an affine morphism given here has variants depending on the categorical context.
[[!redirects affine localization functor]]
[[!redirects affine localizations]] |
affine logic | https://ncatlab.org/nlab/source/affine+logic |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[formal logic]], *affine logic* refers to [[substructural logics]] which omit the [[contraction rule]] but retain the [[weakening rule]]. Conversely, affine logic is [[linear logic]] adjoined with the [[weakening rule]].
In terms of [[categorical semantics]], affine logic is modeled by ([[symmetric monoidal category|symmetric]]) [[monoidal categories]] whose [[tensor unit]] $I$ is [[terminal object|terminal]], also known as *[[semicartesian monoidal categories]]*, an example of which is the category of [[affine spaces]], whence the name.
(In contrast, the category of [[linear spaces]], ie. [[vector spaces]], serves as [[categorical semantics]] for the multiplicative fragment of [[linear logic]] [eg. [Murfet 2014](linear+logic#Murfet14)], where also the [[weakening rule]] is dropped.)
## Categorical semantics
One might imagine that a more general notion of [[categorical semantics]] would be given by monoidal categories equipped with a [[natural transformation|natural]] (in $A$) family $A\to I$ of [[morphisms]] implementing [[weakening rule|weakening]] for each [[object]]. However, such an interpretation is in general badly behaved, unless one additionally requires these [[natural transformations]] to be *[[monoidal transformation|monoidal]]*, but it can be shown that this additional requirement already forces the [[tensor unit]] to be [[terminal object|terminal]] (specifically, this follows from the component at $I$ being the identity).
An example of the badly behaved case -- where the transformation is not monoidal, and the tensorial unit is not terminal -- is given by the category [[Rel]] of [[relations]], with cartesian product as tensor product (i.e., with $Rel$ as [[cartesian bicategory]]). Here a natural family of relations $A\to I$ is given by picking empty relations everywhere. In the corresponding interpretation of affine logic, any weakening yields an empty relation, which contradicts intuitive principles like for example that "adding a dummy variable to a proof and then substituting a closed term" should not change the semantics.
## Examples
\begin{example}
The substructural part of many forms of [[bunched logic]] are affine instead of [[linear type theory|linear]], sometimes inadvertently, ultimately due to former technical problems with formulating [[dependent linear types]] (see review in [Riley 2022 §1.7.2](dependent+linear+type+theory#Riley22Thesis)).
\end{example}
## Related concepts
* Affine **BCK** [[combinatory logic]]
## References
* [[Vaughan Pratt]], *Re: Affine*, mailing list discussion (1997) [[web](https://www.seas.upenn.edu/~sweirich/types/archive/1997-98/msg00135.html)]
* [[Alexei P. Kopylov]], *Decidability of Linear Affine Logic*, Information and Computation **164** 1 (2001) 173-198 [[doi:10.1006/inco.1999.2834](https://doi.org/10.1006/inco.1999.2834)]
* Gianluigi Bellin , *Two Paradigms of Logical Computation in Affine Logic?*, in: *Logic for Concurrency and Synchronisation*, Trends in Logic **15** (2003) [[doi:10.1007/0-306-48088-3_3](https://doi.org/10.1007/0-306-48088-3_3)]
* {#Shulman22} [[Michael Shulman]], *Affine logic for constructive mathematics*, Bulletin of Symbolic Logic **28** 3 (2022) 327-386 [[arXiv:1805.07518](https://arxiv.org/abs/1805.07518), [doi:10.1017/bsl.2022.28](https://doi.org/10.1017/bsl.2022.28)]
See also:
* Wikipedia, *[Affine logic](https://en.wikipedia.org/wiki/Affine_logic)*
[[!redirects affine logics]]
|
affine modality | https://ncatlab.org/nlab/source/affine+modality |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a suitable [[line object]] $\mathbb{A}^1$ in a suitable ambient [[(∞,1)-topos]], then there exists the [[cohomology localization]] at [[morphisms]] that induces [[equivalences]] in [[cohomology]] with [[coefficients]] in $\mathbb{A}^1$.
In this case the [[right adjoint]] to the reflector typically has the interpretation of producing [[spaces]] which are "affine" in that they are entirely characterized by their [[function algebras on ∞-stacks|function ∞-algebra]] with [[coefficients]] in $\mathbb{A}^1$.
Therefore in this case the [[localization]] [[modality]] deserves to be called the _affine modality_.
## Examples
Examples for this in [[higher geometry|higher]] [[algebraic geometry]] and [[synthetic differential geometry]] are discussed at _[[function algebras on ∞-stacks]]_ in the section _[Localization of the (∞,1)-topos at R-cohomology](function+algebras+on+infinity-stacks#Intrinsic)_.
## Related concepts
* [[n-truncation modality]], [[double negation modality]]
* [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]
* [[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]
[[!redirects affine modalities]] |
affine monad | https://ncatlab.org/nlab/source/affine+monad | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Recall that for a [[monoidal monad]] (or equivalently [[commutative monad]]) on a [[monoidal category]], both the [[Kleisli category]] and the [[Eilenberg-Moore category]] inherit a [[monoidal category|monoidal structure]].
An *affine* monoidal monad, on a [[cartesian monoidal category]], preserves the monoidal unit (the [[terminal object]]), and so it makes the [[Kleisli category|Kleisli]] and [[Eilenberg-Moore category|Eilenberg-Moore categories]] [[semicartesian monoidal category|semicartesian]].
## Definition
Let $(C,\times,1)$ be a [[cartesian monoidal category]].
A [[commutative monad]] $(T,\mu,\eta)$ with [[monoidal monad|monoidal structure map]] $\nabla$ is called **affine** if any of the following equivalent conditions hold:
1. The unit map $\eta_1:1\to T1$ is an [[isomorphism]];
2. The following diagram commutes for all objects $A$ and $B$,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
TA \times TB \ar{dr}[swap]{\mathrm{id}} \ar{r}{\nabla} & T(A\times B) \ar{d}{(T\pi_1,T\pi_2)} \\
& TA\times TB
\end{tikzcd}
where $\pi_1:A\times B\to A$ and $\pi_2:A\times B\to B$ are the [[product|product projection maps]].
(Note: This should be generalizable to monads on [[cartesian multicategories]].)
## Strongly and weakly affine monads
A strong monad on a cartesian monoidal category is called **strongly affine** ([Jacobs'16](#strongly_affine)) if and only if for all objects $A$ and $B$, the following diagram is a [[pullback]],
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
A\times TB \ar{r}{\sigma} \ar{d}{\pi_1} & T(A\times B) \ar{d}{T\pi_1} \\
A \ar{r}{\eta} & TA
\end{tikzcd}
where $\sigma$ denotes the [[strong monad|strength]] of the monad, and $\pi_1$ the product projection.
A commutative monad on a cartesian monoidal category is called **weakly affine** ([FGPT'23](#weaklymarkov)) if and only if any of the following equivalent conditions hold:
1. The [[internal monoid]] $T1$ (with its canonical monoid structure) is a [[group]];
2. For all objects $A$, $B$ and $C$, the [[monoidal functor|associativity diagram]] is a [[pullback]].
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
TA\times TB\times TC \ar{r}{\mathrm{id}\times\nabla} \ar{d}{\nabla\times\mathrm{id}} & TA\times T(B\times C) \ar{d}{\nabla} \\
T(A\times B)\times TC \ar{r}{\nabla} & T(A\times B\times C)
\end{tikzcd}
Every (commutative) strongly affine monad is affine, and every affine monad is weakly affine.
## Properties
* The [[Eilenberg-Moore category]] of an affine commutative monad is a [[semicartesian monoidal category]]. In particular, it is a model of [[affine logic]].
* The [[Kleisli category]] of an affine commutative monad is a [[Markov category]] ([Fritz'20](#fritzmarkov)). This makes affine commutative monads suitable as [[probability monads]], modelling [[joint and marginal distributions]].
Similarly:
* The [[Kleisli category]] of a strongly affine commutative monad is a [[Markov category#positivity_and_causality|positive Markov category]] ([FGGPS'23](#dilations)).
* The [[Kleisli category]] of a weakly affine commutative monad is a *weakly Markov category*, see [FGPT'23](#weaklymarkov).
## See also
* [[monoidal monad]], [[strong monad]], [[commutative monad]]
* [[Markov category]]
* [[joint and marginal distributions]]
## References
* [[Anders Kock]], _Bilinearity and cartesian closed monads_, Mathematica Scandinavica 29(2), 1971.
* [[Bart Jacobs]], _Semantics of weakening and contraction_, Annals of Pure and Applied Logic 69(1), 1994. ([full text](https://www.sciencedirect.com/science/article/pii/0168007294900205))
* {#strongly_affine} [[Bart Jacobs]], _Affine Monads and Side-Effect-Freeness_, Proceedings of CMCS, 2016. ([pdf](http://www.cs.ru.nl/B.Jacobs/PAPERS/side-effects.pdf))
* {#bimonoidal_monads} [[Tobias Fritz]], [[Paolo Perrone]], _Bimonoidal Structure of Probability Monads_, Proceedings of MFPS, 2018, ([arXiv:1804.03527](https://arxiv.org/abs/1804.03527))
* {#cd_categories} Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))
* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))
* {#dilations} [[Tobias Fritz]], Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, [[Paolo Perrone]] and Dario Stein, _Dilations and information flow axioms in categorical probability_. Mathematical Structures in Computer Science, 2023. ([arXiv:2211.02507](https://arxiv.org/abs/2211.02507)).
* {#weaklymarkov} [[Tobias Fritz]], [[Fabio Gadducci]], [[Paolo Perrone]] and Davide Trotta, _Weakly Markov categories and weakly affine monads_, Proceedings of CALCO, LIPIcs 10, 2023. ([arXiv](https://arxiv.org/abs/2303.14049))
[[!redirects affine monads]]
[[!redirects strongly affine monad]]
[[!redirects strongly affine monads]]
[[!redirects weakly affine monad]]
[[!redirects weakly affine monads]]
|
affine morphism | https://ncatlab.org/nlab/source/affine+morphism |
# Affine morphisms
* table of contents
{: toc}
## Idea and definition
An affine morphism of [[schemes]] is a relative version of an [[affine scheme]]: given a scheme $X$, the canonical morphism $X \to Spec \mathbb{Z}$ is affine iff $X$ is an affine scheme. By the basics of [[spectrum|spectra]], every morphism of affine schemes $Spec S \to Spec R$ corresponds to a morphism $f^\circ\colon R \to S$ of [[rings]]. The affine morphisms of general schemes are defined as the ones which are locally of that form:
* a morphism $f\colon X\to Y$ of (general) schemes is **affine** if there is a [[cover]] of $Y$ (as a [[ringed space]]) by affines $U_\alpha$ such that $f^{-1} U_\alpha$ is an affine subscheme of $X$.
A seemingly stronger, but in fact equivalent, characterization follows: $f\colon X\to Y$ is affine iff for every affine $U \subset Y$, the [[inverse image]] $f^{-1}(U)$ is affine.
## Relative spectra and affine schemes
[[Grothendieck]] constructed a spectrum of a (commutative unital) [[commutative algebra|algebra]] in the category of quasicoherent $\mathcal{O}X$-modules. The result is a scheme over $X$; [[relative point of view|relative]] schemes of that form are called __relative affine schemes__.
## Functorial point of view
Now notice that a map of (associative) rings, possibly noncommutative (and possibly nonunital), induces an [[adjoint triple]] of functors $f^*\dashv f_*\dashv f^!$ among the categories of (say left) modules where $f^*$ is the extension of scalars, $f_*$ the restriction of scalars and $f^!\colon M \mapsto Hom_R(S,M)$ where the latter is an $R$-module via $(r x) (s) = x (s r)$. In particular, $f_*$ is exact.
In fact, if $f\colon X\to Y$ is a [[quasicompact morphism]] of schemes and $X$ is [[separated scheme|separated]], then $f$ is affine iff it is cohomologically affine, that is, the direct image $f_*$ is exact ([[Serre's criterion of affineness]], cf. EGA II 5.2.2, EGA IV 1.7.17).
An [[affine localization]] is a localization functor among categories of quasicoherent $\mathcal{O}$-modules which is also the inverse image functor of an affine morphism; or an abstraction of this situation.
See also [[monad in algebraic geometry]].
## Extensions
One can extend the notion of an affine morphism to [[algebraic spaces]], the noncommutative schemes of Rosenberg, Durov's generalized schemes, algebraic stacks and so on. The affinity is a local property so for algebraic stacks and the like one looks at the pullback to affine charts and checks if the resulting morphism is affine; for Durov's and Rosenberg's schemes one is basically generalizing the functorial criterium by definition. (more on this later)
## Literature
Some of the material is extracted from MathOverflow <http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide/58486>.
* R. Hartshorne, _Algebraic geometry_, exercise II.5.17
* [[A. L. Rosenberg]], _Noncommutative schemes_, Compositio Math. __112__ (1998) 93--125, [MR99h:14002](http://www.ams.org/mathscinet-getitem?mr=1622759), [doi](http://dx.doi.org/10.1023/A:1000479824211)
category: algebraic geometry, noncommutative geometry
[[!redirects affine morphism]]
[[!redirects affine morphisms]]
[[!redirects affine morphism of schemes]]
[[!redirects affine morphisms of schemes]]
|
affine scheme | https://ncatlab.org/nlab/source/affine+scheme | [[!redirects affine schemes]]
#Contents#
* table of contents
{:toc}
## Definition
### General
An __affine scheme__ is a [[scheme]] that as a [[sheaf]] on the [[opposite category]] [[CRing]]${}^{op}$ of commutative [[ring]]s (or equivalently as a sheaf on the subcategory of finitely presented rings) is [[representable functor|representable]]. In a [[ringed space]] picture an affine scheme is a [[locally ringed space]] which is isomorphic to the [[prime spectrum]] of a commutative ring. Affine schemes form a [[full subcategory]] $Aff\hookrightarrow Scheme$ of the category of schemes.
The correspondence $Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y)$ extends to a [[functor]] $Scheme\to Aff$. The __fundamental theorem on morphisms of schemes__ (see [below](#IsbellDuality)) says that there is a bijection
$$
CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).
$$
In other words, for fixed $Y$, and for varying $R$ there is a restricted functor
$$ Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,$$
and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff$ is [[fully faithful functor|fully faithful]]. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on $Aff$. See at _[[functorial geometry]]_.
There is an analogue of this theorem for relative [[noncommutative scheme]]s in the sense of Rosenberg.
+-- {: .num_remark}
###### Remark
There is no similar equation the other way round, that is "$Ring(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y)$". As a mnemonic, note that with ordinary Galois connections between power sets, one is always [[Galois connection#properties|homming into (not out of)]] the functorial construction. More geometrically, consider the example $Y = \mathbb{P}^n$ and $R = \mathbb{Z}$. Then the left hand side consists of all the $\mathbb{Z}$-valued points of $\mathbb{P}^n$ (of which there are many). On the other hand, the right hand side only contains the unique ring homomorphism $\mathbb{Z} \to \mathbb{Z}$, since $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.
=--
### Relative affine schemes
A __relative affine scheme__ over a scheme $Y$ is a [[relative scheme]] $f:X\to Y$ isomorphic to the spectrum of a (commutative unital) algebra $A$ in the category of quasicoherent $\mathcal{O}_Y$-modules; such a "relative" spectrum has been introduced by Grothendieck. It is characterized by the property that for every open $V\subset Y$ the inverse image $f^{-1}V\subset X$ is an open affine subscheme of $X$ isomorphic to $Spec(A(V))$ and such open affines glue in such a way that $f^{-1}V\hookrightarrow f^{-1}W$ corresponds to the restriction morphism $A(W)\to A(V)$ of algebras.
Relative affine scheme is a concrete way to represent an [[affine morphism]] of schemes.
## Properties
### Isbell duality
{#IsbellDuality}
+-- {: .num_prop #AffineSchemesFullSubcategoryOfOppositeOfRings}
###### Proposition
**([[affine schemes]] form [[full subcategory]] of [[opposite category|opposite]] of [[rings]])**
The [[functor]]
$$
\mathcal{O}
\;\colon\;
Schemes_{Aff}
\longrightarrow
Ring^{op}
$$
from affine schemes to their global [[rings of functions]] is a [[fully faithful functor]].
=--
(e.g. [Hartschorne 77, chapter II, prop. 2.3](#Hartschorne77))
+-- {: .num_remark}
###### Remark
**([[Isbell duality]] between [[geometry]] and [[algebra]])**
Prop. \ref{AffineSchemesFullSubcategoryOfOppositeOfRings} is the analog in [[algebraic geometry]] of similar statements of [[Isbell duality]] between [[geometry]] and [[algebra]], such as [[Gelfand duality]] or [[Milnor's exercise]].
=--
[[!include Isbell duality - table]]
### Affine Serre's theorem
[[affine Serre's theorem|Affine Serre's theorem]]
Given a commutative unital ring $R$ there is an equivalence of categories
${}_R Mod\to Qcoh(Spec R)$ between the category of $R$-modules and the category of quasicoherent sheaves of $\mathcal{O}_{Spec R}$-modules given on objects by $M\mapsto \tilde{M}$ where $\tilde{M}$ is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization $\tilde{M}(D_f) = R[f^{-1}]\otimes_R M$ where $D_f$ is the principal Zariski open set underlying $Spec R[f^{-1}]\subset Spec R$, and the restrictions are given by the canonical maps among the localizations. The action of $\mathcal{O}_{Spec R}$ is defined using a similar description of $\mathcal{O}_{Spec R} = \tilde{R}$. Its right adjoint (quasi)inverse functor is given by the global sections functor $\mathcal{F}\mapsto\mathcal{F}(Spec R)$.
## Related concepts
* [[spectral topological space]]
## References
* {#Hartschorne66} Robin Hartshorne, _Algebraic geometry_, Springer 1977
* Demazure, Gabriel, _Algebraic groups_
For [[affine schemes]] in [[cubical type theory]], see:
* [[Anders Mörtberg]], [[Max Zeuner]], *A Univalent Formalization of Affine Schemes* ([arXiv:2212.02902](https://arxiv.org/abs/2212.02902))
* [[Max Zeuner]], *A univalent formalization of affine schemes*, 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Zeuner-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=nLP7GjL1Buc))
category: algebraic geometry
[[!redirects Zariski duality]] |
affine space | https://ncatlab.org/nlab/source/affine+space |
# Contents
* table of contents
{: toc}
## Idea
An __affine space__ or __affine linear space__ is a [[vector space]] that has forgotten its origin. An __affine linear map__ (a morphism of affine spaces) is a linear map (a morphism of vector spaces) that need not preserve the origin.
Note that the 'linear functions' of elementary algebra ---the total functions whose graphs are lines--- are in fact (precisely) *affine* $\mathbb{R}$-linear maps from $\mathbb{R}$ to itself. (Similarly, the 'linear relations' ---the relations whose graphs are lines--- are precisely the *[[projective space|projective]]* $\mathbb{R}$-linear maps.)
Alternatively, in [[algebraic geometry]], the terminology "$n$-dimensional affine space" $\mathbb{A}^n k$ (affine line, affine plane, etc.) over a [[field]] $k$ refers to, depending on context, the set $k^n$, or the set of maximal ideals of the [[polynomial algebra]] $k[x_1, \ldots, x_n]$ -- these definitions coinciding if $k$ is an [[algebraically closed field]] -- and typically considered as equipped with relevant extra structure such as a [[Zariski topology]] or, going even further, the [[locally ringed space]] structure adhering to the [[affine variety]] or [[affine scheme]] corresponding to the polynomial algebra $k[x_1, \ldots, x_n]$. Whatever the precise sense chosen, the idea is that an affine space $\mathbb{A}^n k$ is a setting in which the study of loci of polynomial equations, i.e. definable sets in the theory of [[commutative algebras]] over $k$, is carried out.
Most of this article concerns affine spaces in the sense of vector spaces that have forgotten their origins or identities; the algebraic geometry sense is very briefly touched upon in the section *Affine spaces as model spaces*.
## Definitions
### Summary
The definition of affine space can be made precise in various (equivalent) ways. We give a name to some of the definitions for later reference.
* An affine space is simply a vector space, but with different morphisms; an affine linear map is a function that is the difference between a linear map and a [[constant function]].
* An affine space is a [[set]] equipped with an equivalence class of vector space structures, where two vector space structures are considered equivalent if the [[identity function]] is affine linear as a map from one structure to the other; whether a map between affine spaces is affine linear is independent of the representative vector space structures.
* An affine space is a set $A$ together with a vector space $V$ and an [[action]] of (the additive group or _[[translation group]]_ of) $V$ on $A$ that makes $A$ into a $V$-[[torsor]] (over the point); an affine linear map is a $V$-equivariant map. For this point of view, see also [[zoranskoda:affine space]].
* An affine space is a [[heap]] whose [[automorphism group]] is equipped with structure making it the additive group of a vector space; an affine linear map is a heap morphism.
* An affine space is an [[inhabited set]] $A$ together with a vector space $V$ and a function $\Lambda\colon A \times A \to V$ (thought of $\Lambda(x,y) \coloneqq x - y$) that satisfies some equations; an affine linear map $A \to A'$ is a function equipped with a linear map $V \to V'$ relative to which it preserves subtraction (the "vector-valued difference" definition).
* An affine space over the [[ground field]] $k$ is an [[inhabited set]] $A$ together with functions $\mu\colon A \times A \times A \to A$ (thought of as $\mu(x,y,z) \coloneqq x - y + z$) and $\Lambda_*\colon k \times A \times A \to A$ (thought of as $\Lambda_r(x,y) \coloneqq x - r x + r y$) that satisfy some equations; an affine linear map is a function that preserves these operations (the "two ternary operations" definition).
* An affine space over $k$ is an inhabited set $A$ together with a function $\mu_*\colon k\times A\times A\times A\to A$ (thought of as $\mu_r(x,y,z) \coloneqq r x - r y + z$) that satisfies some equations; an affine linear map is a function that preserves this operation (the "one quaternary operation") definition.
* Assuming that $2$ is invertible in the field $k$ (i.e. the [[characteristic]] of $k$ is not $2$), an affine space over $k$ is an inhabited set $A$ together with a function $\Lambda_*\colon k \times A \times A \to A$ that satisfies some equations; an affine linear map is a function that preserves this operation (the "one ternary operation" definition).
* An affine space over the field $k$ is an inhabited set $A$ together with, for every [[natural number]] $n \geq 0$ and every $(n+1)$-tuple $(r_0,\dots,r_n)$ of elements of $k$ such that $r_0 + \dots + r_n = 1$, a function $\gamma_{r_0,\ldots,r_n}\colon A^{n+1}\to A$ (thought of as $\gamma_{r_0,\ldots,r_n}(x_0,\ldots,x_n) \coloneqq r_0 x_0 + \cdots + r_n x_n$), satisfying some equations; an affine linear map is a function that preserves these operations (the "[[unbiased]]" definition).
* An affine space over the field $k$ is a vector space $A'$ together with a surjective linear map $\pi:A'\to k$ (the "slice of $Vect$" definition). The affine space itself (the set being regarded as equipped with affine-space structure) is the fiber $\pi^{-1}(1)$.
+-- {: .query}
[[Mike Shulman]]: I think there should also be a definition of the form "an affine space is a [[projective space]]" with a distinguished line called "infinity", which should also be equivalent to a "synthetic" description involving points and lines and incidence axioms. This definition would not fix the field $k$ at the outset, but rather recover it synthetically using cross-ratios. Accordingly, it ought to define an equivalent groupoid to the groupoid of pairs $(k,A)$ where $k$ is a field and $A$ is an affine space over $A$. I don't know how one could recover the non-invertible affine transformations from it directly.
=--
+-- {: .query}
There should be another characterisation, which I don\'t quite see how to phrase, at least when $k = \mathbb{R}$, which is that an affine space is a manifold (perhaps Riemannian) that is sufficiently flat and unbounded in some sense. ---Toby
[[Mike Shulman]]: It'd have to be at least Riemannian, otherwise you don't have enough structure. I don't suppose it's enough to say that a (finitely generated) affine space is a Riemannian manifold isometric to some $\mathbb{R}^n$?
_Toby_: I intended 'that is [...] in some sense' to include the possibility of structure that should be preserved by the morphisms. Note that a Riemannian manifold is *too much* structure, although it allows a definition like the first one above. (A Riemannian manifold isometric to some $\mathbb{R}^n$ is precisely a [[Euclidean space]].) But really, I\'m hoping for some phrasing such that ‹isomorphic to some $\mathbb{R}^n$› actually becomes a (not too obvious) *theorem*. I\'ll keep thinking about it.
[[Mike Shulman]]: Shouldn't a Riemannian manifold isometric to some $\mathbb{R}^n$ be a "Euclidean affine space" (a torsor over a Euclidean space)? Seems that a Euclidean space would be a Riemannian manifold _equipped with_ an isometry to some $\mathbb{R}^n$. It does seem like there should be a natural way to say this, but I don't know what it is.
_Toby_: I guess that this depends on what you think 'Euclidean space' means; I\'ve known people to define it to *be* $\mathbb{R}^n$, but that seems quite ahistorical to me; I like that Urs calls such a thing [[Cartesian space]] instead. Euclid did not have coordinates; he did not even have an origin, so a Euclidean space should be a heap rather than a group. For my comment above, I would define a [[Euclidean space]] to be an affine inner product space; FWIW [Wikipedia agrees](https://secure.wikimedia.org/wikipedia/en/affine_space). (However, Wikipedia doesn\'t go as far as I do when I claim that the inner product should be valued in an $\mathbb{R}$-line rather than in $\mathbb{R}$ itself; then again, I ignored that subtlety myself in my previous comment.)
=--
Clearly every vector space has an underlying affine space (and every linear map is affine linear), giving a [[forgetful functor]] $U:Vect \to Aff$. Conversely, any affine space gives rise to a canonical vector space, sometimes called its space of _displacements_. This is obvious from the definitions that involve a vector space as part of the structure, but a vector space can also be reconstructed from the other definitions as well, analogously to how a group can be reconstructed from a [[heap]]. This gives a functor $D:Aff\to Vect$ in the other direction. One can verify that $D(U(V))\cong V$ and $U(D(A))\cong A$; the first isomorphism is [[natural isomorphism|natural]], but the second is not (otherwise $Vect$ and $Aff$ would be [[equivalent categories]], which they are not).
The category of affine spaces is almost a [[variety of algebras]], as can be seen from the last few definitions, except for the requirement that an affine space be inhabited. To rectify this, sometimes one allows the [[empty set]] to be an affine space, although it does not have any particular vector space of displacements. (See [heap#empty](heap#empty) for discussion.)
Note that there are a few different ways to think about the operations involved in the final three definitions (those not explicitly involving a vector space). The operation $\mu\colon x,y,z \mapsto x - y + z$ is the same as the [[Mal'cev operation]] (i.e. [[heap]] structure) of the [[translation group|additive group]] of a vector space. It can be viewed as the point completing a parallelogram with given vertices $x,y,z$, or equivalently as the result of adding $x$ and $z$, relative to a choice of $y$ as the origin. The operation $\Lambda_*\colon r,x,y \mapsto x - r x + r y$ can be viewed as either a weighted average of $x$ and $y$ (i.e. as $(1-r)x + r y$) or as the result of multiplying the "displacement vector" $y-x$ by $r$, relative to the origin $x$ (i.e. as $x + r(y-x)$).
### Details and comparisons
The first few definitions, which explicitly involve a vector space, make no especial use of the fact that the vector space is a vector space rather than merely an abelian [[group]]. Thus, they are valid (and equivalent) in the more general context of [[torsors]] and [[heaps]]. They are also mostly complete as stated, except for the final one.
#### Vector-valued differences
In this definition, an affine space over a vector space $V$ is a set $A$ together with a "subtraction" function $\Lambda\colon A\times A\to V$, written $\Lambda\colon x,y \mapsto x-y$, such that:
* $\Lambda(x,x) = 0$, or $x-x = 0$, for any $x$ in $A$.
* $\Lambda(x,y) + \Lambda(y,z) = \Lambda(x,z)$, or $(x-y) + (y-z) = (x-z)$, for any $x,y,z$ in $A$.
* For any $x$ in $A$ and $v$ in $V$ there exists a unique $y$ in $A$ such that $\Lambda(y,x) = v$, or $y - x = v$.
If $y - x = v$, then we write $y = x + v$, which we can regard as an operation on $x$ and $v$ by the third axiom. Hence we have $(x + v) - x = v$ and (by uniqueness) $x + (y - x) = y$, and also $x + 0 = x$ and $(x + v) + w = x + (v + w)$ by the first two axioms. Thus, these axioms suffice to make $A$ into a [[torsor]] over the additive group of $V$ with the action $+$, which is one of the previous definitions given.
Note again that this would makes sense if $V$ is any group, not just the additive group of a vector space.
#### Two ternary operations
This definition is an affine version of the usual definition of a vector space in terms of addition and scalar multiplication. However, in each case the affine operation needs to take an extra parameter. In reading the following axioms it helps to think of $\mu(x,y,z)$ as "the sum of $x$ and $z$ relative to the basepoint $y$" and likewise $\Lambda_r(x,y)$ as "the product $r\cdot y$ relative to the basepoint $x$".
* $\mu(x,x,y) = y$ (identity for addition)
* $\mu(x,y,\mu(z,w,v)) = \mu(\mu(x,y,z),w,v)$ (associativity of addition)
* $\mu(x,y,z) = \mu(z,y,x)$ (commutativity of addition)
* $\Lambda_{r s}(x,y) = \Lambda_r(x, \Lambda_s(x,y))$ (associativity of scalar multiplication)
* $\Lambda_1(x,y) = y$ (identity for scalar multiplication)
* $\Lambda_0(x,y) = x$ (left zero for scalar multiplication)
* $\Lambda_{r+s}(x,y) = \mu(\Lambda_r(x,y), x, \Lambda_s(x,y))$ (left distributivity of scalar multiplication)
* $\Lambda_r(w, \mu(x,y,z)) = \mu(\Lambda_r(w,x), \Lambda_r(w,y), \Lambda_r(w,z))$ (right distributivity of scalar multiplication)
+-- {: .query}
_Joost_: Could it be that there is an axiom missing here ? One can go from Vector spaces to the 2 ternary operations definition and back, but I can't see that by starting with the two ternary operations definition, going to vectorspaces and back, you get the same $\Lambda$. I guess you need an extra axiom as $\mu(x,y,\Lambda_r(y,z))=\Lambda_r(x,\mu(x,y,z))$.
_Toby_: Conceptually, there is something missing, which I\'ve inserted as the left zero property of scalar multiplication. (The right zero property $\Lambda_r(x,x) = x$ follows from this using associativity of scalar multiplication with $s = 0$, same as with vector spaces.) But I have to leave, and I haven\'t yet derived your axiom, even with this aid.
=--
#### One quaternary operation
This definition is an affine version of the less standard definition of a vector space in terms of a single operation $r,x,y\mapsto r\cdot x + y$. Here an affine space over $k$ is a set $A$ together with a single operation $\mu\colon k\times A\times A\times A\to A$, written as $(r,x,y,z)\mapsto \mu_r(x,y,z)$ and thought of as the sum "$r\cdot x + z$ relative to the basepoint $y$," such that:
* $\mu_1(x,y,y) = x$
* $\mu_r(x,x,y) = y$
* $\mu_1(x,y,z) = \mu_1(z,y,x)$
* $\mu_r(\mu_s(x,y,z),w,v) = \mu_{r s}(x,y,\mu_r(z,w,v))$
* $\mu_{r+s}(x,y,z) = \mu_r(x,y,\mu_s(x,y,z))$
#### One ternary operation
In the affine case (in contrast to the vector space case), it turns out that if $2$ is invertible the "addition" $(x,y,z)\mapsto x-y+z$ can be recovered from the "scalar multiplication" $(r,x,y)\mapsto r x + (1-r)y$ by $\mu(x,y,z) = \Lambda_2(y,\Lambda_{1/2}(x,z))$. Thus, in this case we can define an affine space over $k$ to be a set $A$ together with a single operation $\Lambda\colon k\times A\times A\to A$ such that the axioms for the two-ternary-operations definition are satisfied with this definition of $\mu$.
However, we can also simplify the requisite axioms in this presentation. The following axioms are easier to state if we write $\Lambda_r(x,y)$ as $(1-r) x + r y$, or equivalently as $r x + s y$, where we require $r+s=1$ for the expression to be defined.
* Idempotence: $r x + s x = x$ whenever $r+s=1$.
* Commutativity: $r x + s y = s y + r x$ whenever $r+s=1$.
* Associativity: whenever $r+s+t=1$, whichever of the following three expressions are defined are equal:
$$
\array{
r x + (1-r)\left(\frac{s}{1-r} y + \frac{t}{1-r} z\right)\\
(1-s)\left(\frac{r}{1-s} x + \frac{t}{1-s} z\right) + s y\\
(1-t)\left(\frac{r}{1-t} x + \frac{s}{1-t} y\right) + t z
}
$$
The first is defined whenever $r\neq 1$, the second whenever $s\neq 1$, and the third whenever $t\neq 1$. Since $k$ has characteristic $\neq 2$, we cannot have $r=s=t=1$ and $r+s+t=1$ at the same time, so at least one of these expressions is always defined. We write $r x + s y + t z$ for the common value of whichever of them are defined.
* Cancellation: for any $r\in k$ and $x,y\in A$, we have $x + r y - r y = x$.
#### Unbiased definition
Let $Th_{vect}$ denote the [[Lawvere theory]] of $k$-vector spaces. For any $n$, its $n$-ary operations are $n$-tuples $(r_1,\dots,r_n)\in k^n$ representing the [[linear combination]] operation $(x_1,\dots,x_n)\mapsto r_1 x_1 +\dots+ r_n x_n$. Composition of operations is by substitution in the obvious way, and the identity operation is $(1)$. A model of this theory is simply a vector space. With this 'unbiased' definition, a vector space comes equipped with, for every integer $n\ge 0$ and $n$-tuple $(r_1,\dots,r_n)$ of elements of $k$, a function $V^n\to V$ (thought of as $(v_1,\dots,v_n)\mapsto r_1 v_1+\dots r_n v_n$), satisfying some axioms.
Let $Th_{aff}$ denote the subtheory of $Th_{vect}$ containing only those operations $(r_1,\dots,r_n)$ such that $r_1+\dots+r_n=1$; an **affine space** is a nonempty model of $Th_{aff}$. (We have to observe that these are closed under the theory operations and thus define a subtheory. Note that this excludes all zero-ary operations, so an affine space has no distinguished constants, and it also excludes all nonidentity unary operations.) The basic operations $r_0x_0+\dots+r_n x_n$, when $r_0+\dots+r_n=1$, are called __affine (linear) combinations__ of elements of $A$.
The axioms for the unbiased definition are most straightforward to see by writing out the operations of $Th_{aff}$. In particular, this includes "substitution" axioms of the form
$$
r_0(s_{00} x_{00} + \dots + s_{0m_0} x_{0m_0}) + \dots + r_n (s_{n0} x_{n0} + \dots + s_{n m_n} x_{n m_n}) = r_0 s_{00} x_{00} + \dots + r_n s_{n m_n} x_{n m_n}.
$$
However, it also includes "permutation" axioms of the form
$$
r_0 x_0 + \dots + r_n x_n = r_{\sigma 0} x_{\sigma 0} + \dots + r_{\sigma n} x_{\sigma n}
$$
and also "duplication" and "omission" axioms. This Lawvere theory can be defined concisely as follows. The Lawvere theory of vector spaces is the opposite of the category of finite-dimensional vector spaces; its operations are all [[linear combinations]]. The Lawvere theory for affine spaces is the sub-theory of this consisting of only the affine combinations. (The Lawvere theory of vector spaces also has other interesting sub-theories, such as that consisting of _[[convex combinations]]_ whose algebras are abstract [[convex space]]s in one sense of the term.) Note that the empty set is a model (algebra) of this Lawvere theory; an affine space is an *inhabited* model.
Given the unbiased definition in terms of a Lawvere theory, the previous three "biased" vector-space-free definitions can then be recovered by finding particular generating operations for the theory. In particular, this Lawvere theory is generated by $2$-ary operations if $char(k)\neq 2$, and by $3$-ary ones if $char(k)=2$. To wit, suppose given $(r_0,\dots,r_n)\in k^{n+1}$ with $n\ge 3$ such that $r_0+\dots+r_n=1$. Suppose for the moment that the $r_i$ are not all $1$, and WLOG suppose that $r_0\neq 1$. (Note that here we use the invariance under permutations.) Then we have
$$
r_0 x_0 + \dots + r_n x_n = r_0 x_0 + (1-r_0)\left(\frac{1}{1-r_0} r_1 x_1 + \dots + \frac{1}{1-r_0}r_n x_n\right)
$$
so we have expressed the given $(n+1)$-ary operation in terms of a $2$-ary one and an $n$-ary one. By induction, in this way we can express any $(n+1)$-ary operation in terms of $2$-ary ones (note that there is only one $1$-ary operation, namely the identity, and no $0$-ary ones) --- as long as we never hit a tuple where every $r_i=1$. But since we always have the requirement $r_0+\dots+r_n=1$, this badness can only happen if the characteristic of $k$ is $n$. Moreover, we still have
$$
x_0 + \dots + x_n = x_0 - x_1 + (2x_1 + x_2 + \dots + x_n)
$$
so we can still write this $(n+1)$-ary operation in terms of a $3$-ary one and an $n$-ary one. So only if $n+1=3$ (i.e. $n=char(k)=2$) are we prevented from getting down to $2$-ary operations only, and in this case we can still get down to $3$-ary ones. Finally, we observe that any $3$-ary operation can be written in terms of $2$-ary ones and the particular $3$-ary operation $x_0 - x_1 + x_2$:
$$
r_0 x_0 + r_1 x_1 + r_2 x_2 = \big(r_0 x_0 + (1-r_0) x_2\big) - x_2 + \big(r_1 x_1 + (1-r_1) x_2\big).
$$
#### Slice of $Vect$
Given an affine space $A$ (with any other definition), the corresponding $\pi:A'\to k$ is constructed as follows. Let $A' = 1 \sqcup A$, where $\sqcup$ is the coproduct in affine spaces (akin to a simplicial [[join]]), $1$ is the [[terminal object|terminal]] affine space, and $\pi$ is the composite of $1 \sqcup !: 1 \sqcup A \to 1 \sqcup 1$ with a natural identification $\mu: 1 \sqcup 1 \cong k$. Both $1 \sqcup !$ and $\mu$ which are morphisms of $Aff$ may be regarded as morphisms of $1 \downarrow Aff \simeq Vect$ (pointed affine spaces are vector spaces) if we let the first inclusion $i_0: 1 \to 1 \sqcup 1$ be the pointing of $1 \sqcup 1$ and $0: 1 \to k$ the pointing of $k$ and define $\mu$ by $\mu \circ i_0 = 0$, $\mu \circ i_1 = 1$ (the element $1 \in k$). (So $\mu$ is like two ends of a meter stick used to set up coordinates on the line $k$.)
Conversely, given $\pi:A'\to k$, the fiber $\pi^{-1}(1)$ naturally acquires a "vector-valued difference" affine space structure by simple subtraction in the vector space $A'$, where the vector space of displacements is $V = \pi^{-1}(0)$.
Note that this definition embeds the category $Aff$ of (inhabited) affine spaces fully-faithfully in the [[slice category]] $Vect/k$. The objects of $Vect/k$ not in $Aff$ are those of the form $0:V\to k$, which form a category equivalent to $Vect$ itself. Moreover, there are no morphisms from objects of $Aff$ to objects not in $Aff$; while by the above construction, a morphism from $0:V\to k$ to an affine space $\pi:A'\to k$ is just a map from $V$ to the vector space of displacements of $A$. Hence, $Vect/k$ is equivalent to the (dual) [[cograph of a functor|cograph]] of $D:Aff\to Vect$.
### Closed monoidal structure {#ClosedMonoidal}
If we allow affine spaces to be empty, then they are the models of an [[algebraic theory]] $Th_{Aff}$. Moreover, like $Th_{Vect}$, the theory $Th_{Aff}$ is a [[commutative theory]]. It follows that if $A, B$ are affine spaces, then the set $\hom(A, B)$ is closed under all affine space operations pointwise defined on the set of all functions from $A$ to $B$. This gives $Aff$ a [[closed category]] structure; on general grounds, it is in fact a [[symmetric monoidal closed category]]. The unit of this structure is the terminal or one-pointed affine space $1$, via the natural isomorphism $\hom(1, B) \cong B$. Thus $Aff$ is a closed [[semicartesian monoidal category]].
Analogous to the case of $Vect$, every affine space is a coproduct of copies of the monoidal unit: an affine space $A$ of dimension $n$ admits an affine basis, which amounts to an isomorphism $1 \sqcup 1 \sqcup \ldots \sqcup 1 \cong A$, represented by $n+1$ [*sic*] points of $A$. Such basis representations allow one to coordinatize spaces of maps $\hom(A, B) \cong B^{n+1}$, with dimension $(n+1)\dim(B)$. If one uses the first of the affine basis elements to give a pointing of the affine space (equivalent to a vector space structure), then the remaining affine basis elements provide a vector space basis, and in those coordinates every element $f \in \hom(A, B)$ may be written in matrix-vector form $f(x) = M x + b$, where again the space of such $(M, b)$ has dimension $m n + m$ if $m$ is the dimension of $B$. (There are also more 'unbiased' coordinate descriptions, not biased in favor of the first basis element playing the role of the origin.)
Similarly, we can coordinatize affine tensor products $A \otimes B$: the tensor distributes over coproducts (as it does in any symmetric monoidal closed category) and so
$$(\bigsqcup_{n+1} 1) \otimes (\bigsqcup_{m+1} 1) \cong \bigsqcup_{(m+1)(n+1)} 1 \otimes 1 \cong \bigsqcup_{(m+1)(n+1)} 1$$
with dimension $m n + m + n$. In other words, if $a_1, \ldots, a_{n+1}$ is an affine basis of $n$-dimensional $A$ and $b_1, \ldots, b_{m+1}$ a basis of $m$-dimensional $B$, then $A \otimes B$ has an affine basis consisting of the $m n + m + n + 1$ many elements $a_i \otimes b_j$.
The embedding $Aff \to Vect/k$ described in the previous subsection, sending $A$ to $1 \sqcup !: 1 \sqcup A \to 1 \sqcup 1$, is a [[strong monoidal functor]] (preserves the tensor product up to coherent isomorphism) if $Vect/k$ is endowed with the obvious tensor product acquired from $Vect$. Note that $Vect/k$ is the coreflection of $Vect$ from monoidal categories to [[semicartesian monoidal categories]]; the embedding $Aff \to Vect/k$ was in fact discovered by one of us in conjunction with this fact, and is the same as the functor induced by universality from the strong monoidal functor $Aff \to Vect$ given by $A \mapsto 1 \sqcup A$.
## Further Remarks
Every finitely-generated affine space is isomorphic to the $n$-fold [[direct sum]] $k^n$, where $k$ is the [[base field]] and $n$ is a [[natural number]] (possibly $0$). In [[algebraic geometry]], an $n$-dimensional affine space is often denoted $\mathbb{A}^n$ and identified with $k^n$. If one accepts the empty set as an affine space, then this is considered to have dimension $-1$ by convention (so $k^{-1} = \empty$).
The notion of affine space may be generalised to __affine module__ by replacing the vector space above by a [[module]] and the base field $k$ by a [[commutative ring]]. Then an affine module over the ring $\mathbb{Z}$ of [[integer]]s is precisely a commutative [[heap]], just like a module over $\mathbb{Z}$ is an [[abelian group]]. Note that the definition involving only one "scalar multiplication" operation works if and only if $2$ is invertible in $k$; it\'s not enough that $2 \ne 0$ in $k$.
+-- {: .query}
[[Mike Shulman]]: I haven't thought much about affine modules, but it seems likely to me that the "biased" module-free definitions won't be right any more, since the Lawvere theory needn't be generated by 2-ary or 3-ary operations (as far as I can see). More explicitly, I don't immediately see how to write an operation like
$$(x_0,x_1,x_2,x_3) \mapsto 4x_0 - 6x_1 - 2x_2 + 5x_3$$
in terms of $A^3\to A$ and $\mathbb{Z}\times A^2\to A$, but it seems to me that this operation should still exist in an affine $\mathbb{Z}$-module.
[[Mike Shulman]]: Well that's rubbish isn't it. The operation $A^3\to A$ is enough to give you a heap, hence an additive group, and then $\mathbb{Z}\times A^2\to A$ gives you the scalar multiplication. And so
$$4x_0 - 6x_1 - 2x_2 + 5x_3 = \big((4x_0 - 3y) - y + (-6x_1+7y)\big) - y + \big((-2x_2+3y) - y + (5x_3 - 4y)\big)$$
for any $y$ at all.
_Toby_: Right. But I find an affine module of a *rig* to be a trickier concept.
[[Mike Shulman]]: Quite so. Perhaps first one should look for a version of a [[heap]] corresponding to a [[monoid]]?
_Toby_: Yes, that would be an affine $\mathbb{N}$-module.
=--
## Affine spaces as model spaces
Affine spaces typically serve as local models for more general kinds of spaces.
For instance a [[manifold]] is a [[topological space]] that is locally isomorphic to an affine space over the [[real numbers]].
Similarly, in [[algebraic geometry]] a [[scheme]] is locally isomorphic to an [[affine scheme]].
Therefore there are attempts to axiomatize properties of categories of affine spaces for the purpose of using these as model spaces for more complicated geometries. One such axiomatization is the notion of [[geometry (for structured (∞,1)-toposes)]]. and in particular that of [pregeometry](/nlab/show/geometry+(for+structured+(infinity%2C1%29-toposes%29#Pregeometry).
## Related concepts
* the [[automorphism group]] of an affine space is an _[[affine group]]_
* [[Cartesian space]]
* [[polydisk]]
* [[projective space]], [[conical space]]
* [[relative affine n-space]]
## References
Textbook accounts:
* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §8 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]
See also:
* [[Aurelio Carboni]], _Categories of Affine Spaces_ , JPAA **61** (1989) pp.243-250.
* [[Aurelio Carboni]], [[George Janelidze]], _Modularity and Descent_ , JPAA **99** (1995) pp.255-265.
* Michel Thiébaud, _Modular Categories_ , pp.386-400 in _Proc. Como conference - Category Theory_ , LNM **1488** Springer Heidelberg 1991.
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[[!redirects affine modules]] |
affine symplectic group | https://ncatlab.org/nlab/source/affine+symplectic+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Given a [[symplectic vector space]] $(V,\omega)$, then its _affine symplectic group_ $ASp(V,\omega)$ (or _inhomogeneous sympelctic group_ $ISp(V,\omega)$) is equivalently
* the [[intersection]] $ASp(V,\omega) = Aff(V)\times_{Diff(V)} Sympl(X,\omega)$ of the [[affine group]] of the [[affine space]] $V$ and the [[symplectomorphism group]] of the [[symplectic manifold]] $(X,\omega)$, i.e.the [[group]] of all those [[affine transformations]] which preserve the [[symplectic form]] $\omega$;
* the [[semidirect product]] $ASp(V,\omega) = V \rtimes Sp(V,\omega)$ of the [[symplectic group]] acting on $V$ regarded as the [[translation group]] over itself.
The further restriction to [[linear functions]] gives the [[symplectic group]] proper.
## Properties
### Extensions
There is a [[circle group|circle]] [[group extension]] $ESp(V,\omega)$ of the affine symplectic group -- the _[[extended affine symplectic group]]_ -- given by restricting the [[quantomorphism group]] of $(V,\omega)$ to affine transformations. The further restriction of that to elements coming from [[translation group|translations]] is the [[Heisenberg group]] $Heis(V,\omega)$.
$$
\array{
Heis(V,\omega)
&\hookrightarrow&
ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
\\
\downarrow && \downarrow && \downarrow
\\
V
&\hookrightarrow&
ASp(V,\omega)
&\hookrightarrow&
HamSympl(V,\omega)
}
$$
## Related concepts
* [[Poincare group]]
## References
Review includes
* {#Low12} Stephen G. Low, section 1 of _Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group_, J. Math. Phys. 55, 022105 (2014) ([arXiv:1207.6787](http://arxiv.org/abs/1207.6787))
[[!redirects affine symplectic groups]]
|
affine variety | https://ncatlab.org/nlab/source/affine+variety |
#Contents#
* table of contents
{:toc}
## Idea
__Affine__ $k$-variety is a locus of zeros of a set of polynomials in the affine $n$-dimensional space $\mathbf{A}^n_k$. Usually $k$ is taken to be a field.
## Definition
Given a field $k$, an __affine $k$-variety__ is a [[maximal spectrum]] (= set of [[maximal ideals]]) of a finitely generated [[noetherian ring|noetherian]] (commutative unital) $k$-[[commutative algebra]] without [[nilpotent element|nilpotents]], equipped with the [[Zariski topology]]; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.
Affine varieties can be embedded as closed subvarieties into an [[affine space]] (in the sense of algebraic geometry). As topological spaces affine varieties are [[noetherian space|noetherian]].
## Properties
### Cohomology
{#Cohomology}
For $X$ an affine variety then its [[abelian sheaf cohomology]]
with [[coefficients]] in the [[structure sheaf]] satisfies
$$
H^{\bullet \geq 1}(X,\mathcal{O}_X) = 0
\,.
$$
The converse requires in addition some finiteness condition. ([Ballico 08](#Ballico08)).
## Related concepts
* [[affine scheme]]
* [[spectrum of a commutative ring]]
* [[Stein space]], [[Cartesian space]]
## References
* {#Ballico08} E. Ballico, _A characterization of affine varieties_ 2008
[[!redirects affine varieties]] |
affinoid | https://ncatlab.org/nlab/source/affinoid |
#Contents#
* table of contents
{:toc}
## Idea
In [[analytic geometry]], an _affinoid_ is a sub-[[space]] of a unit [[polydisc]], formally dual to an [[affinoid algebra]] (see there for more). These are the basic spaces out of which [[analytic spaces]] are built by gluing.
## References
* Wikipedia, _[Rigid analytic space](http://en.wikipedia.org/wiki/Rigid_analytic_space)_
* Johannes Nicaise, Chenyang Xu, Tony Yue Yu, _The non-archimedean SYZ fibration_, [arxiv/1802.00287](https://arxiv.org/abs/1802.00287)
> We construct non-archimedean SYZ fibrations for maximally degenerate Calabi-Yau varieties, and we show that they are affinoid torus fibrations away from a codimension two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt-models along one-dimensional strata.
[[!redirects affinoids]] |
affinoid algebra | https://ncatlab.org/nlab/source/affinoid+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _affinoid algebra_ is a local model for [[analytic spaces]] in [[analytic geometry]] ([[rigid analytic geometry]]).
## Definition
Let $K$ be a [[complete normed field|complete]] [[non-archimedean valued field]].
As a [[ring]], a __standard affinoid algebra__ (or __Tate algebra__) $T_{n,K}$ is the subring of the ring of [[formal power series]] in $K[ [x_1, \ldots, x_n] ]$ consisting of all strictly [[convergence|converging]] series $ c= \sum_I c_I x^I$, that is such that $|c_I|\to 0$ as $I\to \infty$.
There is a [[Gauss norm]] on such series $\|\sum_I c_I x^I \| = max\{|c_I|\}_I$. This is indeed a [[norm]] making $T_{n,K}$ into a Banach $K$-algebra of countable type.
An __affinoid algebra__ is any [[Banach algebra]] which can be represented in a form (Tate algebra)/(closed ideal).
The [[category]] of **$k$-[[affinoid spaces]]** is the [[opposite category]] of the category of $k$-affinoid algebras and bounded [[homomorphisms]] between them.
## Properties
A version of the [[Weierstrass preparation theorem]] in this context implies a version of the
[[Hilbert basis theorem]]: $T_{n,K}$ is a [[noetherian ring]]. Moreover $T_{n,K}$ is a [[unique factorization domain]] of [[Krull dimension]] $n$.
## References
Affinoid algebras were introduced in
* [[John Tate]], (1961)
A standard textbook account is
* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], part B of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))
See the references at _[[analytic geometry]]_ for more details.
Discussion of affinoid algebras as a [[site]] for a more [[topos]]-theoretic formulation of of [[analytic geometry]] is in
* {#BenBassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 7 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))
See also
* {#Bambozzi14} [[Federico Bambozzi]], _On a generalization of affinoid varieties_ ([arXiv:1401.5702](http://arxiv.org/abs/1401.5702))
[[!redirects affinoid algebras]] |
affinoid domain | https://ncatlab.org/nlab/source/affinoid+domain |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[non-archimedean analytic geometry]], affinoid domains are basic model [[spaces]]: a [[Berkovich analytic space]] is, in particular, a [[topological space]] equipped with an [[atlas]] by ([[analytic spectra]] underlying) affinoid domains.
## Definition
+-- {: .num_defn}
###### Definition
An **[[affinoid domain]]** in an [[affinoid space]] $X = Spec_{an} A$ is a [[closed subset]] $V \subset X$ such that there is a [[homomorphism]] of $k$-affinoid spaces
$$
\phi : Spec_{an} A_V \to X
$$
for some $A_V$, whose [[image]] is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.
=--
([Berkovich 09, def. 2.2.1](#Berkovich09))
+-- {: .num_defn}
###### Definition
A morphism $f\colon X\to Y$ of [[affinoid spaces]] is an _affionoid domain embedding_ if it induces an [[isomorphism]] of $X$ with an affinoid domain in $Y$
=--
([Berkovich 09, def. 2.2.7](#Berkovich09))
These are the "admissible morphisms" in the site of affinoid domains. (...)
## Properties
* [[Tate's acyclicity theorem]]
## Related concepts
* [[closed cover]]
* [[Berkovich space]]
## References
* {#Berkovich09} [[Vladimir Berkovich]], section 2.2 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))
* [[Doosung Park]], _Affinoid domains_, lecture notes ([pdf](http://math.berkeley.edu/~sstich/MAT_274/03-07.pdf))
[[!redirects affinoid domains]]
[[!redirects affinoid domain embedding]]
[[!redirects affinoid domain embeddings]]
|
affinoid space | https://ncatlab.org/nlab/source/affinoid+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
The [[category]] of **$k$-[[affinoid spaces]]** is the [[opposite category]] of the category of $k$-[[affinoid algebras]] and bounded [[homomorphisms]] between them.
[Berkovich 09, def. 2.1.11](#Berkovich09)
## Related concepts
* [[Berkovich space]]
## References
* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))
[[!redirects affinoid spaces]]
|
Agda | https://ncatlab.org/nlab/source/Agda |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Constructivism, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Overview
A [[dependent type|dependently typed]] [[functional programming language]] with applications to [[certified programming]]. It is also used as a [[proof assistant]].
Besides [[Coq]], Agda is one of the languages in which [[homotopy type theory]] has been implemented ([Brunerie](#Brunerie)).
Agda can be compiled to [[Haskell]], Epic or Javascript.
## Variants
### Cubical Agda
{#CubicalAgda}
[Cubical Agda](https://agda.readthedocs.io/en/latest/language/cubical.html) is a mode of Agda (turned on by the flag `--cubical`) that implements a type theory similar to CCHM (De Morgan) [[cubical type theory]], and thus a form of [[homotopy type theory]].
Its main difference from CCHM is that instead of an exotype of "cofibrant propositions" it uses the interval itself, replacing cofibrant propositions by statements of the form $r \equiv 1$ for some dimension expression $r$. This change does not prevent the construction of a model for the theory in De Morgan [[cubical sets]], although it doesn't technically fall under the Orton-Pitts axioms since $I$ is not a subobject of $\Omega$, and no one has checked whether this model can be strengthened to a [[Quillen model category]].
More problematically, to support [[identity types]] a la Swan (which are distinct from both cubical "path types" and Martin-Lof "identity types" -- the latter sometimes called "jdentity types" to emphasize their definition relative to the J-eliminator) the type of cofibrant propositions must support a [[dominance]]. Cubical Agda thus assumes that $I$ supports a dominance, but this is not true in De Morgan cubical sets. So the semantics of the entirety of Cubical Agda, with Swan identity types, is unclear. (For this reason, the Cubical Agda library generally avoids using Swan identity types, although Cubical Agda supports them.)
Ordinary Martin-Löf [[identity types]] should, in principle, also be definable in Cubical Agda as an indexed inductive family, with computational behavior as usual for any inductive types in cubical type theory. As of March 2021, however, there is a bug in Cubical Agda that prevents jdentity types from computing correctly.
### Guarded Cubical Agda
The guarded cubical variant extends cubical Agda to support [[guarded recursive]] definitions which can be used to formalize [[synthetic guarded domain theory]].
### Agda-flat
{#AgdaFlat}
`Agda-flat` is a mode of Agda that implements a [[comonad|co-]][[idempotent monad|monadic]] [[modal operator]] $\flat$ ("flat", following the notation used in [[cohesive homotopy type theory]] as introduced in [[schreiber:dcct|dcct]] and type-theorertically developed in [Shulman 15](cohesive+homotopy+type+theory#Shulman15)). This makes Agda model a [[modal type theory]] and hence a [[modal homotopy type theory]], such as used, for instance, in [[schreiber:thesis Wellen|Wellen 2017]].
See:
* [`Agda-flat` documentation](https://agda.readthedocs.io/en/v2.6.2.2/language/flat.html)
* [installation instructions](http://www.cl.cam.ac.uk/~amp12/agda/internal-universes/code/agda-flat/install.txt)
## Little-known features {#LittleKnownFeatures}
Listed here are some little-known or undocumented features of Agda that are sometimes useful. Note that undocumented "features" may change without warning; this list is current as of September 2022, Agda v2.6.2.
* There are a lot of useful documented [keybindings](https://agda.readthedocs.io/en/v2.6.2.2/tools/emacs-mode.html#keybindings) that you may not be aware of.
* This appears not to be documented in the manual (although it is in the Emacs docstring): When in a hole, the commands `C-c C-,` and `C-c C-.` can be prefixed with `C-u C-u` to normalize the type of the hole (and the term, in the second case) before displaying them.
* The manual doesn't document the customizable variables in the Emacs mode. Of particular note are:
* `agda-input-user-translations` allows you to add new bindings to the Unicode input mode
* `agda2-highlight-level`, when set to `interactive`, uses highlighting to display realtime information about which terms and subterms in the buffer Agda is currently typechecking.
* `agda2-program-args` allows you to add command-line arguments to be used every time (e.g. the `-v` options below). This in in addition to the arguments specified by a particular file in the `OPTIONS` line.
To change the values of these variables, run `M-x customize-variable RET` and enter the variable name, change the value, and then "Set and Save".
* The command `-v` (verbose output) accepts various additional options that are, according to the developers, "documented by their implementation". These include:
* `rewriting.rewrite:50` --- displays information about attempted uses of rewrite rules.
* `rewriting.match:60` --- displays information about attempted matches during rewriting.
* `import.chase:2` --- when compiling imported files, displays a notification when each file is completed in addition to when it is started.
Unfortunately, the output produced by these flags appears in the `*Agda debug*` buffer, which is not visible by default, rather than the standard `AgdaInfo` buffer. To turn these flags on, you can add (for instance) `-v import.chase:2` to `agda2-program-args` via Customization, as above, or to the `OPTIONS` line of a particular file.
## Minimizing bugs
Since Agda has many experimental features under active development, bugs in these features are not uncommon. When [reporting a bug](https://github.com/agda/agda/issues/new/choose), it is helpful to "minimize" it to make the shortest possible [MRE](https://en.wikipedia.org/wiki/Minimal_reproducible_example). Some tips for doing this [from the developers](https://github.com/agda/agda/issues/6067#issuecomment-1236313145) include
* Enable `--type-in-type`, `--no-termination-check`, `--no-positivity-check`, `--no-projection-like`, `--no-fast-reduce`
* Disable unused flags
* Remove unused imports
* Copy definitions from imported modules (other than `Agda.Builtin` modules)
* Delete unused definitions
* Turn functions into postulates
* Inline functions
* Remove clauses from definitions
* Replace patterns by `_` if they are not used
* Replace types by Set
* Delete unused arguments to functions or constructors
* Remove dependencies from types
* Replace (sub)terms by `w/e` where `w/e : ∀ {ℓ} {A : Set ℓ} → A` is a postulate
* Make implicit arguments explicit
* Replace wildcards `_` in terms by their solution
## Related concepts
* [[1lab]]
[[!include proof assistants and formalization projects -- list]]
## References
{#References}
### General
Agda landing page:
* [wiki.portal.chalmers.se/agda/pmwiki.php](http://wiki.portal.chalmers.se/agda/pmwiki.php)
Documentation:
* [agda.readthedocs.io](https://agda.readthedocs.io/en/v2.6.3)
Online Agda interface:
* [[Ingo Blechschmidt]]: [agdapad.quasicoherent.io](https://agdapad.quasicoherent.io)
Plain Agda originates with:
* [[Ulf Norell]], _Towards a practical programming language based on dependent type theory_, PhD thesis (2007) [[pdf](https://www.cse.chalmers.se/~ulfn/papers/thesis.pdf), [[Norell-PracticalDTT.pdf:file]]]
* [[Ulf Norell]], *Dependently Typed Programming in Agda*, in: *Advanced Functional Programming* AFP 2008, Lecture Notes in Computer Science **5832** (2009) 230-266 [[doi:10.1007/978-3-642-04652-0_5](https://doi.org/10.1007/978-3-642-04652-0_5), [pdf](https://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf)]
Textbook account:
* [[Aaron Stump]], *Verified Functional Programming in Agda*, Association for Computing Machinery and Morgan & Claypool (2016) [[doi:10.1145/2841316](https://doi.org/10.1145/2841316), ISBN:978-1-970001-27-3]
[[Cubical Agda]] (an implementation of [[cubical type theory]], for [[univalence axiom|univalently]] computing [[homotopy type theory]]) originates with:
* [[Anders Mörtberg]], *Cubical Agda* (2018) [[blog post](https://homotopytypetheory.org/2018/12/06/cubical-agda)]
* {#VMA19} [[Andrea Vezzosi]], [[Anders Mörtberg]], [[Andreas Abel]], *Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types*, Proceedings of the ACM on Programming Languages **3** ICFP 87 (2019) 1–29 [[doi:10.1145/3341691](https://doi.org/10.1145/3341691), [pdf](https://www.cse.chalmers.se/~abela/icfp19.pdf)]
### Libraries
Libraries for/with [[homotopy type theory]]/[[univalent foundations of mathematics]]:
* original HoTT library: [github.com/hott/hott-agda](https://github.com/hott/hott-agda)
* [[UniMath project]]: [unimath.github.io/agda-unimath](https://unimath.github.io/agda-unimath)
* [[1lab]]
> (motivated towards [[univalent categories]])
E.g. on [[group theory]] (cf. *[[Symmetry]]*):
* [[UniMath project]]: [agda-unimath/src/group-theory](https://github.com/UniMath/agda-unimath/tree/master/src/group-theory)
Implementation of [[Cauchy real numbers]] (in [[Errett Bishop|Bishop]]-style [[constructive analysis]]) in [[Agda]] (cf. *[[exact real computer arithmetic]]*):
* [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [<a href="https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4211-Lundfall%20(2015)%20-%20Formalizing%20Real%20Numbers%20in%20Agda.pdf">pdf</a>, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]
* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]
### Introductions
{#Introductions}
#### Quick expositions
{#Expositions}
* _['Hello World!' in Adga](http://progopedia.com/example/hello-world/251/)_
* [[Guillaume Brunerie]], _The Agda proof assistant_ (2012) [slides:[[Brunerie-AgdaProofAssistant.pdf:file]]]
* [[Ulf Norell]], *Programming in Agda*, presentation at *Oregon Programming Languages Summer School* (2014) [lecture 1: [video](https://www.youtube.com/watch?v=NrSW7YsneVg), 2: [video](https://www.youtube.com/watch?v=X0JWsoWTWnI), 3: [video](https://www.youtube.com/watch?v=HjpLZr_sirY), 4: [video](https://www.youtube.com/watch?v=Yw2VGwxn_g8)]
* [[Scott Fleischman]], *Agda from Nothing*, talk at λC (2016) [[video 1](https://www.youtube.com/watch?v=-i-QQ36Nfsk&list=PLCqzQM-xq02Uv4OmcckrFsIyZ20hWEeAx&index=7), [video 2](https://www.youtube.com/watch?v=XprGyVWXwks)]
* [[Philip Wadler]], *(Programming Languages) in Agda = Programming (Languages in Agda)*, talk at *Codegram* (Sep 2019) [[video](https://www.youtube.com/watch?v=R49VgxNLmsY&t=372s)]
* [[Fredrik Nordvall Forsberg]], *The basic syntax of Agda*, talk at CS410 Advanced Functional Programming (2021) [[video](https://www.youtube.com/watch?v=yreLHTXwkts)]
On [[cubical Agda]]:
* [[Andrea Vezzosi]], *Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types* (2019) [[video](https://www.youtube.com/watch?v=AZ8wMIar-_c)]
#### Tutorials & Lectures
General:
* *Learn you an Agda* (2011) [[web](http://learnyouanagda.liamoc.net/pages/introduction.html), [github](https://github.com/liamoc/learn-you-an-agda)]
* [[Dan Licata]], Ian Voysey, _[Programming and proving in Agda](http://www.cs.cmu.edu/~drl/teaching/oplss13/)_, course material (2013)
* [[Peter Selinger]], *Lectures on Agda* (2021) [[web](https://www.mathstat.dal.ca/~selinger/agda-lectures/)]
* [[Ingo Blechschmidt]], *Agda -- a beautiful proof assistant*, course at *Teoria dei Tipi*, Padova (Apr 2023) [[webpage](https://agdapad.quasicoherent.io/~Padova)]
With emphasis on implementing [[homotopy type theory]] and [[univalent foundations of mathematics]]:
* {#Brunerie} [[Guillaume Brunerie]], _Agda for homotopy type theory_ [[github](https://github.com/guillaumebrunerie/HoTT/tree/master/Agda/tutorial)]
* [[Martín Hötzel Escardó]], *Introduction to Univalent Foundations of Mathematics with Agda* (2019) [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580), [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/)]
category: software
[[!redirects agda]]
[[!redirects Cubical Agda]]
[[!redirects cubical Agda]]
[[!redirects cubical agda]]
[[!redirects Agda-flat]]
|
age | https://ncatlab.org/nlab/source/age | In [[model theory]], an age of a [[structure in model theory|structure]] $M$ is the set $Age(M)$ of all finite structures which are embeddable in $M$.
* related entries: [[amalgamation]]
* wikipedia: [age (model theory)](http://en.wikipedia.org/wiki/Age_%28model_theory%29)
* J D Brody, _Model theory of graphs_, thesis, [pdf](http://tarski.fandm.edu/brody/mainthesis.pdf) |
Agnese Bissi | https://ncatlab.org/nlab/source/Agnese+Bissi |
* [webpage](https://katalog.uu.se/profile/?id=N17-1951)
## Selected writings
On [[string scattering amplitudes]] via [[AdS/CFT]] and the [[conformal bootstrap]]:
* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))
category: people |
Agnès Beaudry | https://ncatlab.org/nlab/source/Agn%C3%A8s+Beaudry |
* [Website](https://www.colorado.edu/math/agnes-beaudry)
## Selected writings
On [[chromatic homotopy theory]]:
* [[Tobias Barthel]], [[Agnès Beaudry]], *Chromatic structures in stable homotopy theory*, in [[Handbook of Homotopy Theory]], Chapman and Hall/CRC Press (2019) [[arXiv:1901.09004](https://arxiv.org/abs/1901.09004), [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]
On the [[Steenrod algebra]] and the [[Adams spectral sequence]] (with motivation from [[functorial quantum field theory]]):
* [[Agnès Beaudry]], [[Jonathan A. Campbell]], *A Guide for Computing Stable Homotopy Groups*, in *Topology and Quantum Theory in Interaction*, Contemporary Mathematics **718**, Amer. Math. Soc. (2018) [[arXiv:1801.07530](https://arxiv.org/abs/1801.07530), [ams:conm/718](https://bookstore.ams.org/view?ProductCode=CONM/718)]
On the [[K-theory classification of topological phases of matter]] (and more general classification in [[Whitehead generalized cohomology theories]]):
* {#BHMP23} [[Agnès Beaudry]], Michael Hermele, Juan Moreno, [[Markus Pflaum]], Marvin Qi, Daniel Spiegel, *Homotopical Foundations of Parametrized Quantum Spin Systems* [[arXiv:2303.07431](https://arxiv.org/abs/2303.07431)]
category:people |
AGT conjecture > history | https://ncatlab.org/nlab/source/AGT+conjecture+%3E+history |
see _[[AGT correspondence]]_. |
AGT correspondence | https://ncatlab.org/nlab/source/AGT+correspondence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### Functorial Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _AGT correspondence_ ([AGT 09](#AGT09)) is a relation between
1. the [[instanton]]-[[partition function]] of $SU(2)^{n+3g-3}$-[[N=2 D=4 super Yang-Mills theory]] (Nekrasov's partition function, e.g. [Szabo 15 (2.1)](#Szabo15))
1. the [[conformal blocks]] of [[Liouville theory]] on an $n$-punctured [[Riemann surface]] $C_{g,n}$ of [[genus]] $g$
Here the idea is that $C_{g,n}$ is the [[super Yang-Mills theory]] obtained by [[Kaluza-Klein mechanism|compactifying]] the worldvolume [[6d (2,0)-supersymmetric QFT]] of two [[M5-branes]], see at [[N=2 D=4 super Yang-Mills theory]], the section [Construction by compactification](N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes)). This method generalizes a bit beyond super Yang-Mills, to [[class S]].
In particular, the [[N=2 D=4 super Yang-Mills theory]] is a [[quiver gauge theory]] and the correspondence matches the shape of its [[quiver]]-diagram to the [[genus of a surface|genus]] and punctures of the [[Riemann surface]]:
<center>
<img src="https://ncatlab.org/nlab/files/AGTQuiver.jpg" width="400">
</center>
More generally, this construction yields something like a decomposition of the [[6d (2,0)-superconformal QFT]] into a [[2d SCFT]] "with values in [[super Yang-Mills theory|4d SYM field theory]]" (e.g. [Tachikawa 10, slide 25 (33 of 54)](#Tachikawa10)). Hence composition with any kind of suitable invariant of the 4d field theories yields an actual [[2d SCFT]], for instance taking the superconformal index in 4d yields a [[2d TQFT]] ([GPRR 10](#GPRR10)). In this picture of "4d-SYM field theory-valued [[2d SCFT]]" one has the following correspondences:
* the [[complex structure]] in 2d is the [[coupling constants]] and [[theta angles]] etc in the 4d [[super Yang-Mills theory]];
* the [[mapping class group]] (large [[conformal transformations]]) in 2d is the (generalized) [[S-duality]] of the 4d theory.
## Related concepts
* [[M5-brane elliptic genus]]
* [[duality in physics]], [[duality in string theory]]
* [[Nekrasov function]]
* [[McKay correspondence]]
* [[3d-3d correspondence]]
* [[knots-quivers correspondence]]
* [[class S]]
Wrapping the [[M5-brane]] on a [[3-manifold]] instead yields: [[3d-3d correspondence]].
## References
A [[2d SCFT]] argued to describe the [[KK-compactification]] of the [[M5-brane]] on a [[4-manifold]] (specifically: a [[complex surface]]) originates with
* [[Juan Maldacena]], [[Andrew Strominger]], [[Edward Witten]], _Black Hole Entropy in M-Theory_, JHEP 9712:002, 1997 ([arXiv:hep-th/9711053](https://arxiv.org/abs/hep-th/9711053))
The origin of the AGT correspondence is:
* {#Gaiotto09} [[Davide Gaiotto]], _$N=2$ dualities_, JHEP 08 (2012) 034 [[arXiv:0904.2715](https://arxiv.org/abs/0904.2715)]
* {#AGT09} [[Luis Alday]], [[Davide Gaiotto]], [[Yuji Tachikawa]], _Liouville Correlation Functions from Four-dimensional Gauge Theories_, Lett.Math.Phys.91:167-197, 2010 ([arXiv:0906.3219](https://arxiv.org/abs/0906.3219))
* [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics __234__ (2013) 239--403 ([arXiv:0907.3987](https://arxiv.org/abs/0907.3987) [doi](https://doi.org/10.1016/j.aim.2012.09.027))
The [[2d TQFT]] obtained from this by forming the 4d index is discussed in
* {#GPRR10} Abhijit Gadde, Elli Pomoni, [[Leonardo Rastelli]], Shlomo S. Razamat, _S-duality and 2d Topological QFT_, JHEP 1003:032, 2010 ([arXiv:0910.2225](https://arxiv.org/abs/0910.2225))
Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]]
* Benjamin Assel, [[Sakura Schafer-Nameki]], Jin-Mann Wong, _M5-branes on $S^2 \times M_4$: Nahm's Equations and 4d Topological Sigma-models_, J. High Energ. Phys. (2016) 2016: 120 ([arxiv:1604.03606](https://arxiv.org/abs/1604.03606))
(relating to the [[moduli space of monopoles]])
and to the [[3d-3d correspondence]]:
* [[Clay Cordova]], [[Daniel Jafferis]], _Toda Theory From Six Dimensions_, J. High Energ. Phys. (2017) 2017: 106 ([arxiv:1605.03997](https://arxiv.org/abs/1605.03997))
* Sam van Leuven, Gerben Oling, _Generalized Toda Theory from Six Dimensions and the Conifold_, J. High Energ. Phys. (2017) 2017: 50 ([arxiv:1708.07840](https://arxiv.org/abs/1708.07840))
Brief surveys include
* {#Tachikawa10} [[Yuji Tachikawa]], _M5-branes, 4d gauge theory and 2d CFT_, 2010 ([pdf](http://member.ipmu.jp/yuji.tachikawa/transp/4d-2d-caltech.pdf))
* Abhijit Gadde, _$\mathcal{N}= 2$ Dualities and 2d TQFT_ 2012 ([[Gadde2dTQFT.pdf:file]])
* Nikolay Bovev, _New SCFTs from wrapped branes_, 2013 ([pdf](http://ipht.cea.fr/Meetings/Itzykson2013/Talks/bobev-Itzykson-july2013.pdf))
* Giulio Bonelli, _Variations on AGT Correspondence_, 2013 ([pdf](https://indico.cern.ch/event/217384/attachments/348854/486363/Bonelli.pdf))
* {#Taki16} Masato Taki, _String Theory as an Attempt of PolyMathematics_, talk at 2016.4/28 iTHES-AIMR-IIS ([pdf](https://indico2.riken.jp/event/2215/attachments/4106/4775/Taki_invited.pdf))
More detailed review is in
* Rober Rodger, _A pedagogical introduction to the AGT conjecture_, Master Thesis Utrecht (2013) ([pdf](http://testweb.science.uu.nl/ITF/teaching/2013/R.J.Rodger.pdf))
* {#Szabo15} [[Richard Szabo]], _$N=2$ gauge theories, instanton moduli spaces and geometric representation theory_, Journal of Geometry and Physics Volume 109, November 2016, Pages 83-121 ([arXiv:1507.00685](https://arxiv.org/abs/1507.00685))
* Bruno Le Floch, _A slow review of the AGT correspondence_, J. Phys. A: Math. Theor. __55__ (2022) 353002 ([doi](https://doi.org/10.1088/1751-8121/ac5945)[arXiv:2006.14025](https://arxiv.org/abs/2006.14025))
See also
* [[Alexander Belavin]], M. A. Bershtein, B. L. Feigin, A. V. Litvinov, G. M. Tarnopolsky, _Instanton moduli spaces and bases in coset conformal field theory_ ([arxiv/1111.2803](http://arxiv.org/abs/1111.2803))
* [[Volker Schomerus]], Paulina Suchanek, _Liouville's imaginary shadow_ ([arxiv/1210.1856](http://arxiv.org/abs/1210.1856))
* A.Mironov, A.Morozov, _The power of Nekrasov functions_ ([arxiv/0908.2190](http://arxiv.org/abs/0908.2190))
* D. Galakhov, A. Mironov, A. Morozov, _S-duality as a beta-deformed Fourier transform_ ([arxiv/1205.4998](http://arxiv.org/abs/1205.4998))
* A. Mironov, _Spectral duality in integrable systems from AGT conjecture_ ([arxiv/1204.0913](http://arxiv.org/abs/1204.0913))
* A. Belavin, V. Belavin, _AGT conjecture and integrable structure of conformal field theory for $c=1$_, Nucl.Phys.B850:199-213 (2011) ([arxiv/1102.0343](http://arxiv.org/abs/1102.0343))
* A. Belavin, V. Belavin, M. Bershtein, _Instantons and 2d Superconformal field theory_ ([arxiv/1106.4001](http://arxiv.org/abs/1106.4001))
* Kazunobu Maruyoshi, _Quantum integrable systems, matrix models, and AGT correspondence_, seminar ([slides pdf](http://db.ipmu.jp/seminar/sysimg/seminar/428.pdf))
* Giulio Bonelli, Alessandro Tanzini, _Hitchin systems, N=2 gauge theories and W-gravity_ ([arxiv/0909.4031](http://arxiv.org/abs/0909.4031))
* Giulio Bonelli, Kazunobu Maruyoshi, Alessandro Tanzini, _Quantum Hitchin systems via beta-deformed matrix models_ ([arxiv/1104.4016](http://arxiv.org/abs/1104.4016))
* {#ChacaltanaDistler10} [[Oscar Chacaltana]], [[Jacques Distler]], _Tinkertoys for Gaiotto Duality_, JHEP 1011:099,2010, ([arXiv:1008.5203](http://arxiv.org/abs/arXiv:1008.5203))
* Satoshi Nawata, _Givental J-functions, Quantum integrable systems, AGT relation with surface operator_ ([arXiv/1408.4132](http://arxiv.org/abs/1408.4132))
The AGT correspondence is treated with the help of a [[Riemann-Hilbert problem]] in
* G. Vartanov, [[Jörg Teschner]], _Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory_ ([arxiv/1302.3778](http://arxiv.org/abs/1302.3778))
* [[Andrei Mironov]], A. Morozov, *Superintegrability as the hidden origin of Nekrasov calculus* [[arXiv:2207.08242](https://arxiv.org/abs/2207.08242)]
Proof of the AGT conjecture in special cases:
* [[Andrei Mironov]], [[Andrey Morozov]], [[Sh. Shakirov]], *A direct proof of AGT conjecture at $\beta = 1$*, JHEP 1102:067 (2011) [[arXiv:1012.3137](https://arxiv.org/abs/1012.3137), <a href="https://doi.org/10.1007/JHEP02(2011)067">doi:10.1007/JHEP02(2011)067</a>]
* Qing-Jie Yuan, Shao-Ping Hu, Zi-Hao Huang, Kilar Zhang, *A proof of An AGT conjecture at $\beta = 1$* [[arXiv:2305.11839](https://arxiv.org/abs/2305.11839)]
See also:
* Leszek Hadasz, Błażej Ruba, *Decomposition of $\widehat{sl}_{2,k} \otimes \widehat{sl}_{2,1}$
highest weight representations for generic level $k$ and equivalence between two dimensional CFT models* [[arXiv:2312.14695](https://arxiv.org/abs/2312.14695)]
category: physics
[[!redirects AGT-correspondence]]
[[!redirects AGT conjecture]]
[[!redirects Class S]] |
Agustí Roig | https://ncatlab.org/nlab/source/Agust%C3%AD+Roig | [[!redirects Agusti Roig]]
* [webpage](https://mat.upc.edu/en/people/agustin.roig)
* [FUTUR website](https://futur.upc.edu/AgustinRoigMarti)
## related $n$Lab entries
* [[Grothendieck construction for model categories]]
* [[minimal fibration]]
* [[KS-model]]
* [[circle action]]
* [[4-sphere]]
category: people |
Aharon Casher | https://ncatlab.org/nlab/source/Aharon+Casher |
* [InSpire page](https://inspirehep.net/authors/1014341)
## Selected writings
Early discussion of [[discretized light-cone quantization]]:
* {#Casher76} [[Aharon Casher]], *Gauge fields on the null plane*, Phys. Rev. D **14** (1976) 452 [[doi:10.1103/PhysRevD.14.452](https://doi.org/10.1103/PhysRevD.14.452)]
category: people
|
Aharonov-Bohm effect | https://ncatlab.org/nlab/source/Aharonov-Bohm+effect |
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{:toc}
## Idea
The _Aharonov-Bohm effect_ is a configuration of the [[electromagnetic field]] which has vanishing electric/magnetic [[field strength]] (vanishing [[Faraday tensor]] $F = 0$) and but is nevertheless non-trivial, in that the [[vector potential]] $A$ is non-trivial. Since the vector potential affects the [[phase|quantum mechanical phase]] on the [[wavefunction]] of [[electrons]] moving in an electromagnetic field, in such a configuration classical physics sees no effect, but the phase of quantum particles, which may be observed as a [[quantum interference|interference]] pattern on some screen, does.
More technically, a configuration of the [[electromagnetic field]] is generally given by a [[circle]]-[[principal connection]] and an Aharonov-Bohm configuration is one coming from a [[flat connection]], whose [[curvature]]/[[field strength]] hence vanishes, but which is itself globally non-trivial. This is only possible on [[spaces]] ([[spacetimes]]) which have a non-trivial [[fundamental group]], hence for instance it doesn't happen on [[Minkowski spacetime]].
In practice one imagines an idealized [[electric current]]-carrying [[solenoid]] in [[Euclidean space]]. Away from the solenoid itself the [[magnetic field]] produced by it gives such a configuration.
## Details
Let $\mathbb{R}^2 - \{0\}$ be the [[plane]] with the origin removed, and consider the space $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}$ (thought of as 3d [[Cartesian space]] with the z-axis removed) and [[spacetime]] $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}^2$ (thought of as the previous configuration statically moving in time).
For the following argument only the topological structure of the space matters, and nothing needs to explicitly depend on the $z$-[[coordinate]] and the time-coordinate, so for notational simplicity we may suppress these and consider just $\mathbb{R}^2 - \{0\}$.
On this space minus the x-axis consider the [[polar coordinates]] $(\phi,r)$ with
$$
x = r cos(\phi)\,,\;\;\; y = r sin(\phi)
\,.
$$
Accordingly we have the [[differential 1-forms]]
$$
\mathbf{d}x = cos(\phi)\mathbf{d}r - r sin(\phi) \mathbf{d}\phi
$$
$$
\mathbf{d}y = sin(\phi)\mathbf{d}r + r cos(\phi) \mathbf{d}\phi
$$
hence
$$
\begin{aligned}
\mathbf{d}\phi
& =
\frac{1}{r}cos(\phi)\mathbf{d}y - \frac{1}{r}sin(\phi) \mathbf{d}x
\\
& =
\frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x
\end{aligned}
\,.
$$
Here the expression on the right extends smoothly also to the $x$-axis and this extension we call
$$
\theta \coloneqq
\frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x
\;\;
\in \Omega^1(\mathbb{R}^2 - \{0\})
\,.
$$
From the way this is constructed it is clear that $\theta$ is a closed differential form
$$
\mathbf{d}\theta = 0
\,.
$$
However, on $\mathbb{R}^2 - \{0\}$ this is not an exact form. In other words, if one regards $\theta$ as the [[vector potential]] being the configuration of an [[electromagnetic field]]
$$
A \coloneqq \theta
$$
then:
1. the [[field strength]] vanishes $F = \mathbf{d}A = 0$;
1. but there is no [[gauge transformation]] relating $A$ to the trivial field configuration.
This is possible because $\mathbb{R}^2 - \{0\}$ is not [[simply connected topological space|simply connected]] and hence the [[Poincaré lemma]] does not apply.
## Related concepts
* [[fiber bundles in physics]]
* [[Dirac charge quantization]], [[magnetic monopole]]
* anyonic [[braid group statistics]] as Aharanov-Bohm effect for a *[[fictitious gauge field]]*
## References
The effect was first predicted by
* W. Ehrenberg, R. E. Siday, *The Refractive Index in Electron Optics and the Principles of Dynamics*, Proceedings of the Physical Society. Section B, **62** 8 (1949) 1 [[doi:10.1088/0370-1301/62/1/303](https://iopscience.iop.org/article/10.1088/0370-1301/62/1/303)]
It is named after:
* [[Yakir Aharonov]], [[David Bohm]], *Significance of Electromagnetic Potentials in the Quantum Theory*, Phys. Rev. **115** (1959) 485 [[doi:10.1103/PhysRev.115.485](https://doi.org/10.1103/PhysRev.115.485), [pdf](https://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485)]
Early discussion with emphasis of the role of [[connection on a bundle|connections]] on [[fiber bundles in physics]] and generalization to non-abelian [[Yang-Mills theory]]:
* [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]
See also:
* L. Mangiarotti, [[Gennadi Sardanashvily]], section 6.6 of _Connections in Classical and Quantum Field Theory_, World Scientific, 2000
* [[Mikio Nakahara]], Section 10.5.3 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)
* Wikipedia, _[Aharonov-Bohm effect](https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect)_
|
Ahmadnagar | https://ncatlab.org/nlab/source/Ahmadnagar | |
Ahmed Almheiri | https://ncatlab.org/nlab/source/Ahmed+Almheiri |
* [webpage](https://www.ias.edu/scholars/ahmed-almheiri)
## Selected writings
On the interpretation of [[tensor networks]] encoding [[holographic entanglement entropy]] as [[quantum error correcting codes]]:
* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))
* [[Ahmed Almheiri]], _Holographic Quantum Error Correction and the Projected Black Hole Interior_ ([arXiv:1810.02055](https://arxiv.org/abs/1810.02055))
On the [[black hole firewall problem]]:
* [[Ahmed Almheiri]], [[Donald Marolf]], [[Joseph Polchinski]], James Sully, _Black holes: complementarity or firewalls?_ ([arXiv:1207.3123](http://arxiv.org/abs/arXiv:1207.3123))
On [[AdS/CFT]]-duality for [[AdS2-CFT1|$AdS_2/CFT_1$]] via [[Jackiw-Teitelboim gravity]]:
* {#AlmheiriPolchinski14} [[Ahmed Almheiri]], [[Joseph Polchinski]], _Models of $AdS_2$ Backreaction and Holography_, J. High Energ. Phys. (2015) 2015: 14. ([arXiv:1402.6334](https://arxiv.org/abs/1402.6334))
Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:
* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020) [[arXiv:1911.12333](https://arxiv.org/abs/1911.12333)]
Review:
* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)]
## Activity
The case for a Kavli-like theoretical physics institute in Abu Dhabi, UAE:
* with [[David Gross]]: *Majlis Mohamed bin Zayed lecture: “The Future of Scientific Research”*, Abu Dhabi (Jan 2023) [video: [TW](https://twitter.com/i/broadcasts/1mnxeRYyomQKX), [YT](https://www.youtube.com/watch?v=PpMWrajguzg)]
category: people |