string theory results applied elsewhere

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#### Physics
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#### Quantum field theory
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#### Gravity
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#Contents#
* table of contents
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## Idea
 {#Idea}

Beyond the speculative hypothesized role of [[string theory]] as a physical theory of [[fundamental strings]] that constitute the observed [[fundamental particles]] in the [[standard model of particle physics]], the theory has shed light on many aspects of [[quantum field theory]] as such, both on the conceptual structure of QFT as well as on concrete theories and their concrete properties. This entry lists such instances of string theory results having led to insights in non-stringy physics and in particular into experimentally confirmed physics, such as [[QCD]] in the [[standard model of particle physics]].

> Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.

> [[Jacques Hadamard]] ([source](http://homepage.math.uiowa.edu/~jorgen/hadamardquotesource.html))

\linebreak

> The construction of efficient computational methods for field theory scattering amplitudes has benefited substantially from string theory input.

> [Roiban & Volovich 2004](#RoibanVolovich04)

## Examples

The two basic theories that underlie observed fundamental physics -- and which string theory unifies at least qualitatively and in [[perturbation theory]] -- are [[Yang-Mills theory]] and [[gravity|Einstein gravity]]/[[general relativity]].

1. [Applications to Yang-Mills theory](#ApplicationsToYangMillsTheory)

1. [Applications to Gravity](#ApplicationsToGravity)

Many of the insights are based on the gauge/gravity duality in string theory:

1. [General relation between Yang-Mills theory and gravity]()

### Worldline formalism

The [[worldline formalism]] for expressing QFT [[scattering amplitudes]] in an effective [[gauge invariance|gauge invariant]] way (different from but equivalent to the [[Feynman rules]]) was originally found by taking the point-particle limit of the expressions for [[string scattering amplitudes]]. See at _[[worldline formalism]]_ for more.

Example:

The first calculation along these lines was actually done earlier in ([Metsaev-Tseytlin 88](#MetsaevTseytlin88)), where the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] was obtained as the point-particle limit of the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in ([Nepomechie 83](#Nepomechie83)) that when computed via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].

* {#Nepomechie83} R.I. Nepomechie, _Remarks on quantized Yang-Mills theory in 26 dimensions_, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (doi:10.1016/0370-2693(83)90385-4)

* {#MetsaevTseytlin88} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _On loop corrections to string theory effective actions_, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (doi:10.1016/0550-3213(88)90306-9)

### General relation between Yang-Mills theory and gravity

[[AdS/CFT correspondence]] [[open/closed string duality]]

* Spenta R. Wadia, _Gauge/Gravity Duality and Some Applications_ ([arXiv:1009.0212](http://arxiv.org/abs/1009.0212))

talks at

* [[Zvi Bern]], Thomas Gehrmann, Frank Petriello, Anastasia Volovich, _The Harmony of Scattering Amplitudes_ KITP Program (2011) ([web](http://online.kitp.ucsb.edu/online/qcdscat11/))

### Applications to Yang-Mills theory
 {#ApplicationsToYangMillsTheory}

#### Worldline formalism

The [[worldline formalism]] for expressing QFT [[scattering amplitudes]] in an effective [[gauge invariance|gauge invariant]] way (different from but equivalent to the [[Feynman rules]]) was originally found by taking the point-particle limit of the expressions for [[string scattering amplitudes]]. See at _[[worldline formalism]]_ for more.

Example:

The first calculuation along these lines was actually done earlier in ([Metsaev-Tseytlin 88](#MetsaevTseytlin88)), where the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] from the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in ([Nepomechie 83](#Nepomechie83)) that when computed in via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].

* {#Nepomechie83} R.I. Nepomechie, Phys. Lett. B128 (1983) 177

* {MetsaevTseytlin88} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _On loop corrections to string theory effective actions_, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 ([doi:10.1016/0550-3213(88)90306-9](https://doi.org/10.1016/0550-3213(88)90306-9))

#### QFT Duality and specifically Montonen-Olive electric/magnetic duality

By embedding quantum field theories in string theory (typically as the [[worldvolume]] theories of various [[branes]]) the various [[duality in string theory|dualities of string theory]] will relate different QFTs in ways that are typically far from obvious from just looking at these QFTs themselves.

The investigation specifically of [[N=1 D=4 super Yang-Mills theory]] and [[N=2 D=4 super Yang-Mills theory]] in this fashion has come to be known as _[[geometric engineering of quantum field theory]]_.

[[Montonen-Olive duality]] of ([[super Yang-Mills theory|super]])  [[Yang-Mills theory]] derives from conformal invariance of the [[6d (2,0)-supersymmetric QFT]] (see there) [[Kaluza-Klein mechanism|compactified]] on a [[torus]].

[[!include gauge theory from AdS-CFT -- table]]

See also

* [[Matthew Strassler]], _Millenial Messages for QCD from the Superworld and from the String_ ([arXiv:hep-th/0309140](http://arxiv.org/abs/hep-th/0309140))

#### Application to QCD and experimental particle physics

The realization of [[Yang-Mills theory]] that describes [[quarks]] and their interaction by the [[strong nuclear force]] carried by [[gluons]] is _[[quantum chromodynamics]]_ ([[QCD]]).

1. [QCD Scattering amplitudes](#QCDScatteringAmplitudes)

1. [Quark-gluon plasma](#QuarkGluonPlasma)

##### Scattering amplitudes
 {#QCDScatteringAmplitudes}

The [[string scattering amplitudes]] exhibit certain relations due to the extended nature of the string which are retained in the point particle limit and hence explain and serve to discover subtle relations in [[QFT]] [[scattering amplitudes]].

* [[twistor string theory]] explains some ([[N=4 D=4 super Yang-Mills theory|super]]) [[Yang-Mills theory]] [[scattering amplitudes]]

* Precision scattering amplitudes in [[QCD]] use [[twistor string theory]] [[on-shell recursion]] methods,

This also goes by the term "on-shell methods". See also at _[[amplituhedron]]_.

Reviews include

* [[Matthew Strassler]], _From string theory to the large hadron collider_ ([blog post](http://profmattstrassler.com/2012/08/15/from-string-theory-to-the-large-hadron-collider/))

* {#Dixon13} [[Lance Dixon]], _Calculating Amplitudes_, December 2013 ([web](http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/))

* Rutger Boels, _On-shell recursion for string theory amplitudes on the disk and the sphere_ ([pdf](http://www.itp.uni-hannover.de/nordic/slides/Boels.pdf))

Original articles include

* [[Zvi Bern]], [[Lance Dixon]], [[David Kosower]], _On-Shell Methods in Perturbative QCD_ ([arXiv:0704.2798](http://arxiv.org/abs/0704.2798))

* [[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))

See also

* {#BDDPT19} Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi, _Elliptic polylogarithms and Feynman parameter integrals_ ([arXiv:1902.09971](https://arxiv.org/abs/1902.09971))

  reviewed in

  Lorenzo Tancredi, _Feynman integrals and higher genus surfaces_, talk at _[Amplitudes 2019](https://indico.cern.ch/event/750565/)_  ([pdf](https://indico.cern.ch/event/750565/contributions/3438642/attachments/1871838/3080501/Tancredi.pdf))

See also below _[Application to gravity -- Scattering amplitudes](#GravityScattetingAmplitudes)_.

##### Quark-gluon plasma
 {#QuarkGluonPlasma}

Properties of [[quark-gluon plasma]] from [[AdS/CFT|AdS/CFT-dual]] [[type II string theory]]

* 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))

#### To the confinement problem

Discussion of [[confinement]] in the context of the [[AdS-CFT correspondence]] is in

* [[David Berman]], Maulik K. Parikh, _Confinement and the AdS/CFT Correspondence_, Phys.Lett. B483 (2000) 271-276 ([arXiv:hep-th/0002031](http://es.arxiv.org/abs/hep-th/0002031))

* Henrique Boschi Filho, _AdS/QCD and confinement_, Seminar at the _Workshop on Strongly Coupled QCD: The confinement problem_, November 2011 ([pdf](http://www.if.ufrj.br/~boschi/pesquisa/seminarios/AdS_QCD_Confinement_UERJ_2011.pdf))

#### To super Yang-Mills theory

[[Seiberg duality]] in [[super Yang-Mills theory]] is conceptually explained by [[type II string theory]] on certain [[D-brane]] configurations (...)

### Applications to gravity
 {#ApplicationsToGravity}

#### Scattering amplitudes
 {#GravityScattetingAmplitudes}

* [[open/closed string duality]] in [[string scattering amplitudes]] allows to compute [[gravity]] [[scattering amplitudes]] in terms of [[Yang-Mills theory]] scattering amplitudes: the _[[KLT relations]]_

* [[Zvi Bern]], _Perturbative Quantum Gravity and its Relation to Gauge Theory_, Living Rev Relativ. 2002; 5(1): 5. ([arXiv:gr-qc/0206071](https://arxiv.org/abs/gr-qc/0206071), [doi:10.12942/lrr-2002-5](https://dx.doi.org/10.12942%2Flrr-2002-5))

* more on this string-organizatioon of [[graviton]] [[scattering]] amplitudes is in

  David C. Dunbar, Paul S. Norridge, _Calculation of Graviton Scattering Amplitudes using String-Based Methods_, Nucl.Phys. B433 (1995) 181-208 ([arXiv:hep-th/9408014](http://arxiv.org/abs/hep-th/9408014))

* [[KLT relations]] used for instance to demonstrate:

  * multi-loop finiteness of [[N=8 d=4 supergravity]]

#### Gravitational wave signatures
 {#GravitationalWaveSignatures}

Application of the [[KLT relation]]/[[double copy]]-technique to computation of [[gravitational wave]]-signature of [[relativistic binary]] mergers:

* {#BCRSSZ19} [[Zvi Bern]], Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng, _Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order_, Phys. Rev. Lett. 122, 201603 (2019) ([arXiv:1901.04424](https://arxiv.org/abs/1901.04424))

#### Black hole entropy

Semi-classical QFT computations suggest that there should be [[entropy]] associated with  [[black holes]], the _[[Bekenstein-Hawking entropy]]_, without however providing microscopic degrees of freedom of which this would be an entropy in the ordinary sense.

Since the quantum dynamics of general black holes is outside the reach of perturbative methods in string theory, certain [[BPS state|supersymmetric black hole]] solutions in [[supergravity]] have properties independent of the coupling and are known to be the strong-coupling limit of what at weak coupling is a certain configuration of [[branes]] in flat space. Therefore the ordinary entropy of these brane configurations should match the [[Bekenstein-Hawking entropy]] of the corresponding black holes, and this has been confirmed to good precision.

While this argument does not give direct information about the origin of the BH-entropy of physically observed black holes, it does show conceptually, in the general context of black holes in theories of gravity, BH-entropy can be accounted for by microscopic degrees of freedom in a theory of [[quantum gravity]].

Reviews include

* {#AGMOO} {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], Chapter 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))

### Application to pure mathematics

> string theory "knows more mathematics than we do" and its study has led to the development of a large quantity of new mathematics: [[mirror symmetry]] (enumerative formulas, [[homological mirror symmetry]], [[Bridgeland stability]]), new invariants ([[Gromov-Witten invariants|Gromov-Witten]], [[Donaldson-Thomas invariant|Donaldson-Thomas]], [[Gopakumar-Vafa invariant|Gopakumar-Vafa]], etc.), [[topological field theory]] and topological [[quantum gravity]], etc. ([Douglas 19, slide 30](#Douglas19))

Reviews include

* [[Gregory Moore]], _[[The Impact of D-Branes on Mathematics]]_ (2014), _[[Physical Mathematics and the Future]]_ (2014)

* [[Mina Aganagic]], _String Theory and Math: Why This Marriage May Last_, ([arXiv:1508.06642](https://arxiv.org/abs/1508.06642))

* Ibrahima Bah, [[Daniel Freed]], [[Gregory Moore]], Nikita Nekrasov, Shlomo S. Razamat, Sakura Schafer-Nameki, _A Panorama Of Physical Mathematics c. 2022_ ([arXiv:2211.04467](https://arxiv.org/abs/2211.04467))

For a review of prospects of computerized support for string theory, see

* {#Douglas19} [[Michael Douglas]], _Physics and Formal Methods_, ([slides](https://www.icms.org.uk/downloads/bigproof/Douglas.pdf))

See also [Fields medal (and other) work related to string theory](string+theory#FieldMedalWork)

* [[twisted K-theory]]

* [[monstrous moonshine]]

* The [[string orientation of tmf]] was directly motivated from the index of the heterotic string ("[[Witten genus]]").

* The relation between [[Bruhat-Tits buildings]] and zeros of the [[Riemann zeta function]], as described in [Huang-Stoica-Yau](https://ncatlab.org/nlab/show/p-adic+string+theory#HuangStoicaYau19).

* [[mirror symmetry]] took complex geometers by complete surprise, but is easily seen from the [[2d (2,0)-superconformal QFT|2d (2,0)-sigma model]] with [[target space]] the respective CY manifolds.

* [[Morse theory]] of loop spaces via loop space [[supersymmetric quantum mechanics]] describing superstring propagation.

(...)


string topology

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#Contents#
* table of contents
{:toc}

## Idea

In **string topology** one studies the [[BV-algebra]]-structure on the [[ordinary homology]] of the [[free loop space]] $X^{S^1}$ of an [[oriented]] [[manifold]] $X$, or more generally the [[framed little 2-disk operad|framed little 2-disk algebra]]-structure on the singular [[chain complex]]. This is a special case of the general algebraic structure on higher order [[Hochschild cohomology]], as discussed there.

The study of _string topology_ was initated by [[Moira Chas]] and [[Dennis Sullivan]].

## The string operations

Let $X$ be a [[smooth manifold]], write $L X$ for its [[free loop space]] (for $X$ regarded as a [[topological space]]) and $H_\bullet(L X)$ for the [[ordinary homology]] of this space (with coefficients in the [[integer]]s $\mathbb{Z}$).

### The string product

+-- {: .num_defn }
###### Definition

The **string product** is a morphism of [[abelian group]]s

$$
  (-)\cdot(-) :
  H_\bullet(L X) \otimes H_\bullet(L X)
  \to
  H_{\bullet - dim X}(L X)
  \,,
$$

where $dim X$ is the [[dimension]] of $X$, defined as follows:

Write $ev_* : L X \to X$ for the [[evaluation map]] at the basepoint of the loops.

For $[\alpha] \in H_i(L X)$ and $[\beta] \in H_j(L X)$ we can find representatives $\alpha$ and $\beta$ such that $ev(\alpha)$ and $ev(\beta)$ intersect [[transversal map|transversally]]. There is then an $((i+j)-dim X)$-chain $\alpha \cdot \beta$ such that $ev(\alpha \cdot \beta)$ is the chain given by that intersection: above $x \in ev(\alpha \cdot \beta)$ this is the loop obtained by concatenating $\alpha_x$ and $\beta_x$ at their common basepoint. The _string product_ is then defined using such representatives by

$$
  [\alpha] \cdot [\beta] := [\alpha \cdot \beta]
  \,.
$$

=--

+-- {: .num_theorem }
###### Theorem

The string product is [[associativity|associative]] and graded-commutative.

=--

This is due to ([ChasSullivan](#ChasSullivan)).
There is is a more elegant way to capture this, due to ([CohenJones](#CohenJones)):

Let

$$
  S^1 \coprod S^1 \to 8 \leftarrow S^1
$$

be the [[cospan]] that exhibts the inner and the outer circle of the figure "8" topological space. By forming [[hom space]]s this induces the [[span]]

$$
  \array{
      && X^8
      \\
      & {}^{\mathllap{in}}\swarrow && \searrow^{\mathrlap{out}}
     \\
     L X \times L X &&&& L X
  }
  \,.
$$

Write $in^!$ for the "pullback" in [[ordinary homology]] along $in$ (the dual [[fiber integration]]) and $out_*$ for the ordinary pushforward.

+-- {: .num_theorem }
###### Theorem

The string product is the pull-push operation

$$
  out_* \circ in^! : H_\bullet(L X \times L X)
  \simeq H_\bullet(L X) \otimes H_\bullet(L X)
  \to
  H_{\bullet - dim X}(L X)
  \,.
$$

=--

This is due to ([CohenJones](#CohenJones)).

### The BV-operator

+-- {: .num_defn }
###### Definition

Define a morphism of [[abelian group]]s

$$
  \Delta : H_\bullet(L X) \to H_{\bullet + 1}(L X)
$$

as follows. Consider first the rotation map

$$
  \rho : S^1 \times L X \to L X
$$

that sends $(\theta, \gamma) \mapsto (t \mapsto \gamma(\theta + t))$.  Then take

$$
  \Delta : a \mapsto \rho_* ([S^1] \times a)
  \,,
$$

where $[S^1] \in H_1(S^1)$ is the [[fundamental class]] of the [[circle]].

=--

This is called the **BV-operator** for string topology.

+-- {: .num_prop }
###### Proposition

The [[Goldman bracket]] on $H_0(L X)$ is equivalent to the string product applied to the image of the BV-operator

$$
  \{[\gamma_1], [\gamma_2]\}
  =
  \Delta[\Gamma_1] \cdot \Delta[\Gamma_2]
  \,.
$$

=--

This is due to ([ChasSullivan](#ChasSullivan)).

## Properties

### As a TQFT
 {#InTermsOfTQFTs}

The structures studied in the _string topology_ of a [[smooth manifold]] $X$ may be understood as being essentially the data of a 2-dimensional [[topological field theory]] [[sigma model]] with [[target space]] $X$, or rather its linearization to an [[HQFT]] (with due care on some technical subtleties).

The idea is that the [[configuration space]] of a closed or open [[string]]-[[sigma-model]] propagating on $X$ is the [[loop space]] or path space of $X$, respectively. The space of [[state]]s of the string is some space of sections over this configuration space, to which the (co)homology $H_\bullet(L X)$ is an approximation. The string topology operations are then the [[cobordism]]-representation with [[coefficients]] in the [[category of chain complexes]]

$$
  H_\bullet(Bord_2) \to Ch_\bullet
$$

given by the [[FQFT]] corresponding to the $\sigma$-modelon these state spaces, acting on these state spaces.

$\,,$

Let $X$ be an [[orientation|oriented]] [[compact space|compact]] [[manifold]] of dimension $d$.

For $\mathcal{B} = \{A, B , \cdots\}$
a collection of oriented compact submanifolds write
$P_X(A,B)$ for the [[path space]] of paths in $X$ that start in $A \subset X$ and end in $B \subset X$.

+-- {: .num_theorem }
###### Theorem

The  tuple $(H_\bullet(L M, \mathbb{Q}), \{H_\bullet(P_X(A,B), \mathbb{Q})\}_{A,B \in \mathcal{B}})$ carries the
structure of a $d$-dimensional [[HCFT]] with _positive boundary_ and set of [[branes]] $\mathcal{B}$, such that the correlators in the closed sector are the standard string topology operation.

=--

For [[closed strings]] this is discussed in ([Cohen-Godin 03](#CohenGodin03), [Tamanoi 07](#Tamanoi07)).
For [[open strings]] on  a single [[brane]] $\mathcal{B} = \{*\}$ this was shown in ([Godin 07](#Godin)), where the general statement for arbitrary branes is conjectured. A detailed proof of this general statement is in ([Kupers 11](#Kupers)).

+-- {: .num_remark }
###### Remark

These constructions work by regarding the [[mapping spaces]] from 2-dimensional [[cobordisms]] with maps to the base space as [[correspondences]] and then applying pull-push (pullback followed by [[push-forward in generalized cohomology|push-forward in cohomology]]/[[Umkehr maps]]) to these. Hence these quantum field theory realizations of string topology may be thought of as arising from a [[quantization]] process  of the form _[[path integral as a pull-push transform]]/[[motivic quantization]]_.

=--

## Related concepts

* [[Goldman bracket]]

* [[Sullivan chord diagram]]

* [[path integral as a pull-push transform]]

## References

The original references include the following:

*  {#ChasSullivan} [[Moira Chas]], [[Dennis Sullivan]], _String topology_, Ann. Math. [math.GT/9911159](http://arxiv.org/abs/math/9911159)

*  [[Ralph Cohen]], John R. Klein, [[Dennis Sullivan]], _The homotopy invariance of the string topology loop product and string bracket_, J. of Topology 2008 __1__(2):391-408; [doi](http://dx.doi.org/10.1112/jtopol/jtn001)

* [[Ralph Cohen]], _Homotopy and geometric perspectives on string topology_, [pdf](http://math.stanford.edu/~ralph/skyesummary.pdf)

In

* {#CohenJones} [[Ralph Cohen]], [[John David Stuart Jones]], _A homotopy theoretic realization of string topology_ , Math. Ann. 324
(2002), no. 4, ([arXiv:0107187](http://arxiv.org/abs/math/0107187))

the string product was realized as genuine pull-push (in terms of dual [[fiber integration]] via [[Thom isomorphism]]).

The interpretation of closed string topology as an [[HQFT]] is discussed in

* {#CohenGodin03} [[Ralph Cohen]], [[Veronique Godin]], _[[A Polarized View of String Topology]]_ ([arXiv:math/0303003](http://arxiv.org/abs/math/0303003))

* {#Tamanoi07} Hirotaka Tamanoi, _Loop coproducts in string topology and triviality of higher genus TQFT operations_ (2007) ([arXiv](http://arxiv.org/abs/0706.1276))

A detailed discussion and generalization to the open-closed [[HQFT]] in the presence of a single space-filling [[brane]] is in

* {#Godin} [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))

The generalization to multiple [[D-branes]] is discussed in

* {#Kupers} [[Sander Kupers]], _String topology operations_ MS thesis (2011) ([pdf](http://math.stanford.edu/~kupers/thesis7thjune2011.pdf))

Exposition of the perspective of regarding [[string topology]]-operations as the [[TQFT]] of a [[topological string]] [[sigma model]]:

* [[Ralph Cohen]], [[Alexander Voronov]], _Notes on string topology_, in: [[Ralph Cohen]], [[Kathryn Hess]], [[Alexander Voronov]], _String topology and cyclic homology_, Advanced courses in mathematics CRM Barcelona, Birkhäuser 2006 ([math.GT/05036259](http://arxiv.org/abs/math/0503625), [doi:10.1007/3-7643-7388-1](https://doi.org/10.1007/3-7643-7388-1), [pdf](http://gen.lib.rus.ec/get?md5=adde9464705ede0fea6b435edb58fbe7))

* [[Dennis Sullivan]], _Sigma models and string topology_, in: Mikhail Lyubich, Leon Takhtajan (eds.), _Graphs and Patterns in Mathematics and Theoretical Physics_, Proc. Symp. Pure Math. 73 (2005) ([doi:10.1090/pspum/073](https://doi.org/10.1090/pspum/073), [spire:1697823](http://inspirehep.net/record/1697823))

* [[Dennis Sullivan]], _Open and closed string field theory interpreted in classical algebraic topology_, chapter 11 in: [[Ulrike Tillmann]] (ed.) _Topology, Geometry and Quantum Field Theory_, Cambridge University Press (2005) ([doi:10.1017/CBO9780511526398.014](https://doi.org/10.1017/CBO9780511526398.014))

For target space a [[classifying space]] of a [[finite group]] or [[compact space|compact]] [[Lie group]] this is discussed in

* David Chataur, [[Luc Menichi]], _String topology of classifying spaces_ ([pdf](http://math.univ-angers.fr/perso/lmenichi/String_Classifiant09.pdf))

Arguments that this string-topology [[HQFT]] should refine to a chain-level theory -- a [[TCFT]] -- were given in

* {#Costello} [[Kevin Costello]],  _Topological conformal field theories and Calabi-Yau $A_\infty$-categories_ (2004) , ([arXiv:0412149](https://arxiv.org/abs/math/0412149))

and

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
 {#Lurie}
(see example 4.2.16, remark 4.2.17).

For the string product and the BV-operator this extension has been known early on, it yields a [[homotopy BV algebra]] considered  in

* [[Scott Wilson]], around page 101 of: _On the Algebra and Geometry of a Manifold's Chains and Cochains_ (2005) ([pdf](http://qcpages.qc.cuny.edu/~swilson/main.pdf))

Evidence for the existence of the [[TCFT]] version by exhibiting a [[dg-category]] that looks like it ought to be the dg-category of string-topology [[branes]] (hence ought to correspond to the TCFT under the suitable version of the [[TCFT]]-version of the [[cobordism hypothesis]]) is discussed in

* {#BlumbergCoheneTeleman09} [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], _Open-closed field theories, string topology, and Hochschild homology_, pp. 53--56 in Alpine Perspectives on Algebraic Topology, Contemp. Math. 504, Amer. Math. Soc. 2009
([arXiv:0906.5198](https://arxiv.org/abs/0906.5198))

Refinements of string topology from [[homology groups]] to the full [[ordinary homology]]-[[spectra]] is discussed in ([Blumberg-Cohen-Teleman 09](#BlumbergCoheneTeleman09)) and in

* [[Ralph Cohen]], [[John Jones]], _A homotopy theoretic realization of string topology_, Mathematische Annalen ([arXiv:math/0107187](https://arxiv.org/abs/math/0107187))

* [[Ralph Cohen]], [[John Jones]], _Gauge theory and string topology_ ([arXiv:1304.0613](https://arxiv.org/abs/1304.0613))

* Kate Gruher, Paolo Salvatore, _Generalized string topology operations_, Proc. London Math. Soc. __96__:1
(2008) 78--106 [doi](https://doi.org/10.1112/plms/pdm030)
[math.AT/0602210](https://arxiv.org/abs/math/0602210)

Further generalization to target spaces that are more generally [[differentiable stacks]]/[[Lie groupoids]] is discussed in

* [[Kai Behrend]], [[Gregory Ginot]], [[Behrang Noohi]], [[Ping Xu]], _String topology for stacks_, (89 pages) [arxiv/0712.3857](http://arxiv.org/abs/0712.3857); _String topology for loop stacks_, C. R. Math. Acad. Sci. Paris, __344__ (2007), no. 4, 247--252, (6 pages, [pdf]())

* Po Hu, _Higher string topology on general spaces_, Proc. London Math. Soc. __93__ (2006) 515-544, [doi](http://dx.doi.org/10.1112/S0024611506015838), [ps](http://www.math.wayne.edu/~po/koszul04.ps)

The relation between string topology and [[Hochschild cohomology]] in dimenion $\gt 1$ is discussed in

* [[Dmitry Vaintrob]], _The String topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces_ ([arXiv:0702859](http://arxiv.org/abs/math/0702859))

More developments are in

* Eric Malm, _String topology and the based loop space_, 2009 ([arXiv:1103.6198](http://arxiv.org/abs/1103.6198), [slides](http://math.ucr.edu/~jbergner/ucr-st-present.pdf))


string-localized quantum field

#Contents#
* table of contents
{:toc}

## Idea

Traditional [[DHR superselection theory]] considers charges localized in compact regions. One can also generalize DHR theory to the case of representations localized on neighbourhoods of semi-infinite rays. This is now usually going by the term "string localization" (though there is no known relation to [[string theory|other uses]] of the word "string" in physics, except maybe the suggestion ([Buchholz-Fredenhagen 82](#BuchholzFredenhagen)), that one can understand these rays as reflecting [[flux tubes]] between two charged particles).

## References

Massive interacting string-localized objects in [[AQFT]] were first considered (though not constructed) in

* [[Detlev Buchholz]], [[Klaus Fredenhagen]], _Locality and the structure of particle states_, Comm. Math. Phys. 84, 1-54 (1982)
 {#BuchholzFredenhagen}

Massless string localized fields motovated by a long-ignored sector in the [[Wigner classification]] of [[unitary representations of the Poincare group]] are considered in

* J. Mund, [[Bert Schroer]], J. Yngvasond, _String-localized quantum fields and modular localization_ ([arXiv:math-ph/0511042](http://arxiv.org/abs/math-ph/0511042))

and reviewed for instance in

* Christian Köhler, _String-localized fields and point-localized currents in massless Wigner representations with infinite spin_, Diplom(Master) thesis (2005) ([pdf](http://www.theorie.physik.uni-goettingen.de/forschung/qft/theses/dipl/Koehler.pdf))

On [[perturbative quantum gravity]] via string-localized quantum fields:

* Christian Gaß, [[José M. Gracia-Bondía]], Karl-Henning Rehren, *Quantum general covariance* [[arXiv:2304.09843](https://arxiv.org/abs/2304.09843)]

[[!redirects string-localized quantum fields]]


string-net model

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### Context
#### Solid state physics
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=--
#### Quantum systems
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[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[condensed matter theory]], *string-net models* were introduced as [[lattice model|lattice]] [[model (in theoretical physics)|models]] for 2-dimensional [[topological phases of matter]] exhibiting [[topological order]]. The name essentially derives from the fact that the main [[degrees of freedom]] are located on the lattice [[edges]].

## Related concept

* [[fusion category]], [[anyons]]

* [[topological order]]

* [[string diagrams]]

## References

Original articles:

* [[Michael Levin]], [[Xiao-Gang Wen]], *String-net condensation: A physical mechanism for topological phases*, Phys.Rev. B **71** (2005) 045110 $[$[arXiv:cond-mat/0404617](http://arxiv.org/abs/cond-mat/0404617)$]$

  > (see also at *[[Levin-Wen model]]*)

* [[Michael Levin]], [[Xiao-Gang Wen]], *Detecting topological order in a ground state wave function*, Phys. Rev. Lett., 96, 110405 (2006) ([arXiv:cond-mat/0510613](https://arxiv.org/abs/cond-mat/0510613))

  > (introducing [[topological entanglement entropy]] with regards to string-net models)

In terms of [[tensor networks]]:

* O. Buerschaper, M. Aguado, G. Vidal, *Explicit tensor network representation for the ground states of string-net models*, Phys. Rev. B **79** (2009) 085119 $[$[arXiv:0809.2393](https://arxiv.org/abs/0809.2393), [doi:10.1103/PhysRevB.95.195110](https://doi.org/10.1103/PhysRevB.95.195110)$]$

* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$

Further development:

* Ling-Yan Hung, Yidun Wan, *String-Net Models with $\mathbb{Z}_N$ Fusion Algebra*, Phys. Rev. B **86** (2012) 235132 $[$[arXiv:1207.6169](https://arxiv.org/abs/1207.6169)$]$

* Chien-Hung Lin, *Multi-flavor string-net models*, Phys. Rev. B **95** 195110 (2017) $[$[arXiv:1611.08288](https://arxiv.org/abs/1611.08288), [doi:10.1103/PhysRevB.95.195110](https://doi.org/10.1103/PhysRevB.95.195110)$]$

Review and exposition:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sections 6.4-6.9 in: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

* Chien-Hung Lin, [[Michael Levin]], Fiona J. Burnell, *Generalized string-net models: A thorough exposition*, Phys. Rev. B **103** (2021) 195155  [[arXiv:2012.14424](https://arxiv.org/abs/2012.14424), [doi:10.1103/PhysRevB.103.195155](https://doi.org/10.1103/PhysRevB.103.195155)]

Constructing [[correlators]] in [[rational conformal field theory]] via [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-Net Construction of RCFT Correlators*, Springer Briefs in Mathematical Physics **45**, Springer (2023) [[doi:10.1007/978-3-031-14682-4](https://doi.org/10.1007/978-3-031-14682-4), [arXiv:2112.12708](https://arxiv.org/abs/2112.12708)]

Constructing [[modular functors]] from [[pivotal category|pivotal]] [[bicategories]] using [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-net models for pivotal bicategories* [[arXiv:2302.01468](https://arxiv.org/abs/2302.01468)]

Exposition and review:

* [[Christoph Schweigert]], *String-net methods for CFT correlators*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#SchweigertMar2023)]

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], pp. 18 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

[[!redirects string-net models]]

[[!redirects string net model]]
[[!redirects string net models]]

[[!redirects string-net]]
[[!redirects string-nets]]

[[!redirects string net]]
[[!redirects string nets]]


string-string duality

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#Contents#
* table of contents
{:toc}

## Idea

The term _string/string duality_ ([Duff-Khuri 93](#DuffKhuri93)) refers to [[duality in string theory]] which relates a [[perturbative string theory]] (at small [[string coupling constant]]) with another string theory, but now at large [[string coupling]] ([[non-perturbative effect|non-perturbative]]).

The archetypical example is the [[duality between heterotic and type II string theory]], and often this is taken to be synonymous with "string-string duality".

This is in contrast to [[duality in string theory|dualities]] which relate weakly coupled strings to strongly coupled [[branes]] of other dimension, such as the [[electric-magnetic duality]] which relates the [[heterotic string]] to the [[NS5-brane]] in [[dual heterotic string theory]], or the [[duality between type IIA string theory and M-theory]], which relates the string to the [[membrane]] ([[M2-brane]])

## References

Original articles

* {#DuffKhuri93} [[Michael Duff]], [[Ramzi Khuri]], _Four-Dimensional String/String Duality_, Nucl.Phys. B411 (1994) 473-486 ([arXiv:hep-th/9305142](https://arxiv.org/abs/hep-th/9305142))

Discussion in view of the [[I8]]-anomaly term

* [[Michael Duff]], [[Ruben Minasian]], _Putting string/string duality to the test_, Nucl.Phys. B436 (1995) 507-528 ([arXiv:hep-th/9406198](https://arxiv.org/abs/hep-th/9406198))

* [[Michael Duff]], [[James Liu]], [[Ruben Minasian]], _Eleven Dimensional Origin of String/String Duality: A One Loop Test_, Nucl.Phys. B452 (1995) 261-282 ([arXiv:hep-th/9506126](https://arxiv.org/abs/hep-th/9506126))

Further discussion

* {#Witten95} [[Edward Witten]], p. 3 of _String Theory Dynamics In Various Dimensions_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](https://arxiv.org/abs/hep-th/9503124))

* [[Ashoke Sen]], _String string duality conjecture in six dimensions and charged solitonic strings_, Nucl.Phys.B450:103-114, 1995 ([arXiv:hep-th/9504027](https://arxiv.org/abs/hep-th/9504027))

* K. Davis, _M-Theory and String-String Duality_ ([arXiv:hep-th/9601102](https://arxiv.org/abs/hep-th/9601102))

[[!redirects string-string dualities]]


string^c 2-group

> under construction

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=--
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+-- {: .hide}
[[!include string theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In analogy to how the [[Lie group]] [[spin^c]] is obtained by twisting the lift through the second stage of the [[Whitehead tower]] of $\mathbf{B}O$ by the [[first Chern class]]

$$
  \array{
    \mathbf{B}Spin^c &\to& \mathbf{B}(SO \times U(1))
    \\
    && \downarrow^{\mathrlap{\mathbf{w}_1 - \mathbf{c}_1}}
    \\
    && \mathbf{B}^2 \mathbb{Z}
  }
$$

there is a similar twist by the [[second Chern class]] of the lift through the next stage of the Whitehead tower

$$
  \array{
    \mathbf{B}String^{\mathbf{c}_2} &\to& \mathbf{B}(Spin \times SU(n))
    \\
    && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - \mathbf{c}_2}}
    \\
    && \mathbf{B}^3 U(1)
  }
  \,.
$$

Accordingly a [[reduction and lift of structure groups|lift of the structure group]] to $String^c$ is a **$String^c$-structure**.

For the moment see at _[[twisted smooth cohomology in string theory]]_ for more.

## Related concepts

* [[string 2-group]]

* [[twisted differential string structure]]

* [[twisted differential c-structures|(twisted, differential) c-structures]]

  * [[orientation]]

  * [[spin structure]], [[twisted spin structure]], [[differential spin structure]]

    [[spin^c structure]], [[twisted spin^c structure]],

  * [[string structure]], [[twisted differential string structure]],

  * **$string^c$ structure**

  * [[fivebrane structure]], [[twisted differential fivebrane structure]]

## References
 {#References}

Topological $string^c$-structures were introduced

* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_. J. Noncommut. Geom., 2(4):497–552, 2008.

and shown to induce a twisted [[Witten genus]] in

* {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))

* Jianqing Yu, Bo Liu, _On the Witten Rigidity Theorem for $String^c$ Manifolds_, Pacific Journal of Mathematics 266.2 (2013): 477-508. ([arXiv:1206.5955](http://arxiv.org/abs/1206.5955))

* Haibao Duan, [[Fei Han]], Ruizhi Huang, _$String^c$ Structures and Modular Invariants_, Trans. AMS 2020 ([arXiv:1905.02093](https://arxiv.org/abs/1905.02093))

The [[push-forward in generalized cohomology|push-forward]] in [[twisted cohomology|twisted]] [[tmf]] induced by a $string^c$-structure is discussed in

* {#ABG} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], Section 11 of  _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

A discussion explicitly in the context of [[string theory]] is in

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]] II: Twisted $String$ and $String^c$ structures_, J. Australian Math. Soc. 90 (2011), 93-108 ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419))

Their smooth refinement and their [[smooth infinity-groupoid|smooth moduli 2-stacks]] were introduced in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane|Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]_

see also

* [[Urs Schreiber]], section 5.2 of _[[schreiber:differential cohomology in a cohesive topos]]_

[[!redirects string^c 2-groups]]

[[!redirects String^c 2-group]]
[[!redirects String^c 2-groups]]

[[!redirects String^c-structure]]

[[!redirects String^c structure]]
[[!redirects String^c structures]]

[[!redirects string^c structure]]
[[!redirects string^c structures]]


stringor bundle

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[higher structure|higher]] version of how a [[spin structure]] induces [[spinor bundles]],
so [[string structure]] on a [[manifold]] [[transgression|transgresses]] into a kind of [[spin structure]] on its [[free loop space]], which is "fusive" with respect to fusion of loops (along [[trinions]]). Equipped with this structure this should be a [[2-vector bundle]] on the base space, and this is what is called the *stringor bundle*.

## Related concepts

* [[spinor bundle]]

* [[fusion bundle]]

* [[Dirac operator on loop space]]

* [[What is an elliptic object?]]

## References

The idea that [[String structure]] on a manifold is a kind of [[spin structure]] on its [[loop space]] is due to

* {#Witten87} [[Edward Witten]], _The Index Of The Dirac Operator In Loop Space_ Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) ([spire](http://inspirehep.net/record/245523))

The idea and the terminology of "stringor bundles" originates with:

* [[Stephan Stolz]], [[Peter Teichner]], _The spinor bundle on loop space_ (2005) [[pdf](http://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/MPI.pdf), [[Stolz-Teichner-SpinorOnLoopSpace.pdf:file]]]

An actual construction appears in:

* {#KristelLudewigWaldorf22} [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]

Reviewed in:

* [[Konrad Waldorf]], *The stringor bundle*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#WaldorfMar2023), video:[YT](https://www.youtube.com/watch?v=wX99ZbMuyMA)]

* [[Matthias Ludewig]], *The spinor bundle on loop space and its fusion product*, talk at [[CQTS]] (Apr 2023) [[web](Center+for+Quantum+and+Topological+Systems#LudewigApr2023), video: [YT](https://www.youtube.com/watch?v=iiRm47HavRU)]

* [[Konrad Waldorf]], *String structures and loop spaces*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12998](https://arxiv.org/abs/2312.12998)]

Compare also

* Alessandra Capotosti, *[[From String structures to Spin structures on loop spaces]]*, Ph. D. thesis, Università  degli Studi Roma Tre, Rome (2016)

and other references at *[[smooth loop space]]*.

[[!redirects stringor bundles]]


Strings 2022

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#### Gravity
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[[!include AQFT and operator algebra contents]]
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#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

The meeting

* **[Strings 2022](https://indico.cern.ch/event/1085701)**

  Vienna, Austria.

  18-22 July (2022)

  [indico.cern:1085701](https://indico.cern.ch/event/1085701)

was an installment of the annual [[string theory]]-conference, on the first two days with some emphasis on [[quantum gravity]]-topics via [[holographic principle|holography]]/[[AdS-CFT duality]] (such as [[black hole entropy]] [[black holes in string theory|in string theory]]).

#Contents#
* table of contents
{:toc}

## Talks

### 18 July

* [[Emil Martinec]], _The branes behind black holes_ ([slides](https://indico.cern.ch/event/1085701/contributions/4940814/attachments/2481320/4259653/Emil%20Martinec%20Monday-Review-Martinec.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=6645d5b4-ecc8-4849-a870-e1211e85376f))

  > on [[black holes in string theory]] via [[D1-D5 brane bound states]]

* [[Natalie Paquette]], _4d physics from 2d chiral correlators_ ([slides](https://indico.cern.ch/event/1085701/contributions/4940742/attachments/2481322/4259658/Natalie%20Paquette%20Monday-Research-Paquette.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=967d7949-dc4e-47e8-befb-8aa53c589ba9))

  > on 4d [[scattering amplitudes]] via [[holography as Koszul duality]]

* [[Edward Witten]], _Algebras and Entropies_ [[indico:4940799](https://indico.cern.ch/event/1085701/contributions/4940799), [slides](https://indico.cern.ch/event/1085701/contributions/4940799/attachments/2479685/4260630/Slides_Witten.pdf), [video](https://indico.cern.ch/event/1085701/contributions/4940799/attachments/2479685/4259987/go)]

* [[Shira Chapman]], _Holographic Complexity and de Sitter Space_ [[indico:4940811](https://indico.cern.ch/event/1085701/contributions/4940811), [slides](https://indico.cern.ch/event/1085701/contributions/4940811/attachments/2481532/4260404/Slides_Chapman.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e3ddea36-2f95-42ac-8b26-98c41f1d800d)]

  > on [[computational complexity and physics|computational complexity]] in [[quantum gravity]]

* [[Vijay Balasubramanian]], _Complexity, chaos, and black holes_ ([slides](https://indico.cern.ch/event/1085701/contributions/4940812/attachments/2481535/4260044/Vijay%20Balasubramanian%20Monday-Research-Balasubramanian.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e6fa0823-d9bf-42a9-aa52-694abe79a65a))

* [[Monica Guica]], _$T\bar T$ deformations and holography: review and open questions_ [[indico:4940816](https://indico.cern.ch/event/1085701/contributions/4940816/), [slides](https://indico.cern.ch/event/1085701/contributions/4940816/attachments/2481655/4260285/slides.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=bfe514d2-b4c8-4f03-a2d1-33ee630b74f6)]

  > on [[TT deformation|$T \bar T$ deformations]]

* [[Daniel Harlow]], _A theory of the black hole interior_ ([slides](https://indico.cern.ch/event/1085701/contributions/4940817/attachments/2481657/4260295/slides.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=efcf6b09-b948-4d73-a075-f25346f541aa))

  > on [[black hole entropy]] via [[quantum error correcting codes]]

* [[Maxim Pospelov]], _Dark matter identification efforts_ [[indico:4940821](https://indico.cern.ch/event/1085701/contributions/4940821), [slides](https://indico.cern.ch/event/1085701/contributions/4940821/attachments/2481749/4261004/Slides_Pospelov.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=118664ce-20a5-4b62-a3b3-735ce5deea20)]

  > on [[dark matter]]

* [[Tom Rudelius]], _Consequences of No Global Symmetries in Quantum Gravity_, ([slides](https://indico.cern.ch/event/1085701/contributions/4940824/attachments/2481780/4260540/Tom%20Rudelius%20Strings_2022_Consequences_No_Global_Symmetries.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=ddcba7cd-3342-48dc-aff7-520053645549))

* [[Liam McAllister]], _Small Cosmological Constants in String Theory_ [[indico:4940826](https://indico.cern.ch/event/1085701/contributions/4940826), [slides](https://indico.cern.ch/event/1085701/contributions/4940826/attachments/2481781/4260542/Liam%20McAllister%20Monday-Research-McAllister.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=1e5171ec-597a-4c34-bfb7-58d243b39ece)]

  > on realistic values of the [[cosmological constant]] in [[string theory]]

### 19 July

* [[Andreas Karch]], *Randall-Sundrum Branes and Holography*, talk at *[[Strings 2022]]* [[indico:4940841](https://indico.cern.ch/event/1085701/contributions/4940841), [slides](https://indico.cern.ch/event/1085701/contributions/4940841/attachments/2482053/4261118/Slides_Karch.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=19f3c464-448d-44ca-907b-91a1bfa5d0ce)]

  > on [[Randall-Sundrum models]] in [[AdS-CFT duality]] and [[black hole entropy]]

* [[Amit Sever]], _Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions_, [[indico:4940843](https://indico.cern.ch/event/1085701/contributions/4940843), [slides](https://indico.cern.ch/event/1085701/contributions/4940843/attachments/2482058/4261226/Slides_Sever.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=a9f7872c-1603-4954-9030-192a72e4f456)]

  > on [[Wilson line]]-[[quantum observables]] and [[bosonization]] in 3d [[Chern-Simons theory|Chern-Simons]]/[[matter]]-theories (such as the [[ABJM model]])

* [[Hong Liu]], *The emergence of space and time in holography*, [[indico:4940845](https://indico.cern.ch/event/1085701/contributions/4940845), [slides](https://indico.cern.ch/event/1085701/contributions/4940845/attachments/2482132/4261244/Hong%20Liu%20Vienna1_Hong.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=9b1a2a73-1a2f-4882-a222-4e5ab52b077f)]

  > on [[holographic principle|holography]]

* [[Nick Evans]], *Holography for Multi-Representation and Chiral Matter*, [[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)]

  > on [[holographic QCD]], related non-conformal [[holographic principle|holography]] and [[DBI action]]-effects

* [[Luca Iliesiu]], *From statistical mechanics to microstate counting for extremal and near-extremal black holes*, [[indico:4940850](https://indico.cern.ch/event/1085701/contributions/4940850), [slides](https://indico.cern.ch/event/1085701/contributions/4940850/attachments/2482365/4261646/Luca%20Iliesiu%20Tuesday-Reserach-Iliesiu-v2.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=96645a86-6d44-40ab-9abf-dd5a1920d881)]

  > on [[black hole entropy]] for (near-)[[extremal black holes]]

* [[Shlomo Razamat]], *News from the higher $\mathcal{D}$ SQFT frontier*, [[indico:4940856](https://indico.cern.ch/event/1085701/contributions/4940856), [slides](https://indico.cern.ch/event/1085701/contributions/4940856/attachments/2482369/4261653/Shlomo%20Razamat%20Tuesday-Review-Razamat.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=ad1f1a6b-5018-4b69-9e38-e499daabda1c)]

  > on [[SCFTs]] in higher dimensions ([[D=4 SYM]], [[D=5 SYM]], [[D=6 N=(1,0) SCFT|D=6 N=1]], [[D=6 N=(2,0) SCFT|D=6, N=2]]) their [[moduli]] ([[Higgs branches]]/[[Coulomb branches]]) and their [[duality in string theory|dualities]] ([[KK-compactification]], [[AGT correspondence]], [[AdS/CFT duality]], ...)

* [[James Sparks]], *Black hole entropy functions and spinning spindles*, [[indico:4940857](https://indico.cern.ch/event/1085701/contributions/4940857), [slides](https://indico.cern.ch/event/1085701/contributions/4940857/attachments/2482432/4261772/James%20Sparks%20Tuesday-Research-Sparks.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=758eace7-ef2a-4b54-b385-0fc244285886)]

  > on [[black hole entropy]]

### 20 July

* [[Daniel Waldram]], *Fluxes, holography and uses of Exceptional Generalized Geometry* [[indico:4940880](https://indico.cern.ch/event/1085701/contributions/4940880), [slides](https://indico.cern.ch/event/1085701/contributions/4940880/attachments/2482886/4262576/Daniel%20Waldram%20Strings2022-waldram.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e222a3a9-ebf4-445c-ada0-f7b1d592d908) ]

  > on generalized [[G-structures]] and [[exceptional generalized geometry]] of [[flux compactifications]] and in [[AdS/CFT]]

(...)

### 21 July

* [[Umut Gürsoy]], *Recent developments in gauge-gravity duality applied to quantum many-body systems* [[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) ]

  > on applications of [[AdS/CFT duality]] to many-body [[quantum systems]], such as to high-$T_c$ [[superconductivity]] ([[AdS/CMT correspondence]]) and to [[QCD]] ([[AdS/QCD correspondence]])

(...)

### 22 July

* [[Emily Nardoni]], *From $\mathcal{N} = 2$ Supersymmetry to adjoint QCD*, talk at *[[Strings 2022]]* [[indico:4940894](https://indico.cern.ch/event/1085701/contributions/4940894), [pdf](https://indico.cern.ch/event/1085701/contributions/4940894/attachments/2484206/4265102/Slides_Nardoni.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=035dfeb0-ef40-4530-8f20-01e5b890f43d)]

  > on [[supersymmetry breaking]] of [[D=4 N=2 SYM]] to [[confinement|confining]] [[QCD]] with [[adjoint representation|adjoint]] [[fermions]]:

(...)

category: reference


stringy weight system

[[!redirects stringy weight systems]]

+-- {: .rightHandSide}
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###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Where a [[weight system]] on [[chord diagrams]] is equivalently a [[weight system]] on [[Jacobi diagrams]] (since [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]) and hence an assignment of [[Feynman amplitude]]-factors to [[Chern-Simons theory|Chern-Simons-]][[Feynman diagrams]] (in the presence of a [[Wilson loop]]), a _stringy weight system_ assigns such amplitude factors instead to the [[ribbon graph]] [[surfaces]] of [[open string]] [[worldsheets]] which become [[Jacobi diagrams]] in the [[worldline formalism|point particle limit]] ([Bar-Natan 95, Theorem 10](#BarNatan95), see also [Kneissler 97, Section 1.3](#Kneissler97))

## Details

Given any [[ground field]] $\mathbb{F}$, consider

\[
  \label{LinearSpanOfMarkedSurfaces}
  Span\big( MarkedSurfaces_{/\sim} \big)
  \;\in\;
  Vect
\]

the [[linear span]] of the [[set]] of [[diffeomorphism]] [[equivalence class|classes]] of [[surfaces]] [[manifold with boundary|with boundary]] equipped with a [[finite number]] of chosen oriented points on their boundary, at least one on each [[connected component]] ([Bar-Natan 95, Def. 1.12](#BarNatan95)).

A _stringy weight system_ is a [[linear map]] from this space to the [[ground field]], hence the space of stringy weight systems is the [[dual vector space]]

\[
  \label{SpaceOfStringyWeightSystems}
  \mathcal{S}
  \;\coloneqq\;
  \Big(
    Span\big( MarkedSurfaces_{/\sim} \big)
  \Big)^\ast
  \,.
\]

Of course a [[linear function]] $Span\big( MarkedSurfaces_{/\sim}\big) \to \mathbb{F}$ is equivalently a plain [[function]] $MarkedSurfaces \to \mathbb{F}$, but for the following construction it is necessary to consider [[formal linear combinations]] of marked surfaces:

The [['t Hooft double line construction]] constitutes a [[linear map]] from the [[linear span]] of [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU-relations]] to the linear span of marked surfaces (eq:LinearSpanOfMarkedSurfaces)

([Bar-Natan 95, Theorem 10 with Theorem 8](#BarNatan95))

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Under passage to [[dual vector spaces]] this yields a linear map

\[
  \label{PointParticleLimitOfStringyWeightSystems}
  \mathcal{S}
  \overset{
    (tH \,\circ\, perm)^\ast
  }{\longrightarrow}
  \underset{n \in \mathbb{N}}{\prod}
  \big(
    \mathcal{W}^n
  \big)
\]

from stringy weight systems (eq:SpaceOfStringyWeightSystems) to ordinary [[weight systems]]. This may be regarded as the [[worldline formalism|point particle limit]] of stringy weight systems.

## Properties

### Relation to Lie algebra weight systems

+-- {: .num_remark}
###### Remark
**([[stringy weight systems span classical Lie algebra weight systems]])**

The [[image]] of the [[vector space]] (eq:SpaceOfStringyWeightSystems) of [[stringy weight systems]] under the point particle limit map (eq:PointParticleLimitOfStringyWeightSystems) inside the space of ordinary [[weight systems]] coincides with the [[direct sum]] of the linear span of [[Lie algebra weight systems]] for the [[general linear Lie algebras]] $\mathfrak{gl}(N)$ and the [[special orthogonal Lie algebras]] $\mathfrak{so}(N)$, and contains that of [[symplectic Lie algebras]] $\mathfrak{sp}(N)$:

=--

([Bar-Natan 95, Theorem 11](#BarNatan95))

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Related concepts

* [['t Hooft double line notation]]

* [[Chern-Simons theory as a topological string theory]]

[[!include chord diagrams and weight systems -- table]]

## References

The concept (not by this name, though) is due to

* {#BarNatan95} [[Dror Bar-Natan]], p. 9 and section 6 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

where it is attributed (Remark 1.13) to private communication with [[Maxim Kontsevich]].

See also

* {#Kneissler97} [[Jan Kneissler]], Section 1.3 of: _The number of primitive Vassiliev invariants up to degree 12_ ([arXiv:q-alg/9706022](https://arxiv.org/abs/q-alg/9706022))

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 6.2.4 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))


stringy weight systems span classical Lie algebra weight systems

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#### Knot theory
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[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[linear span]] of the [[image]]  of [[stringy weight systems]] inside [[weight systems]] on [[Jacobi diagrams]] (under passage to the [[worldline formalism|point particle limit]]) coincides with that of [[Lie algebra weight systems]] coming from [[metric Lie representations]] of the [[general linear Lie algebras]] $\mathfrak{gl}(n)$
and [[special orthogonal Lie algebras]] $\mathfrak{so}(n)$, which in turn contains also the [[Lie algebra weight systems]] coming from abelian and [[symplectic Lie algebras]] $\mathfrak{sp}(n)$.

[Bar-Natan 95, Theorem 11](#BarNatan95)

## Ingredients

## Details

Given any [[ground field]] $\mathbb{F}$, consider

\[
  \label{LinearSpanOfMarkedSurfaces}
  Span\big( MarkedSurfaces_{/\sim} \big)
  \;\in\;
  Vect
\]

the [[linear span]] of the [[set]] of [[diffeomorphism]] [[equivalence class|classes]] of [[surfaces]] [[manifold with boundary|with boundary]] equipped with a [[finite number]] of chosen oriented points on their boundary, at least one on each [[connected component]] ([Bar-Natan 95, Def. 1.12](#BarNatan95)).

A _stringy weight system_ is a [[linear map]] from this space to the [[ground field]], hence the space of stringy weight systems is the [[dual vector space]]

\[
  \label{SpaceOfStringyWeightSystems}
  \mathcal{S}
  \;\coloneqq\;
  \Big(
    Span\big( MarkedSurfaces_{/\sim} \big)
  \Big)^\ast
  \,.
\]

Of course a [[linear function]] $Span\big( MarkedSurfaces_{/\sim}\big) \to \mathbb{F}$ is equivalently a plain [[function]] $MarkedSurfaces \to \mathbb{F}$, but for the following construction it is necessary to consider [[formal linear combinations]] of marked surfaces:

The [['t Hooft double line construction]] constitutes a [[linear map]] from the [[linear span]] of [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU-relations]] to the linear span of marked surfaces (eq:LinearSpanOfMarkedSurfaces)

([Bar-Natan 95, Theorem 10 with Theorem 8](#BarNatan95))

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Under passage to [[dual vector spaces]] this yields a linear map

\[
  \label{PointParticleLimitOfStringyWeightSystems}
  \mathcal{S}
  \overset{
    (tH \,\circ\, perm)^\ast
  }{\longrightarrow}
  \underset{n \in \mathbb{N}}{\prod}
  \big(
    \mathcal{W}^n
  \big)
\]

from stringy weight systems (eq:SpaceOfStringyWeightSystems) to ordinary [[weight systems]]. This may be regarded as the [[worldline formalism|point particle limit]] of stringy weight systems.

## Statement

+-- {: .num_remark}
###### Remark
**([[stringy weight systems]] [[linear span|span]] [[classical Lie group|classical]] [[Lie algebra weight systems]])**

The [[image]] of the [[vector space]] (eq:SpaceOfStringyWeightSystems) of [[stringy weight systems]] under the point particle limit map (eq:PointParticleLimitOfStringyWeightSystems) inside the space of ordinary [[weight systems]] coincides with the [[direct sum]] of the linear span of [[Lie algebra weight systems]] for the [[general linear Lie algebras]] $\mathfrak{gl}(N)$ and the [[special orthogonal Lie algebras]] $\mathfrak{so}(N)$, and contains that of [[symplectic Lie algebras]] $\mathfrak{sp}(N)$:

=--

([Bar-Natan 95, Theorem 11](#BarNatan95))

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Related theorems

[[!include facts about chord diagrams and weight systems -- contents]]

## References

* {#BarNatan95} [[Dror Bar-Natan]], p. 9 and section 6 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

See also

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 6.2.4 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))


strong adjoint functor

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[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $C$ a [[cartesian closed category]] and $L,R : C \to C$ two [[endomorphism|endo]][[functors]], they are called **strong adjoints** to each other if there is a [[natural isomorphism]]

$$
  [L X, A] \simeq [X, R A]
$$

for all [[objects]] $X,A \in C$ and for $[-,-]$ the [[internal hom]].

## Properties

Notice that for $* $ the [[terminal object]] of $C$ we have that the [[global points]] of the internal hom give the external [[hom set]]

$$
  \Gamma [X,A] := C(*, [X, A]) \simeq C(X,A)
  \,.
$$

Therefore strongly adjoint functors are in particular [[adjoint functors]] in the ordinary sense.

## References

For instance appendix 6 of

* [[Bill Lawvere]], _Outline of synthetic differential geometry_ ([pdf](http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf))

[[!redirects strong adjoint functors]]

[[!redirects strong adjunction]]
[[!redirects strong adjunctions]]


strong dinatural transformation

[[!redirects paranatural transformation]]
* table of contents
{:toc}

## Idea

A **strong dinatural transformation** is a notion of [[natural transformation]] between two functors $C^{op}\times C\to D$ that is stronger than a [[dinatural transformation]].

Unlike dinatural transformations, strong dinatural transformations can always be composed.  They have close connections to [[parametricity]] in [[computer science]].

## Definition

Let $F,G:C^{op}\times C\to D$ be functors.  A **strong dinatural transformation** $\alpha :F\to G$ consists of, for each $c\in C$, a component $\alpha_c : F(c,c) \to G(c,c)$, such that for any morphism $f:c\to c'$ in $C$ and any [[span]] $F(c',c') \leftarrow W \to F(c,c)$ in $D$, if the square on the left commutes, then the outer hexagon also commutes:

\[
\begin{array}{ccccccc}
 &  & F(c,c) & \overset{\alpha_{c}}{\to} & G(c,c)\\
 & \nearrow & & \overset{\mathclap{F(c,f)}}{\searrow}  &  & \overset{\mathclap{G(c,f)}}{\searrow}\\
W &  &  &  & F(c,c') &  & G(c,c')\\
& \searrow & & \underset{\mathclap{F(f,c')}}{\nearrow} &  & \underset{\mathclap{G(f,c')}}{\nearrow}\\
 &  & F(c',c') & \underset{\alpha_{c'}}{\to} & G(c',c')
\end{array}\,.
\]

If the pullback $F(c,c) \times_{F(c,c')} F(c',c')$ exists in $D$, it suffices to assert this when the square on the left is the defining one of that pullback.  On the other hand, if $D$ has a [[separator]] (such as $1\in Set$), it suffices to assert this when $W$ is the separator.

By comparison, a [[dinatural transformation]] asserts this condition *only* when $W = F(c',c)$ with the span consisting of $F(c',f)$ and $F(f,c)$.

## References

Originally introduced (as strong dinatural transformations) in Definition 2.7 of

* [[Philip S. Mulry]], _Strong Monads, Algebras and Fixed Points_ (1991), in: _Applications of Categories in Computer Science_, London Mathematical Society Lecture Note Series 177, 202–216, [doi](https://doi.org/10.1017/CBO9780511525902.012).

Further developments:

* Philip S Mulry. *Strong monads, algebras and fixed points*. Applications of Categories in Computer Science 177 (1992), 202–216

* Robert Paré and Leopoldo Román. *Dinatural numbers*.  Journal of Pure and Applied Algebra 128, 1 (1998), 33–92

* A. Eppendahl, *Parametricity and Mulry's Strong Dinaturality*, <https://qmro.qmul.ac.uk/xmlui/bitstream/handle/123456789/4501/768_Eppendahl_12-1999.pdf?sequence=1>

* Varmo Vene. *Parametricity and Strong Dinaturality.* (2006). <https://www.ioc.ee/~tarmo/tday-voore/vene-slides.pdf>

* Jennifer Hackett and Graham Hutton. *Programs for cheap!*. In 2015 30th Annual ACM/IEEE Symposium on Logic in
Computer Science. IEEE, 115–126.

* Tarmo Uustalu. *Strong dinaturality and initial algebras.* 12th Nordic Wksh. on Programming Theory, NWPT 2 (2000).

* Tarmo Uustalu. *A Note on Strong Dinaturality, Initial Algebras and Uniform Parameterized Fixpoint Operators*. In
FICS. 77–82.

Strong dinatural transformations are called **paranatural transformations** in:

* [[Jacob Neumann]], *Paranatural Category Theory*, 2023, [arxiv:2307.09289](https://arxiv.org/abs/2307.09289)

[[!redirects paranatural transformations]]
[[!redirects paranatural]]
[[!redirects paranaturality]]
[[!redirects strong dinatural transformation]]
[[!redirects strong dinatural transformations]]
[[!redirects strong dinaturality]]


strong epimorphism

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### Context
#### Category theory
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[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **strong epimorphism** in a [[category]] $C$ is an [[epimorphism]] which is left [[orthogonality|orthogonal]] to any [[monomorphism]] in $C$.

## Properties

* The composition of  strong epimorphisms is a strong epimorphism. If $f\circ g$ is a strong epimorphism, then $f$ is a strong epimorphism.

* If $C$ has [[equalizers]], then any morphism which is left orthogonal to all monomorphisms must automatically be an epimorphism.

* Every [[regular epimorphism]] is strong.  The converse is true if $C$ is [[regular category|regular]].

* Every strong epimorphism is [[extremal epimorphism|extremal]].  The converse is true if $C$ has [[pullback]]s.

## In higher category theory

A [[monomorphism in an (∞,1)-category]] is a [[n-truncated|(-1)-truncated]] morphism in an [[(∞,1)-category]] $C$.

Therefore it makes sense to define an **strong epimorphism in an $(\infty,1)$-category** to be a morphism that is part of the left half of an [[orthogonal factorization system in an (∞,1)-category]] whose right half is that of $(-1)$-[[truncated]] morphisms.

If $C$ is an [[(∞,1)-topos]] then it has an [[n-connected/n-truncated factorization system]] for all $n$. The $(-1)$-[[connected]] morphisms are also called [[effective epimorphism]]s. Therefore in an $(\infty,1)$-topos strong epimorphisms again coincide with effective epimorphisms.

## References

Strong epimorphisms were introduced in:

* [[Gregory Maxwell Kelly]]. *Monomorphisms, epimorphisms, and pull-backs*. Journal of the Australian Mathematical society 9.1-2 (1969): 124-142.

Textbook accounts:

* [[Francis Borceux]], Def. 4.3.5 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

[[!redirects strong epimorphisms]]


strong homology

Strong homology theory is a homology theory of topological spaces which in addition to satisfying Eilenberg-Steenrod axioms is an invariant of [[strong shape]]. It has been developed in a series of papers by Lisica and Mardešić.

Unlike [[Čech cohomology]], [[Čech homology]] is not a homology theory in the sense of Eilenberg-Steenrod: the exactness axiom (long exact sequence in homology) does not hold. A correction to the basic Čech definition was given by [[Sibe Mardešić]]. The resulting "strong homology theory" agrees with [[singular homology]] on the spaces having homotopy type of [[CW complexes]], and does give long exact sequence of pairs $(X,A)$ where $X$ is paracompact and $A$ closed in $X$; moreover for metric compacta it satisfies not only all the axioms of Eilenberg-Steenrod, but also the relative homeomorphism axiom and the [[wedge axiom]] ("additivity"). However this axiom is not satisfied for general spaces; the counterexamples have been found by Prasolov and Mardešić.

The only homology theory on the metric compacta satisfying not only the Eilenberg-Steenrod but also the wedge axiom is the [[Steenrod-Sitnikov homology]] theory, hence the strong homology agrees with it.

###References###

A comprehensive reference is

* [[Sibe Mardešić]], _Strong shape and homology_, Springer monographs in mathematics, 2000. xii+489 pp.

Sensitivity of the results on vanishing of (higher) [[derived limit functor]]s on additional axioms of set theory, and consequences on strong homology are discussed in

* Jeffrey Bergfalk, _Strong homology, derived limits, and set theory_, [arXiv/1509.09267](http://arxiv.org/abs/1509.09267)

Extraordinary strong homology theories (represented by usual spectra; in the case of [[Eilenberg-MacLane spectrum]] agree with the usual strong homology) are constructed in

* Andrei Prasolov, _Extraordinary strong homology_, Topology and its Applications 113(1):249-291 (2001)

[[!redirects strong homology theory]]


strong monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

* table of contents
{:toc}

## Idea

A _left-strength_ and a _right-strength_ for a [[monad]] on a [[monoidal category]] are [[structures]] relating the monad with the [[tensor product]] of the category "in one direction". A monad equipped with a (left/right) strength is called a _(left/right) strong monad_. For a [[symmetric monoidal category]], the concepts of left-strength and right-strength coincide, and we simply talk about _strengths_.
The notion of strength can be motivated at least in the following ways.

* A strong monad over a [[monoidal category]] $V$ is a [[monad]] in the [[bicategory]] of $V$-[[actegory|actegories]], on the category $V$ (acting on itself from the left for a left-strength, and from the right for a right-strength).

* A left-strength for a monad is a compatibility condition between the monad and the tensor product "from the left", less symmetric than being a [[monoidal monad]], as it lacks the [[commutative monad|commutativity]] property. A right-strength is the analogous notion, where the action is from the right.

* A left-strength for a monad is a way of creating "slices", expressions over tensor products which are trivial in the first component. (The right-strength gives expressions which are trivial in the second component.)

* A strong monad is a [[tensorial strength|strong endofunctor]] with a monad structure which is compatible with the strength.

* If $V$ is a [[monoidal closed category]], then a left-strong monad is the same thing as a $V$-[[enriched monad]], and a right-strength is the same thing as a pointwise structure for the monad on the [[internal homs]]. In particular, every monad on [[Set]] is canonically left-strong.

Under some conditions, having a strength is a [[property-like structure]].

A more general notion of strength allows for general $V$-[[actegories]] instead of just the canonical action of $V$ on itself (see below).

## Definition

### Abstractly

+-- {: .num_defn}
###### Definition

For $V$ a [[monoidal category]], a **left-strong monad** over $V$ is a [[monad]] in the $2$-[[2-category|category]] $V\text{-}Act$ of left $V$-[[actegory|actegories]] on [[category|categories]].
=--

The classical definition of left-strength is on the category $V$ itself, regarded as equipped with the canonical $V$-action on itself. The definition is useful in the general case too.

The extra [[structure]] that a left-strong monad has as opposed to the underlying monad on [[Cat]] is called **strength**.

Dually, a **right-strength** is the analogous concept where $V$ is instead acting on the right.

**Details:**

If we write $\mathbf{B}V$ for the one-object [[bicategory]] obtained by [[delooping]] $V$ once, we have

$$
  V\text{-}Act \simeq Lax2Funct(\mathbf{B}V, Cat)
  \,,
$$
where on the right we have the $2$-category of [[lax functor|lax 2-functors]] from $\mathbf{B}V$ to [[Cat]], lax [[natural transformations]] of and [[modifications]].

The category $V$ defines a canonical functor $\hat V : \mathbf{B}V \to Cat$.

The left-strong monad, being a [[monad]] in this [[lax functor]] [[bicategory]] is given by

* a lax [[natural transformation]] $T : \hat V \to \hat V$;

* [[modifications]]

  * unit: $\eta : Id_V \Rightarrow T$

  * product: $\mu : T \circ T \Rightarrow T$

* satisfying the usual uniticity and associativity constraints.

By the general logic of $(2,1)$-[[(n,k)-transformation|transformations]] the components of $T$ are themselves a certain [[functor]].

Then the [usual diagrams](#concrete_definition) that specify a strong monad

* unitalness and functoriality of the component functor of $T$;

* naturalness of unit and product modifications.

### Concretely
 {#ConcreteDefinition}

A _left-strong monad_ over a [[monoidal category]] $(C,\otimes,1)$ is a [[monad]] $(T, \eta, \mu)$ equipped with a [[natural transformation]] of the form

\[
  \label{LeftStrengthNaturalTrafo}
  t_{A,B}
  \;\colon\;
  A \otimes T B
  \longrightarrow
  T(A\otimes B)
  \,,
\]

called the *left-strength*, and required to make the following [[commuting diagram|diagrams commute]]:

"Strengthening with 1 is irrelevant" (and plays well with the [[unitors]]):

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
column sep=small,%
]
& TA \ar{ddr}{\cong} \ar{ddl}[swap]{\cong} \\ \\
1 \otimes TA \ar{rr}{t_{1,A}} && T(1\otimes A)
\end{tikzcd}

+-- {: .query}
  I think it only plays well with the left unitor?
=--

"Consecutive applications of strength commute" (and play well with the [[associators]]):
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
(A\otimes B) \otimes TC \ar{d}{\cong} \ar{rr}{t_{A\otimes B, C}} && T((A\otimes B)\otimes C) \ar{d}{\cong} \\
A \otimes (B\otimes TC) \ar{r}{A\otimes t_{B,C}} & A\otimes T(B\otimes C) \ar{r}{t_{A,B\otimes C}} & T(A\otimes(B\otimes C))
\end{tikzcd}

"Strength commutes with the [[monad unit]]":
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
column sep=small,%
]
& A \otimes B \ar{ddl}[swap]{A\otimes\eta_B} \ar{ddr}{\eta_{A\otimes B}} \\ \\
A\otimes TB \ar{rr}{t_{A,B}} && T(A\otimes B)
\end{tikzcd}

"Strength commutes with the monad multiplication":
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
A\otimes TTB \ar{d}{A\otimes\mu_B} \ar{r}{t_{A,TB}} & T(A\otimes TB) \ar{r}{Tt_{A,B}} & TT(A\otimes B) \ar{d}{\mu_{A\otimes B}} \\
A\otimes TB \ar{rr}{t_{A,B}} && T(A\otimes B)
\end{tikzcd}

{#LinearMonads} The notion of strong monads for which the (left) strength $t$ (eq:LeftStrengthNaturalTrafo) is even a [[natural isomorphism]] has been called *very strong* ([Ratkovic (2013), Def. 3.1.6](#Ratkovic13)) or *linear* ([Berger & Ratkovic (2019), §1.2](#BergerRatkovic19)).

More generally, if a monoidal category $V$ acts on a category $C$
\[
\bullet : V\times C\to C
\]
then a $V$-strength for a monad $T$ on $C$ is a family of morphisms
$t_{A,B}:A\bullet T(B)\to T(A\bullet B)$ satisfying similar commutative diagrams.

In particular, the underlying [[endofunctor]] of a strong monad is a [[strong functor]].

### Right-strength

A __right-strength__ for the monad $T$ is a natural transformation $s_{A,B}:T A \otimes B \to T(A\otimes B)$ satisfying analogous diagrams to those for the left-strength.

**Remark.** In some papers, the term "strength" is used for what is here called a "left-strength" and the term "costrength" is used for what is here called a "right-strength". However, this terminology is confusing, as it is not in line with the usual convention in category theory of prefixing a term by "co-" when it is the same concept in the [[opposite category]]: instead, a left-strength is a right-strength with respect to the [[reverse monoidal category|reversal of the tensor product]] $A\otimes^{rev} B = B\otimes A$. See [[tensorial costrength]] for more details.

A __strength__ (on a monoidal category which is not necessarily symmetric) is a left-strength and a right-strength such that the two induced maps $(X\otimes T Y) \otimes Z \to T(X\otimes (Y\otimes Z))$ agree.  If the category is symmetric monoidal, then $T$ is a [[commutative monad]] if it is equipped with a left-strength (hence also a right-strength) such that the two induced maps $T X \otimes T Y \to T(X\otimes Y)$ agree.

## Examples

Every monad on [[Set]] with its [[cartesian product]] is canonically left-strong (see [below](#example-in-set) for why).

* The left-strength for the [[list monad]] works in the following way. Given sets $X$ and $Y$, the left-strength is the map
$$
(x,[y_1,\dots,y_n]) \; \mapsto \; [(x,y_1),\dots,(x,y_n)] .
$$
In other words, it turns an element of $X$ and a list of elements of $Y$ into a list of elements of $X\times Y$, canonically.

* For the left [[action monad]] (on [[Set]] or on any other [[monoidal category]]) relative to a [[monoid]] or [[monoid object]] $M$, the right-strength is inherited by the [[associator]] as follows (see [Brandenburg '14, Example 6.3.4](#Brandenburg2014)),
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
(M \otimes X) \otimes Y \ar{r}{a_{M,X,Y}}[swap]{\cong} & M \otimes (X \otimes Y) .
\end{tikzcd}
* On a [[symmetric monoidal category]], a left-strength for a left action monad can be obtained analogously, using the [[braiding]]. (For right actions, the braiding is not needed.)
In that case, this left-strength is [[commutative monad|commutative]] precisely if $M$ is [[commutative monoid|commutative as a monoid]] (see [Brandenburg '14, Example 6.3.12](#Brandenburg2014)).

* For [[probability monads]] on [[cartesian monoidal categories]], the left-strength creates "slices", or probability distributions which are deterministic in one variable. For the [[Giry monad]], for example, the left-strength for measurable spaces $X$ and $Y$ is the map
$$
(x,p) \;\mapsto\; \delta_x\otimes p
$$
for all $x\in X$ and $y\in P Y$. This left-strength is [[commutative monad|commutative]] and it allows to define easily the [[product of measures]]. Most probability monads behave analogously.
(Note that the category [[Meas]] has another [[monoidal structure]], which is closed, different from the [[cartesian product]]. For that monoidal structure, the Giry monad is not left-strong - see [Sato '16](#sato).)

## Interaction with the Kleisli category

+-- {: .query}
Is what's written here somehow the same as the abstract definition?
Can we be more general than the Kleisli category and pick an arbitrary adjunction?
=--

Given a monad $M : C \to C$ on a monoidal category $(C, \otimes, I)$, let $Kl(M)$ be the [[Kleisli category]] of $M$ with $J : C \to Kl(M)$. If $X \in Obj(C)$, write $[X] \in Obj(Kl(M))$ for the corresponding object in the Kleisli category. Moreover, given $f : X \to MY$, write $[f] : [X] \to [Y]$.

Then there is a correspondence between

* left-strengths of $M$,
* functorial operations $\odot : C \times Kl(M) \to Kl(M)$ that are
  * left unital: $I \odot A \cong A$ (naturally),
  * right unital: $X \odot [I] \cong JX$ (naturally),
  * mixed associative: $(X \otimes Y) \odot [Z] \cong X \odot (Y \odot [Z])$ (naturally),
  * subject to [[coherence laws]] (see at *[[action of a monoidal category]]*)

+-- {: .query}
Right unitality entails that $X \odot [Y] \cong X \otimes Y \odot [I] \cong [X \otimes Y]$ so basically $\odot$ is fixed on objects. Is it better to fix this on the nose? Does this violate the [[principle of equivalence]]?
=--

This correspondence is used in the formulation of [[call-by-push-value]] as an adjoint logic.

Indeed, assume that $M$ is left-strong. Then we define

* $X \odot [Y] := [X \otimes Y]$
* $f^{X \to X'} \odot [g]^{[Y] \to [Y']} := [t_{X', Y'} \circ (f \otimes g)]$

This preserves

* identity because the left-strength commutes with the unit,
* composition because the left-strength commutes with monad multiplication.

This is

* left unital on objects (by mapping the left unitor into the Kleisli category),
* naturally so because the left-strength is compatible with the left unitor,
* right unital on objects (by mapping the right unitor into the Kleisli category),
* naturally so because the left-strength commutes with the unit (!),
* mixed associative on objects (by mapping the assiciator into the Kleisli category),
* naturally so because the left-strength is compatible with the associator.

The coherence laws will be satisfied.

Conversely, assume $\odot$ is given. Then we define
\[
  [t_{X, y}] : J(X \otimes MY) \cong X \otimes MY \odot [I] \cong X \odot JMY \xrightarrow{1_X \odot [1_{MY}]} X \odot JY \cong X \otimes Y \odot [I] \cong J(X \otimes Y).
\]
This is natural and

* commutes with the unit because $\odot$ preserves identity:

\begin{centre}
\begin{xymatrix}
J(X \otimes Y)
  \ar@{}[r]|{\cong}
  \ar[d]_{J(1_X \otimes \eta_Y)}
&
X \odot JY
  \ar@{=}[rd]^{1_X \odot J1_Y = 1_{X \odot JY}}
  \ar[d]_{1_X \otimes J\eta_{Y}}
\\
J(X \otimes MY)
  \ar@{}[r]|{\cong}
&
X \odot JMY
  \ar[r]_{1_X \odot [1_{MY}]}
&
X \odot JY
  \ar@{}[r]|{\cong}
&
J(X \otimes Y)
\end{xymatrix}
\end{centre}

* commutes with multiplication because $\odot$ preserves composition (similar reasoning),
* is compatible with the left unitor by naturality and the coherence laws,
* is compatible with the associator by naturality and the coherence laws.

## On closed and monoidal closed categories

If the category $C$ is [[closed monoidal category|monoidal closed]], left-strength and right-strength induce particular structures on the [[internal homs]].

In this section, let $(C,\otimes 1)$ be monoidal closed, with internal hom denoted by $[-,-]$. Denote the [[unit of an adjunction|unit of the hom-tensor adjunction]] by $\eta_X:X \to [Y,X\otimes Y]$ and the counit (or _evaluation map_, see [[internal hom#EvaluationMap|internal hom - evaluation map]]) by $\epsilon_Y:X\otimes [X,Y]\to Y$.

+-- {: .query}
The unit is the unit of $- \otimes Y \dashv [Y, -]$ applied to $X$. The co-unit is the co-unit of $X \otimes - \dashv [X, -]$ applied to $Y$. The swapping of the variable names is already confusing, but more importantly, we're tensoring on the other side. Either we're assuming that the category is symmetric monoidal (then we should mention so), or we're not but then we should be consistent.
=--

### Left-strong monads are enriched monads
 {#LeftStrongMonadsAreEnrichedMonads}

The original reference for this part is [Kock '72](#kock72). For more references see at *[enriched monad -- Relation to strong and monoidal monads](enriched+monad#RelationToStrongMonads)*.

Beware that [Kock '72](#kock72) uses somewhat different terminology:

* The left-strength as defined here is called _[[tensorial strength]]_;

* The [[enriched monad|enrichment]] of the monad is called _strength_.

Since the two notions are equivalent under mild conditions see ([there](enriched+monad#RelationToStrongMonads)), the different terminology does not lead to confusion (once the equivalence is established).

\linebreak

First of all, a [[tensorial strength|strong endofunctor]] $T$ on a closed monoidal category $C$ is "the same thing" as a $C$-[[enriched functor|enriched endofunctor]]. To see this, note that an enriched endofunctor amounts to a [[natural transformation]]
$$
[X,Y] \;\to\; [T X, T Y]
$$
which satisfies the usual [[identity]] and [[composition]] rules (see [[enriched functor]]).
Notice also that a map as above, by the hom-tensor adjunction (or [[currying]]) is equivalently specified by a map
$$
[X,Y] \otimes T X \;\to\; T Y.
$$
The latter is provided by the strength $t$ as follows,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
{[X,Y]} \otimes T X \ar{r}{t} & T\big({[X,Y]}\otimes X \big) \ar{r}{T\epsilon} & TY.
\end{tikzcd}

Conversely, given a map $t':[X,Y] \otimes T X \to T Y$ natural in $Y$ and [[extranatural]] in $X$, one can define a left-strength as
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
X \otimes T Y \ar{r}{\eta\otimes TY} & {[Y,X\otimes Y]} \otimes T Y \ar{r}{t'} & T(X\otimes Y).
\end{tikzcd}

One can verify that these assignments are inverse to each other. Moreover, the [strength diagrams](#concrete_definition) for $t$ commute if and only the resulting assigment is indeed an [[enriched monad]], and vice versa.

#### Example in Set {#example-in-set}

_Every monad on [[Set]] is [[Set]]-enriched. In particular, it is left-strong._

Given a monad $T$ on [[Set]], the assignment
$$
[X,Y] \;\to\; [T X, T Y]
$$
is just the action of $T$ on the [[morphisms]],
$$
\mathrm{Hom}(X,Y) \;\to\; \mathrm{Hom}(T X, T Y) .
$$
Under [[currying]], this corresponds to the map
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
column sep=large,%
]
\mathrm{Hom}(X,Y) \times T X \ar{r} & T Y \\
(f,k) \ar[mapsto]{r} & Tf (k) .
\end{tikzcd}
For example, for the [[list monad]], the map is given by
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
column sep=large,%
]
\mathrm{Hom}(X,Y) \times L X \ar{r} & L Y \\
(f,{[x_1,\dots,x_n]}) \ar[mapsto]{r} & {[f(x_1),\dots,f(x_n)]} .
\end{tikzcd}

The strength obtained this way is the usual left-strength of the list monad,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
column sep=large,%
]
X \times T Y \ar{r}{\eta\times \mathrm{id}} & \mathrm{Hom}(Y,X\times Y) \times T Y \ar{r}{t'} & T(X\times Y) \\
(x,{[y_1,\dots,y_n]}) \ar[mapsto]{r} & \big( y\mapsto (x,y) , {[y_1,\dots,y_n]}\big) \ar[mapsto]{r} & {[(x,y_1),\dots,(x,y_n)]}.
\end{tikzcd}
(Compare with the [example above](#examples).)

### Right-strength and pointwise structure

The reference for this part can be found in [Kock '71, Section 1](#kock71).

Let's give a preliminary definition. A _pointwise structure_ or _cotensorial strength_ ([Kock '72](#kock72)) of a [[monad]] $T$ on a [[closed category]] $C$ is a [[natural transformation]]
$$
s' _{X,Y}: T[X,Y] \to [X, T Y]
$$
such that the following diagrams commute.

* Compatibility with the unit of the monad $\eta^T$:
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=large,%
column sep=small,%
]
& {[X,Y]} \ar{dl}[swap]{\eta^T} \ar{dr}{{[X,\eta^T]}} \\
T{[X,Y]}  \ar{rr}[swap]{s'} && {[X,TY]}
\end{tikzcd}

* Compatibility with the multiplication of the monad $\mu^T$:
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=large,%
]
TT{[X,Y]} \ar{dd}{\mu^T} \ar{rr}{Ts'} && T{[X,TY]} \ar{d}{s'} \\
&& {[X,TTY]} \ar{d}{{[X,\mu^T]}} \\
T{[X,Y]} \ar{rr}{s'} && {[X,TY]}
\end{tikzcd}

* Compatibility with the "unitor of the closed structure", the natural isomorphism $i:X\to [1,X]$ (see [[closed category#basic_definition]]):
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=large,%
column sep=small,%
]
& T X \ar{dl}[swap]{Ti} \ar{dr}{i} \\
T{[1,X]} \ar{rr}{s'} && {[1,T X]}
\end{tikzcd}

* Compatibility with the (curried) composition morphism of the closed structure, the morphism $l:[Y,Z]\to [[X,Y],[Y,Z]]$ (see [[closed category#basic_definition]]):
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=large,%
column sep=large,%
]
T{[Y,Z]} \ar{rr}{s'} \ar{d}{Tl} && {[Y, T Z]}  \ar{d}{l} \\
T{[{[X,Y]}, {[X,Z]}]} \ar{r}{s'} & {[{[X,Y]}, T{[X,Z]}]} \ar{r}{{[\mathrm{id}, s']}} & {[{[X,Y]}, {[X, T Z]}]}
\end{tikzcd}

#### Equivalence with the notion of right-strength

Suppose now that $C$ is a [[monoidal closed category]]. In this case, a pointwise structure and a costrength coincide. Let's see how.
Given a right-strength $s:T X\otimes Y \to T(X\otimes Y)$ we can obtain a pointwise structure as follows,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
column sep=large,%
]
T{[X,Y]} \ar{r}{\eta} & {[X,T{[X,Y]}\otimes X]} \ar{r}{{[X,s]}} & {[X,T({[X,Y]}\otimes X)]} \ar{r}{{[X,T\epsilon]}} & {[X,TY]} .
\end{tikzcd}

Conversely, given a pointwise structure $s':T[X,Y]\to [X,TY]$ we can obtain a right-strength as
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
column sep=large,%
]
TX \otimes Y \ar{r}{\eta\otimes Y} & T{[Y,X\otimes Y]} \otimes Y \ar{r}{s'\otimes Y} & {[Y,T(X\otimes Y)]} \otimes Y \ar{r}{\epsilon} & T(X\otimes Y ) .
\end{tikzcd}

By [[currying]] it can be shown that the respective diagrams of costrength and pointwise structure are equivalent (the unit condition corresponds to the unit condition, and so on).

#### Algebra structure on the internal homs

A pointwise structure allows in particular to _form pointwise algebra structures_, in the following sense. Given a $T$-[[algebra over a monad|algebra]] $(A,a)$, and any object $X$, we can canonically define an algebra structure on $[X,A]$, given by

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
column sep=large,%
]
T{[X,A]} \ar{r} & {[X,TA]} \ar{r}{{[X,a]}} & {[X,A]} .
\end{tikzcd}

The diagrams above assure us that

* This map gives indeed a $T$-algebra;
* $A$ and $[1,A]$ are canonically isomorphic as $T$-algebras too;
* The internal precomposition with a morphism $f:W\to X$ gives a morphism of $T$-algebras $[X,A]\to [W,A]$.

#### Example in Set

Consider the [[list monad]] on [[Set]]. The usual right-strength gives the following pointwise structure:
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
column sep=small,%
]
L{[X,Y]} \ar{r}{\eta} & {[X,L{[X,Y]}\times X]} \ar{r}{{[X,s]}} & {[X,L({[X,Y]}\times X)]} \ar{r}{{[X,L\epsilon]}} & {[X,LY]} \\
{[f,g,h]} \ar[mapsto]{r} & \big( x\mapsto ({[f,g,h]},x) \big) \ar[mapsto]{r} & \big( x\mapsto {[(f,x), (g,x), (h,x)]}\big) \ar[mapsto]{r} & \big( x\mapsto {[f(x),g(x),h(x)]} \big)
\end{tikzcd}

Indeed, this turns a list of functions into a function into lists, that maps $x\in X$ to the list of the results.

If $(A,a)$ is an $L$-[[algebra over a monad|algebra]], i.e. a [[monoid]], this allows to equip function spaces with a monoid structure as follows:
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
column sep=small,%
]
L{[X,A]} \ar{r} & {[X,LA]} \ar{r}{{[X,a]}} & {[X,A]} \\
{[f,g,h]} \ar[mapsto]{r} & \big( x\mapsto {[f(x),g(x),h(x)]} \big) \ar[mapsto]{r} & \big( x\mapsto f(x) + g(x) + h(x) \big)
\end{tikzcd}

In other words, functions into a monoid form canonically a monoid under _pointwise addition_ (or multiplication).

## Moggi's typing rules and parameterized definition

Moggi ([Moggi '89](#Moggi89)) proposed the following typing rules for a sequence operator:
\[
\frac{\Gamma \vdash t:X}{\Gamma\vdash \eta(t):T(X)}
\qquad
\frac{\Gamma,x:X \vdash u: T(Y)\quad \Delta\vdash t:T(X)}
{\Gamma,\Delta\vdash let\,x=t\,in\,u:T(Y)}
\]
To interpret these rules in a category, where types are interpreted as objects and judgements are interpreted as morphisms, we require

* For each object $X$, an object $T(X)$;

* a morphism $
\eta_X:X\to T(X)$ for each $X$ (equivalently,
a natural family of functions $C(\Gamma,X)\to C(\Gamma, T(X))$);

* a family of functions $*_{\Gamma,\Delta}:C(\Gamma\otimes X, T(Y))\times C(\Delta,T(X))\to C(\Gamma\otimes \Delta,T(Y))$ natural in $\Gamma$ and $\Delta$,

such that

* $f * \eta =f$, $\eta * f=f$, and $h*(g*f)=(h*g)*f$.

With the subscripts:

* $f *_{\Gamma,X} \eta_X = f$ and $\eta *_{I,\Delta} f=f$, and
$h *_{\Gamma,\Delta\otimes \Xi} (g *_{\Delta,\Xi} f) = (h *_{\Gamma,\Delta\otimes X} g) *_{\Gamma\otimes\Delta,\Xi} f$.

Such a structure is the same thing as a strong monad. One way to see this is to notice that it is essentially the same as the Kleisli triple form of an $\hat C$-[[enriched monad]] on $C$, where $\hat C$ is the category of [[presheaves]] on $C$ regarded with the [[Day convolution]] monoidal structure. More concretely,

\[\eta_{X\otimes Y}*_{X,T(Y)} id_{T(Y)}:X\otimes T(Y)\to T(X\otimes Y)\]

is a left-strength map.

## Uniqueness of left-strength with enough points

+-- {: .num_theorem}
###### Theorem
**(Moggi)**

Let $C$ be a category with finite products and let $T$ be a left-strong monad on $C$.
For any points $x:1\to X$, $y:1\to T(Y)$, we have
\[
t\circ(x,y)\ = \
T((x\circ !_Y),id_Y)\circ y\ : 1 \to T(X\times Y)
\]
Hence if $1$ is a [[generator]], i.e. $C(1,-):C\to Set$ is faithful, then there is at most one left-strength for any ordinary monad on $C$.
=--

In other words, a monad being left-strong is a [[property-like structure]] in a category with enough points.

## Related concepts

* [[tensorial strength]]

* [[costrong monad]], [[cotensorial strength]]

* [[monoidal monad]]

* [[commutative monad]]

* [[enriched monad]]

* [[monad (in computer science)]]

## References

The notion of strong monads was introduced (as an alternative description of [[enriched monads]] and related to [[monoidal monads]] and [[commutative monads]]) in:

* [[Anders Kock]], *Monads on symmetric monoidal closed categories*, Arch. Math. **21** (1970) 1-10 [[doi:10.1007/BF01220868](https://doi.org/10.1007/BF01220868)]

* {#kock71} [[Anders Kock]], *Closed categories generated by commutative monads*, Journal of the Australian Mathematical Society **12** 4 (Nov 1971) 405-424  [[doi:10.1017/S1446788700010272](https://doi.org/10.1017/S1446788700010272), [[KockMonoidalMonads.pdf:file]]]

* {#kock72} [[Anders Kock]], *Strong functors and monoidal monads*, Arch. Math **23** (1972) 113–120 [[doi:10.1007/BF01304852](https://doi.org/10.1007/BF01304852), [pdf](http://home.imf.au.dk/kock/SFMM.pdf)]

See also:

* H. Lindner, _Commutative monads_ in _Deuxiéme colloque sur l'algébre des catégories_. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

* {#Keigher78} [[William Keigher]], _Symmetric monoidal closed categories generated by commutative adjoint monads_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293  ([NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1978__19_3_269_0), [pdf](http://archive.numdam.org/article/CTGDC_1978__19_3_269_0.pdf))

The notion of strong monads is picked up in the context of [[monads in computer science]]:

* {#Moggi91} [[Eugenio Moggi]], Def. 3.2 in: *Notions of computation and monads*, Information and Computation, **93** 1 (1991) [<a href="https://doi.org/10.1016/0890-5401(91)90052-4">doi:10.1016/0890-5401(91)90052-4</a>, [pdf](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf)]

* Jean Goubault-Larrecq, Slawomir Lasota, David Nowak, *Logical Relations for Monadic Types*, Mathematical Structures in Computer Science, **18** 6 (2008) 1169-1217 [[arXiv:cs/0511006](https://arxiv.org/abs/cs/0511006), [doi:10.1017/S0960129508007172](https://doi.org/10.1017/S0960129508007172)]

More resources and surveys:

* {#Seal12} Gavin J. Seal, _Tensors, monads and actions_ ([arXiv:1205.0101](http://arxiv.org/abs/1205.0101))

* {#Brandenburg2014} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](https://arxiv.org/abs/1410.1716))

* {#McDermottUustalu22} [[Dylan McDermott]], [[Tarmo Uustalu]], *What Makes a Strong Monad?*, EPTCS **360** (2022) 113-133 [[arXiv:2207.00851](https://arxiv.org/abs/2207.00851), [doi:10.4204/EPTCS.360.6](https://doi.org/10.4204/EPTCS.360.6)]

Usually strong monads are described explicitly in terms of data and coherence conditions. The above repackaging of the definition in terms of a 2-category of actions appears in the comments of the blog post:

* [[John Baez]], _The Monads Hurt My Head -- But Not Anymore_ ([blog](http://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html#c025476))

More in-text references:

* {#sato} Tetsuya Sato, _The Giry monad is not strong for the canonical symmetric monoidal closed structure on Meas_,   Journal of Pure and Applied Algebra, Volume 222, Issue 10, October 2018, Pages 2888-2896. ([arXiv:1612.05939](https://arxiv.org/abs/1612.05939))

The terminology of "very strong" monads:

* {#Ratkovic13} [[Kruna S. Ratkovic]], Def. 3.1.6 in: *Morita theory in enriched context* (2013) [[arXiv:1302.2774](https://arxiv.org/abs/1302.2774)]

* Daniel Schmitter, Def. 8.10 in: *On Operations with Binders and Operations with Equations* (2020) [[web](https://researchportal.bath.ac.uk/en/studentTheses/on-operations-with-binders-and-operations-with-equations), [pdf](https://purehost.bath.ac.uk/ws/portalfiles/portal/210538297/ThesisCorrectionsDanielSchmitter.pdf)]

also called "linear monads" in:

* {#BergerRatkovic19} [[Clemens Berger]], [[Kruna Ratkovic]], §1.2 in: *Gabriel-Morita theory for excisive model categories*, Appl Categor Struct **27** (2019) 23–54 [[arXiv:1705.03863](https://arxiv.org/abs/1705.03863), [doi;10.1007/s10485-018-9539-x](https://doi.org/10.1007/s10485-018-9539-x)]

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strong monoidal functor > history

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