holographic entanglement entropy

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### Context
#### Quantum systems
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[[!include quantum systems -- contents]]
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#### Measure and probability theory
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[[!include measure theory - contents]]
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#Contents#
* table of contents
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## Idea

The term *holographic entanglement entropy* refers to expressions of [[entanglement entropy]] in [[boundary field theory|boundary]] [[quantum field theories]], via a version of [[AdS-CFT duality]] ("holography"), in terms of the [[geometry]] of a higher-dimensional [[bulk]] [[spacetime]].

The essential idea is that the [[entanglement entropy]] of a region $U$ in the domain of a [[boundary field theory]] should be proportional to the [[volume]] ([[length]], [[area]], ...) of the [[hypersurface]] $\Sigma$ in the [[bulk]] [[spacetime]] which shares the same [[boundary]] $\partial U = \partial \Sigma$ and which has minimal volume with this property.
A more precise version of this idea is given by the *Ryu-Takayanagi formula*, see [below](#RyuTakayanagiFormula).

This kind of relation between [[entropy]] of [[quantum systems]] and [[area]]/[[volumes]] of critical surfaces in a [[curved spacetime]] is akin to the [[Bekenstein-Hawking entropy]]-formula for [[black holes]], and indeed it is meant to *subsume* the [[black hole entropy]]-formula for black holes in [[anti de Sitter spacetimes]].

While in the original context of the [[AdS-CFT correspondence]] the Ryu-Takayanagi formula remains ill-defined, or at least intractable in detail (not the least because it is, ultimately, a statement about [[non-perturbative quantum field theory|non-perturbative]] [[string theory]], hence about [[M-theory]], which [remains elusive](M-theory#TheOpenProblem)), its general idea led to the discovery of fully well-defined discretized ("toy") models  of holography in terms of [[tensor networks]] that express ([[code subspaces]]) of [[quantum error correcting codes]] (such as the [[HaPPY code]] and [[Majorana dimer codes]]). In these holographic [[tensor network]] models the behaviour of holographic entanglement entropy, and of several other expected aspects of [[holography]], turn out to have a faithful reflection (up to lattice effects caused by the discretization) amenable to explicit analysis, by tools

1. from [[quantum information theory]],

1. from [[condensed matter physics]] (see also at *[[AdS/CMT]]*).

The power of these [[quantum information theory]]-methods in making precise and detailed sense of [[quantum gravity]] in a [[holography|holographic]] [[bulk]] [[spacetime]] has re-inforced the earlier idea in [[AdS/CFT]] that notions of [[spacetime]] and of [[quantum gravity]] may be holographically defined and in fact *emerge* from non-gravitational [[quantum physics]], and here specifically from [[quantum entanglement]] -- an idea that has become known under the slogan "[It from Qbit](#SimonsFoundationItFromQBit)" (see also [Georgescu 19](#Georgescu19)). Just to keep in mind that, a priori, this applies "only" to the *extra* (higher) [[bulk]] [[dimension|dimensions]] of [[spacetime]], while the definition of the [[boundary field theory]] typically relies on some notion domain space(-time) already (maybe unless one considers bulk duals of [[D(-1)-branes]]...).

### Ryu-Takayanagi formula
 {#RyuTakayanagiFormula}

For [[quantum field theories]] that are exhibited as [[boundary field theories]] on the [[asymptotic boundary]] $A$ of an approximately [[anti de Sitter spacetime]] via some approximation to [[AdS-CFT duality]] (for instance for [[QCD]] via [[AdS-QCD duality]]) their [[entanglement entropy]] of a given [[bounded set|bounded domain]] $B\subset A$ turns out to be proportional to the [[volume]] ([[area]]) of the minimal-area [[codimension]]-2 [[hypersurface]] inside the [[bulk]] spacetime that has the same [[boundary]] $\partial B$ (see [Nishioka-Ryu-Takayanagi 09 (3.3)](#NishiokaRyuTakayanagi09) for review of the formula and [Lewkowycz-Maldacena 13](#LewkowyczMaldacena13) for a conceptual explanation).

> graphics grabbed from [Nishioka-Ryu-Takayanagi 09](#NishiokaRyuTakayanagi09)

This relation is known as the _Ryu-Takayanagi formula_ ([Ryu-Takayanagi 06a](#RyuTakayanagi06a), [Ryu-Takayanagi 06b](#RyuTakayanagi06b)) for holographic computation of entanglement entropy, or _holographic entanglement entropy_, for short.

This is a generalization of the proportionality of [[black hole entropy]] to the area of its [[event horizon]]. Indeed, [[AdS-CFT duality]] applies to the [[near horizon geometry]] of [[black branes]], the higher-dimensional generalizations of [[black holes]] and reduces 4d black holes under suitable [[KK-compactification]] (see also at _[[black holes in string theory]]_)

> graphics grabbed from [Nishioka-Ryu-Takayanagi 09](#NishiokaRyuTakayanagi09)

In fact quantum corrections to the [[black hole entropy]] in the presence of matter fields is equal to the [[entanglement entropy]]. ([Ryu-Takayanagi 06a, p. 13](#RyuTakayanagi06a))

Various properties of [[entanglement entropy]] find immediate geometric interpretations this way, for instance subadditivity

> graphics grabbed from [Nishioka-Ryu-Takayanagi 09](#NishiokaRyuTakayanagi09)

### Tensor network models
 {#TensorNetworkModels}

Further discussion of implications of the Ryu-Takayanagi formula in [van Raamsdonk 10](#vanRaamsdonk10) suggested that the logic may also be turned around: Instead of computing [[entanglement entropy]] of a given [[boundary field theory]] from known [[bulk]] [[geometry]], conversely the [[bulk]] [[spacetime]] may be reconstructed from knowledge of the [[entanglement entropy]] of a boundary field theory.

Talking this perspective to the extreme suggests a description of [[bulk]] [[spacetimes]] entirely in terms of [[quantum information theory]]/[[entanglement]]-relations of a boundary [[QFT]] ("[[tensor networks]]", [Swingle 09](#Swingle09), [Swingle 12](#Swingle12), and [[quantum error correction codes]] [ADH 14](#ADH14), [PYHP 15](#PYHP15), [HNQTWY 16](#HNQTWY16), see [Harlow 18](#Harlow18), [Jahn-Eisert 21](#JahnEisert21) for review).

Here a [[tensor network]] [[Poincaré duality|Poincaré dual]] to a finite-stage [[hyperbolic tesselation]] of the [[hyperbolic plane]] is interpreted as a discretized model for a [[bulk]] [[spacetime]] with holographic [[quantum state]] on its [[asymptotic boundary]] being given by the [[string diagram]]-evaluation of the [[tensor network]] as a [[linear map]] to the [[tensor product]] of spaces corresonding to the [[edges]] crossing the boundary of the [[tesselation]].

\begin{imagefromfile}
    "file_name": "HaPPYCodeTesselation.jpg",
    "float": "right",
    "width": 480,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 20
    },
    "caption": "from [Harlow 18](#Harlow18)"
\end{imagefromfile}

One shows ([ADH 14](#ADH14)) that this is a decent ("toy") model of expected properties of [[bulk]]/[[boundary field theory|boundary]] [[holography]] if this [[tensor network]] gives the [[code subspace]] of a [[quantum error correcting code]].

For example, the [[tensor network]] corresponding to the [[pentagon|pentagonal]] $\{5,4\}$ [[tesselation]] with a [[perfect tensor]] of rank $5+1$ assigned to each tile yields a [[quantum error correcting code]] now known as the [[HaPPY code]].

The idea is that the [[code subspace]] of this [[HaPPY code]] is that subspace of the boundary [[space of quantum states]] which holographically corresponds to all bulk states *without* a [[black hole]] in the bulk.

\begin{imagefromfile}
    "file_name": "BlackHoleInHaPPYCodeTensorNetworkII.jpg",
    "float": "right",
    "width": 360,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Jahn & Eisert 21](#JahnEisert21)"
\end{imagefromfile}

More generally, deleting any of the tiles in the interior of the [[tesselation]] is interpreted as the appearance of a bulk [[black hole]], whose [[Bekenstein-Hawking entropy]] is carried by the [[quantum states]] carried by the loose [[edges]] that cross over the "[[event horizon]]" to the now deleted tiles.

The extreme case that *all* tiles are deleted from the [[tensor network]], is thus interpreted describing a [[black hole]] that has swallowed up the entire [[bulk]] spacetime such that its [[horizon]] now coincides with the previous [[asymptotic boundary]], whence its quantum states and entropy now coincide with that of the holographic boundary theory.

\linebreak

In this discrete [[HaPPY code]] model for the [[AdS-CFT correspondence]] the Ryu-Takayanagi formula for holographic entanglement entropy ([above](#RyuTakayanagiFormula)) has an exact proof [PYHP 15, Theorem 2](#PYHP15).

### Chord diagram representation
 {#ChordDiagramRepresentation}

In fact, holographic entanglement entropy in the [[HaPPY code]] turns out to be entirely encoded by the *[[dimer]] network* in the underlying [[Majorana dimer code]] ([JGPE 19](#JGPE19)):

\begin{imagefromfile}
    "file_name": "HaPPYCodesAsDimerCode.jpg",
    "width": 680,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [JGPE 19](#JGPE19)"
\end{imagefromfile}

in that the entanglement entropy $S_A$ of any [[Majorana dimer code]]-[[tensor network]] [[quantum state|state]] turns out to count the number of [[dimers]] that cross between the ([[connected topological space|connected]]) subregion $A$ and its [[complement]]:

\begin{imagefromfile}
    "file_name": "ChordDiagramEntanglementEntropy.jpg",
    "width": 460,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [JGPE 19](#JGPE19)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "HanUniversalHolographicCode.jpg",
    "float": "right",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Yan 20](#Yan20)"
\end{imagefromfile}

In the case that the [[dimers]] correspond to the [[hyperbolic tesselation]] $\{5,4\}$ from the [[HaPPY code]], this entropy formula

$$
  S_A
  \;=\;
  \tfrac{1}{2}
  \ln
  \big(
    2^{ \left\vert Chords(A, \bar A) \right\vert }
  \big)
$$

recovers the Ryu-Takayanagi formula ([JGPE 19 (78)](#JGPE19)), as here the number of chords crossing any hyperbolic [[geodesic]] grows linearly with the [[length]] of this geodesic.

In the [[continuum limit]] this says that holographic entanglement entropy is exhibited by the [[geodesic]] [[flow]] out of the given subregion through its corresponding minimal bulk surface, an idea that was previously proposed in "bit-thread models" for holographic entanglement entropy ([Freedman & Headrick 16](#FreedmanHeadrick16)):

\begin{imagefromfile}
    "file_name": "HyperbolicGeodesicFlowAndEntanglementEntropy.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [Freedman & Headrick 16](#FreedmanHeadrick16)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "ChordDiagram.jpg",
    "float": "right",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [Sati & Schreiber 19c](#SatiSchreiber19c)"
\end{imagefromfile}

Following [Sati-Schreiber 19c](#SatiSchreiber19c) we recognize the above [[Majorana dimer]]/bit-thread networks from [JGPE 19](#JGPE19), [Yan 19](#Yan19) as *[[chord diagram]]*-encodings of holographic bulks ([p. 38](https://arxiv.org/pdf/1912.10425v3.pdf#page=38)).

Notice how each [[chord]] here reflects at the same time

1. one [[entanglement|entangled pair]] of [[qbits]] in the [[boundary field theory|boundary]] [[quantum system]];

1. a [[geodesic]] through the [[hyperbolic plane]] [[bulk]] spacetime.

This is a (rigorous) state of affairs reminiscent of the "[[ER = EPR]]" slogan.

\linebreak

## Related concepts

* [[holographic Renyi entropy]]

* gravitational entropy

  * [[Bekenstein-Hawking entropy]]

  * [[generalized second law of thermodynamics]]

  * [[black holes in string theory]]

  * [[ER = EPR]]

* [[p-adic AdS-CFT]]

* [[quantum error correcting code]]

  * [[HaPPY code]]

  * [[Majorana dimer code]]

## References
 {#References}

### General

The original articles are

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Holographic Derivation of Entanglement Entropy from AdS/CFT_, Phys. Rev. Lett. 96:181602, 2006 ([arXiv:hep-th/0603001](https://arxiv.org/abs/hep-th/0603001))

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Aspects of Holographic Entanglement Entropy_, JHEP 0608:045, 2006 ([arXiv:hep-th/0605073](https://arxiv.org/abs/hep-th/0605073))

A proposal for a conceptual explanation:

* {#LewkowyczMaldacena13} Aitor Lewkowycz, [[Juan Maldacena]], _Generalized gravitational entropy_, J. High Energ. Phys. (2013) 2013: 90 ([arXiv:1304.4926](https://arxiv.org/abs/1304.4926))

* [[Nele Callebaut]], *Entanglement in conformal field theory and holography* [[arXiv:2303.16827](https://arxiv.org/abs/2303.16827)]

Review:

* [[Jens Eisert]], M. Cramer, M.B. Plenio, _Area laws for the entanglement entropy - a review_, Rev. Mod. Phys. 82, 277 (2010) ([arXiv:0808.3773](https://arxiv.org/abs/0808.3773))

* {#NishiokaRyuTakayanagi09} Tatsuma Nishioka, [[Shinsei Ryu]], [[Tadashi Takayanagi]], *Holographic Entanglement Entropy: An Overview*, J. Phys. A **42** 504008 (2009) [[arXiv:0905.0932](https://arxiv.org/abs/0905.0932), [doi:10.1088/1751-8113/42/50/504008](https://iopscience.iop.org/article/10.1088/1751-8113/42/50/504008)]

* Matthew Headrick, _Lectures on entanglement entropy in field theory and holography_ ([arXiv:1907.08126](https://arxiv.org/abs/1907.08126))

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

  Sec. 5 of: *[[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)$]$

* Bin Chen, *Holographic Entanglement Entropy: A Topical Review*, Communications in Theoretical Physics, **71** (2019) 837 [[doi:10.1088/0253-6102/71/7/837](https://iopscience.iop.org/article/10.1088/0253-6102/71/7/837)]

* [[Bowen Chen]], [[Bartlomiej Czech]], [[Zi-zhi Wang]], *Quantum Information in Holographic Duality*, Rept. Prog. Phys. **85** (2022) 4, 046001 [[arXiv:2108.09188](https://arxiv.org/abs/2108.09188), [doi:10.1088/1361-6633/ac51b5](https://doi.org/10.1088/1361-6633/ac51b5)]

Textbook account:

* [[Mukund Rangamani]], [[Tadashi Takayanagi]], *Holographic Entanglement Entropy*, Lecture Notes in Physics **931**, Springer (2017) [[doi:10.1007/978-3-319-52573-0](https://doi.org/10.1007/978-3-319-52573-0), [arXiv:1609.01287](https://arxiv.org/abs/1609.01287)]

Survey talks:

* [[Robert Myers]], _Holographic entanglement entropy_,  ([pdf slides](http://www.lpt.ens.fr/IMG/pdf/Myers.pdf))

* Shinsei Ryu, _Holographic geometry in Entanglement Renormalization_ ([pdf slides](http://icmt.illinois.edu/Workshops/JointWorkshopPerimeter/Ryu-PI-ICMT-2012.pdf))

* Juan Jottar, _(Entanglement) Entropy in three-dimensional higher spin theories_ ([pdf slides](http://www.hip.fi/holograv13/talk_folder/Jottar-HologravWorkshop-March2013.pdf))

* Matthew Headrick, _Entanglement entropies in holographic field theory_ ([pdf slides](http://www.ggi.fi.infn.it/talks/talk1853.pdf))

* {#Hartmann14} Tom Hartmann, _Entanglement entropy and geometry_, talk slides, 2014 ([pdf](http://online.kitp.ucsb.edu/online/qft14/hartman/pdf/Hartman_QFT14_KITP.pdf))

An influential argument that this relation implies that [[entanglement]] in the boundary theory is what makes spacetime as such appear in the bulk theory is due to

* {#vanRaamsdonk10} [[Mark Van Raamsdonk]], _Building up spacetime with quantum entanglement_, Gen. Rel. Grav. **42** (2010) 2323-2329; Int. J. Mod. Phys. **D19** (2010) 2429-2435 ([arXiv:1005.3035](https://arxiv.org/abs/1005.3035))

* [[Mark Van Raamsdonk]], _Building up spacetime with quantum entanglement II: It from BC-bit_ ([arXiv:1809.01197](https://arxiv.org/abs/1809.01197))

reviewed in

* [[Mark Van Raamsdonk]], _Lectures on Gravity and Entanglement_, chapter 5 in  New Frontiers in Fields and Strings
TASI 2015 Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics 2015 Theoretical Advanced Study Institute in Elementary Particle Physics ([arXiv:1609.00026](https://arxiv.org/abs/1609.00026))

Discussion of the corresponding continuum theory, formulated via [[local nets of observables]] in [[algebraic quantum field theory]]:

* [[Edward Witten]], _Notes on Some Entanglement Properties of Quantum Field Theory_, Rev. Mod. Phys. 90, 45003 (2018) ([arXiv:1803.04993](https://arxiv.org/abs/1803.04993))

* {#Faulkner20} [[Thomas Faulkner]], _The holographic map as a conditional expectation_ ([arXiv:2008.04810](https://arxiv.org/abs/2008.04810))

See also:

* Matthew Headrick, Veronika E. Hubeny, *Covariant bit threads* [[arXiv:2208.10507](https://arxiv.org/abs/2208.10507)]

* Eugenia Colafranceschi, [[Xi Dong]], [[Donald Marolf]], Zhencheng Wang, *Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without invoking holography* [[arXiv:2310.02189](https://arxiv.org/abs/2310.02189)]

### Via holographic tensor networks and quantum codes

Relation to [[renormalization]] of [[entanglement]] and [[tensor networks]] is due to

* {#Swingle09} [[Brian Swingle]], _Entanglement Renormalization and Holography_, Phys. Rev. D 86, 065007 (2012) ([arXiv:0905.1317](https://arxiv.org/abs/0905.1317))

* {#Swingle12} [[Brian Swingle]], _Constructing holographic spacetimes using entanglement renormalization_ ([arXiv:1209.3304](https://arxiv.org/abs/1209.3304), [spire:1185813](http://inspirehep.net/record/1185813))

and further in terms of [[quantum error correcting codes]] due to

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))

* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

* {#HNQTWY16} Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, Zhao Yang, _Holographic duality from random tensor networks_, JHEP 11 (2016) 009 ([arXiv:1601.01694](https://arxiv.org/abs/1601.01694))

* [[Daniel Harlow]], *The Ryu–Takayanagi Formula from Quantum Error Correction*, Comm. Math. Phys. **354** (2017) 865–912 [[doi:10.1007/s00220-017-2904-z](https://doi.org/10.1007/s00220-017-2904-z), [arXiv:1607.03901](https://arxiv.org/abs/1607.03901)]

* Helia Kamal, [[Geoffrey Penington]], _The Ryu-Takayanagi Formula from Quantum Error Correction: An Algebraic Treatment of the Boundary CFT_ ([arXiv:1912.02240](https://arxiv.org/abs/1912.02240))

reviewed in

* [[John Preskill]], *Is spacetime a quantum error-correcting code?*, talk at KITP 2015 ([pdf](http://theory.caltech.edu/~preskill/talks/Preskill-KITP-holographic-2015.pdf), [[Preskill_SpacetimeQEC.pdf:file]])

* {#Harlow18} [[Daniel Harlow]], _TASI Lectures on the Emergence of Bulk Physics in AdS/CFT_ ([arXiv:1802.01040](https://arxiv.org/abs/1802.01040))

* [[Pratik Rath]], *Aspects of Holography And Quantum Error Correction*, 2020 ([pdf](https://digitalassets.lib.berkeley.edu/etd/ucb/text/Rath_berkeley_0028E_19810.pdf), [[RathHolographyAndQEC.pdf:file]])

* {#Harlow20} [[Daniel Harlow]], *Computation and Holography*, talk at [Snowmass Computational Frontier Workshop 2020](https://indico.fnal.gov/event/43829/) ([pdf](https://indico.fnal.gov/event/43829/contributions/193566/attachments/132763/163346/snowmasscomp.pdf), [[HarlowComputationHolography.pdf:file]])

* Tanay Kibe, Prabha Mandayam, Ayan Mukhopadhyay, *Holographic spacetime, black holes and quantum error correcting codes: A review* ([arXiv:2110.14669](https://arxiv.org/abs/2110.14669))

See also

* [[Ahmed Almheiri]], _Holographic Quantum Error Correction and the Projected Black Hole Interior_ ([arXiv:1810.02055](https://arxiv.org/abs/1810.02055))

Further development of these tensor networks:

* Tamara Kohler, Toby Cubitt, *Toy Models of Holographic Duality between local Hamiltonians*, 	J. High Energy Phys. 2019:17 (2019) ([arXiv:1810.08992](https://arxiv.org/abs/1810.08992))

* {#BaoPeningtonSorceWall18} [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_,  JHEP 2019:69 ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))

* Matthew Steinberg, Sebastian Feld, [[Alexander Jahn]], *Holographic Codes from Hyperinvariant Tensor Networks* [[arXiv:2304.02732](https://arxiv.org/abs/2304.02732)]

The bit-thread proposal for associating holographic entanglement entropy with [[geodesic]] [[flow]] in [[hyperbolic space]]:

* {#FreedmanHeadrick16} [[Michael Freedman]], [[Matthew Headrick]], *Bit threads and holographic entanglement*, Comm. Math. Phys. 352, 407 (2017) ([arXiv:1604.00354](https://arxiv.org/abs/1604.00354))

Embedding of the [[HaPPY code]] and the bit-thread models in more general [[Majorana dimer codes]]:

* {#JGPE19} [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], _Majorana dimers and holographic quantum error-correcting codes_, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

* {#KahnZimborasEisert19} [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

reviewed in:

* {#JahnEisert21} [[Alexander Jahn]], [[Jens Eisert]], _Holographic tensor network models and quantum error correction: A topical review_, Quantum Sci. Technol. 6 033002 ([arXiv:2102.02619](https://arxiv.org/abs/2102.02619), [doi:10.1088/2058-9565/ac0293](https://iopscience.iop.org/article/10.1088/2058-9565/ac0293))

* {#Yan19} [[Han Yan]], *Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models*, Phys. Rev. B 102, 161119 (2020) ([arXiv:1911.01007](https://arxiv.org/abs/1911.01007))

See also:

* {#HMPSR19}  Felix M. Haehl, Eric Mintun, Jason Pollack, Antony J. Speranza, [[Mark Van Raamsdonk]], _Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories_, J. High Energ. Phys. (2019) 2019: 005 ([arxiv:1904.01584](https://arxiv.org/abs/1904.01584), [talk recording](https://youtu.be/kRCwzyliJ1M))

*  [[Han Yan]], _Hyperbolic Fracton Model, Subsystem Symmetry, and Holography_, Phys. Rev. B 99, 155126 (2019) ([arxiv:1807.05942](https://arxiv.org/abs/1807.05942))

* [[Chris Akers]], [[Geoff Penington]], *Quantum minimal surfaces from quantum error correction* ([arXiv:2109.14618](https://arxiv.org/abs/2109.14618))

* [[Alexander Jahn]], [[Marek Gluza]], Charlotte Verhoeven, Sukhbinder Singh, [[Jens Eisert]]. *Boundary theories of critical matchgate tensor networks* ([arXiv:2110.02972](https://arxiv.org/abs/2110.02972))

* Pablo Basteiro, Giuseppe Di Giulio, Johanna Erdmenger, Jonathan Karl, René Meyer, Zhuo-Yu Xian, *Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings* $[$[arXiv:2205.05693](https://arxiv.org/abs/2205.05693)$]$

* Lucas Daguerre, Matias Ginzburg, Gonzalo Torroba, *Holographic entanglement entropy inequalities beyond strong subadditivity* [[arXiv:2208.03334](https://arxiv.org/abs/2208.03334)]

* [[Scott Aaronson]], [[Jason Pollack]], *Discrete Bulk Reconstruction* [[arXiv:2210.15601](https://arxiv.org/abs/2210.15601)]

Computation of [[black hole entropy]] in 4d via [[AdS4-CFT3 duality]] from [[holographic entanglement entropy]] in the [[ABJM theory]] for the [[M2-brane]] is discussed in

* Jun Nian, Xinyu Zhang, _Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole_ ([arXiv:1705.01896](https://arxiv.org/abs/1705.01896))

Discussion in terms of [[DHR superselection theory]]:

* {#CHMP19} Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, _Entanglement entropy and superselection sectors I. Global symmetries_ ([arXiv:1905.10487](https://arxiv.org/abs/1905.10487))

Discussion of [[gauge symmetry]] of [[holographic tensor networks]] and their [[quantum error correcting codes]]:

* Kfir Dolev, Vladimir Calvera, Sam Cree, Dominic J. Williamson,
*Gauging the bulk: generalized gauging maps and holographic codes* ([arXiv:2108.11402](https://arxiv.org/abs/2108.11402))

In relation to [[renormalization group flow]]:

* Keiichiro Furuya, Nima Lashkari, Mudassir Moosa, *Renormalization group and approximate error correction* ([arXiv:2112.05099](https://arxiv.org/abs/2112.05099))

Musings on possible implications on relations between [[quantum gravity]] and [[quantum information]]:

* {#SimonsFoundationItFromQBit} [[Simons Foundation]], *[It from Qubit: Simons Collaboration on Quantum Fields, Gravity and Information](https://www.simonsfoundation.org/mathematics-physical-sciences/it-from-qubit/)*

  Final meeting: *[It from QBit 2023](https://events.perimeterinstitute.ca/event/43/)*

* [[Tom Banks]], *Holographic Space-time and Quantum Information* ([arXiv:2001.08205](https://arxiv.org/abs/2001.08205))

* {#Georgescu19} Iulia Georgescu, *Strings and qubits*, Nature Reviews Physics volume 1, page 477 (2019) ([doi:s42254-019-0087-6](https://www.nature.com/articles/s42254-019-0087-6))

* IAS Workshops on Quantum Information and Spacetime:

  * [Workshop on Quantum Information and Black Holes, December 5-6, 2017](https://www.ias.edu/sns/quantum-information-workshop-2017)

  * [Workshop on Quantum Information and the Structure of Spacetime, December 4-5, 2018](https://www.ias.edu/sns/quantum-information-workshop-2018)

  * [Workshop on Qubits and Spacetime, December 3 - 4, 2019](https://www.ias.edu/sns/quantum-information-workshop-2019)

  * [Online Workshop on Qubits and Black Holes, December 7-9, 2020](https://www.ias.edu/sns/quantum-information-workshop-2020)

  * [IAS Workshop on Quantum Information and Spacetime, December 6-8, 2021](https://www.ias.edu/sns/quantum-information-workshop-2021)

[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]

### Application to AdS/QCD

Application to [[AdS/QCD]]:

* Zhibin Li, Kun Xu, Mei Huang, _The entanglement properties of holographic QCD model with a critical end point_ ([arXiv:2002.08650](https://arxiv.org/abs/2002.08650))

[[!include entanglement island proposal for black hole paradox -- references]]

[[!include weight systems on chord diagrams in physics]]

[[!redirects Ryu-Takayanagi formula]]

[[!redirects bit thread]]
[[!redirects bit threads]]

[[!redirects holographic tensor network]]
[[!redirects holographic tensor networks]]


holographic light front QCD

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[semiclassical approximation]] to [[bottom-up model building|bottom-up]] [[holographic QCD]] is _light front holographic QCD_, named after the [[light cone]]-[[coordinate function|coordinates]] in terms if which the effective [[Schrödinger equation]] for the [[quarks]] is written.

## Related concept

* [[light-cone gauge quantization]]

* [[flat space holography]]

## References

### General

Background on [[light-front QCD]] for [[quantum hadrodynamics]]:

* International Light Cone Advisory Committee, *Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics*,  Nuclear Physics B - Proceedings Supplements
Volumes 251–252, June–July 2014, Pages 165-174 ([arXiv:1309.6333](https://arxiv.org/abs/1309.6333), [doi:10.1016/j.nuclphysbps.2014.05.004](https://doi.org/10.1016/j.nuclphysbps.2014.05.004))

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front I: Instanton effects and quark-antiquark effective potentials* ([arXiv:2110.15927](https://arxiv.org/abs/2110.15927))

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front II: QCD strings, Wilson lines and potentials* ([arXiv:2111.01775](https://arxiv.org/abs/2111.01775))

Original articles on holographic LFQCD:

* [[Stanley Brodsky]], [[Guy de Teramond]], _Light-Front Hadron Dynamics and AdS/CFT Correspondence_, Phys. Lett. B **582** (2004) 211-221 [[arXiv:hep-th/0310227](https://arxiv.org/abs/hep-th/0310227), [doi:10.1016/j.physletb.2003.12.050](https://doi.org/10.1016/j.physletb.2003.12.050)]

* [[Guy de Teramond]], [[Stanley Brodsky]], _Light-Front Holography: A First Approximation to QCD_, Phys. Rev. Lett. **102** (2009) 081601 [[doi:10.1103/PhysRevLett.102.081601](https://doi.org/10.1103/PhysRevLett.102.081601), [arXiv:0809.4899](https://arxiv.org/abs/0809.4899)]

Relation to [[renormalization]] ([[field-strength renormalization]]):

* [[Stanley Brodsky]], [[Guy de Teramond]], Alexandre Deur, _AdS/QCD, Light-Front Holography, and the Nonperturbative Running Coupling_,  International Journal of Modern Physics A Vol. 25, No. 27n28, pp. 5009-5024 (2010)  ([arXiv:1002.4660](https://arxiv.org/abs/1002.4660), [doi:10.1142/S0217751X10050822](https://doi.org/10.1142/S0217751X10050822))

Review:

* [[Stanley Brodsky]], [[Guy de Teramond]],  [[Hans Günter Dosch]], [[Joshua Erlich]], _Light-Front Holographic QCD and Emerging Confinement_, Physics Reports Volume 584, 8 July 2015, Pages 1-105 ([arXiv:1407.8131](https://arxiv.org/abs/1407.8131))

* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], Section 4.2 of: *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))

* {#ZhouDosch18} Liping Zou, [[Hans Günter Dosch]], _A very Practical Guide to Light Front Holographic QCD_, ([arXiv:1801.00607](https://arxiv.org/abs/1801.00607))

* [[Stanley Brodsky]], _Color Confinement and Supersymmetric Properties of Hadron Physics from Light-Front Holography_, International Conference on Beauty, Charm and Hyperon Hadrons (BEACH 2018) 17–23 June 2018, Peniche, Portugal ([arXiv:1912.12578](https://arxiv.org/abs/1912.12578))

* Ruben Sandapen, _An overview of light-front holography_ ([arXiv:2001.03479](https://arxiv.org/abs/2001.03479))

* [[Stanley Brodsky]], [[Guy de Teramond]], [[Hans Günter Dosch]], _Light-Front Holography and Supersymmetric Conformal Algebra: A Novel Approach to Hadron Spectroscopy, Structure, and Dynamics_ ([arXiv:2004.07756](https://arxiv.org/abs/2004.07756))

* [[Stanley Brodsky]], *Supersymmetric and Other Novel Features of Hadron Physics from Light-Front Holography*,  Proceedings of the 24th Workshop, "What Comes Beyond the Standard Models" ([arXiv:2112.02453](https://arxiv.org/abs/2112.02453))

* [[Guy de Téramond]], *Emergent phenomena in QCD: The holographic perspective* [[arXiv:2212.14028](https://arxiv.org/abs/2212.14028)]

Further concerning [[confinement]] and [[chiral symmetry breaking]]:

* [[Guy de Teramond]], [[Stanley Brodsky]], _Longitudinal dynamics and chiral symmetry breaking in holographic light-front QCD_ ([arXiv:2103.10950](https://arxiv.org/abs/2103.10950))

Specifically regarding [[rho-mesons]]:

* Satvir Kaur, Chandan Mondal, Harleen Dahiya, _Light-front holographic $\rho$-meson distributions in the momentum space_ ([arXiv:2009.04288](https://arxiv.org/abs/2009.04288))

See also

* Harun Omer, _Embedding LFHQCD in Worldsheet String Theory_ ([arXiv:1909.12866](https://arxiv.org/abs/1909.12866))

* Alfredo Vega, Miguel Angel Martin Contreras, _Two-body light front wave functions from general AdS/QCD models_ ([arXiv:2005.04501](https://arxiv.org/abs/2005.04501))

* Dmitriy N. Kim, Gerald A. Miller, *Light-Front Holography Model of the EMC Effect* [[arXiv:2209.13753](https://arxiv.org/abs/2209.13753)]

Application to [[B-meson]] physics:

* [[Mohammad Ahmady]], _Holographic light-front QCD in B meson phenomenology_ ([arXiv:2001.00266](https://arxiv.org/abs/2001.00266))

### Relation to hadron supersymmetry

Discussion of [[hadron supersymmetry]] via [[light cone gauge|light cone]] [[supersymmetric quantum mechanics]] in [[holographic light front QCD]]:

* {#dTDB14}  [[Guy de Teramond]], [[Hans Günter Dosch]], [[Stanley Brodsky]], _Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding_, Phys. Rev. D 91, 045040 (2015) ([arXiv:1411.5243](https://arxiv.org/abs/1411.5243))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum_, Phys. Rev. D 92, 074010 (2015) ([arXiv:1504.05112](https://arxiv.org/abs/1504.05112))

* {#BTDL16}  [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Cédric Lorcé, _Meson/Baryon/Tetraquark Supersymmetry from Superconformal Algebra and Light-Front Holography_, International Journal of Modern Physics AVol. 31, No. 19, 1630029 (2016) ([arXiv:1606.04638](https://arxiv.org/abs/1606.04638))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II_, Phys. Rev. D 95, 034016 (2017) ([arXiv:1612.02370](https://arxiv.org/abs/1612.02370))

* {#NielsenBrodsky18} [[Marina Nielsen]], [[Stanley Brodsky]], _Hadronic Superpartners from Superconformal and Supersymmetric Algebra_, Phys. Rev. D 97, 114001 (2018) ([arXiv:1802.09652](https://arxiv.org/abs/1802.09652))

* [[Marina Nielsen]], [[Stanley Brodsky]], Guy F. de Téramond, [[Hans Günter Dosch]], Fernando S. Navarra, [[Liping Zou]], _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))

In view of [[tetraquarks]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_ ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

[[!include Polyakov gauge-string duality -- references]]

[[!redirects light-front QCD]]


holographic principle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] what has come to be known as the _holographic principle_ is the fact that the [[correlators]]/[[partition functions]] of some [[quantum field theories]] of [[dimension]] $n$ may be identified with [[states]] of a [[TQFT]] of dimension $n + 1$.

[[!include holographic principle -- table]]

### Some details

Notice that for $\Sigma$ an $(n+1)$-dimensional manifold with $n$-dimensional [[boundary]] $\partial \Sigma$, regarded as a [[cobordism]] $\Sigma : \emptyset \to \partial \Sigma$, an $(n+1)$-dimensional TQFT assigns a morphism

$$
  Z(\Sigma) : 1 \to Z(\partial \Sigma)
  \,,
$$

hence an element of the space $Z(\partial \Sigma)$. Under holography, this element is identified with the [[partition function]] of an $n$-dimensional QFT evaluated on the manifold (without boundary) $\partial \Sigma$.

The idea that some systems in physics are governed by other systems "localized at a boundary" in this kind of way was originally suggested by the behaviour of [[black holes]] in [[general relativity]]: their [[black hole entropy]] is proportional to their "surface", as reflected by the [[generalized second law of thermodynamics]]. This made [[Gerard 't Hooft]] suggest a general principle, called the _holographic principle_, which however remained somewhat vague ([t'Hooft 93](#tHooft93), [Susskind 94](#Susskind94)).

Later, two more precise classes of correspondences were identified, that are regarded now as precise examples of the general idea of the holographic principle:

1. Systems of [[Chern-Simons theory]] and [[higher dimensional Chern-Simons theory]] can be shown explicitly to have spaces of states that are canonically identified with correlator spaces of [[CFT]]s ([[conformal block]]s) and [[self-dual higher gauge theory]] on their boundary.

   (The relation of traditional 3d [[Chern-Simons theory]] to the 2d [[WZW model]] originates in ([Witten 89](#Witten89)) and hence precedes the proposal of ([t'Hooft 93](#tHooft93), [Susskind 94](#Susskind94)), but this relation was not recognized from this perspective earlier.)

1. Systems of [[quantum gravity]] in various dimensions as given by [[string theory]] on asymptotically [[anti de Sitter spacetime]]s have been checked not in total but in a multitude of special aspects in special cases to be dual to [[supersymmetric]] [[CFT]]s on their asymptotic boundary -- this is called [[AdS/CFT correspondence]].

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

In view of these two classes of examples it is maybe noteworthy that one can see that also closed [[string field theory]], which is supposed to be one side of the [[AdS/CFT correspondence]], has the form of an [[schreiber:infinity-Chern-Simons theory]], as discussed there, for the [[L-infinity algebra]] of closed string correlators. So maybe the above two different realizations of the holographic principle are really aspects of one single mechanism for $\infty$-Chern-Simons theory.

Evidence for this also comes from the details of the AdS/CFT mechanism. In ([Witten98](#Witten98)) it is discussed how the [SYM/IIB duality](#SYMAds5) is carried by the Chern-Simons term $\int B_{NS} \wedge d B_{RR}$ in the [[type II string theory]] action, the [6d(2,0)/AdS7 duality](#6dAdS7) - is induced by the Chern-Simons term $\int C_3 \wedge d C_3 \wedge d C_3$ of the [[11-dimensional supergravity]] action.

=--

Below at _[Examples](Examples)_ we list some systems for which something along these lines is known.

### More details
 {#MoreDetails}

We discuss in a bit more detail the central idea of holography, roughly for the case of Chern-Simons type theories and making some simplifications, but giving a precise statement.

The general idea is that [[field (physics)|fields]] $\phi$ in the bulk theory are identified with [[source fields]] in the [[correlation functions]] of the boundary theory.

The archetypical example is the relation between the [[correlators]] of the [[WZW model]] on a [[Lie group]] $G$ with the [[space of quantum states]] of 3d $G$-[[Chern-Simons theory]], as reviewed for instance on page 30 of ([Gawędzki 99](#Gawedzki99)):

a [[correlator]] for the WZW model with source field $A$ has to satisfy a conformal transformation property called a [[Ward identity]]. The space of all suitable functionals satisfying these identities is the space of [[conformal blocks]]. That space is equivalently identified with the space of [[wave functions]] of Chern-Simons theory depending on the fields $A$, hence the quantum states of the CS theory.

More generally, consider some $n$-dimensional [[FQFT]] $Z_B$ and assume that the [[spaces of states]] that it assigns to any $(n-1)$-dimensional manifold $X$ are of finite [[dimension]] (over some ground field $\mathbb{C}$):

$$
  dim Z_B(X) \lt \infty
  \,.
$$

Then for $\Sigma : \partial_{in}{\Sigma} \to \partial_{out}{\Sigma}$ any [[cobordism]] of dimension $n$, the [[correlator]]

$$
  Z_B(\Sigma) : Z_B(\partial_{in} \Sigma) \to Z_B(\partial_{out} \Sigma)
$$

that $Z_B$ assigns may naturally be identified, under the [[closed monoidal category|closed monoidal structure]] of [[Vect]], as an element

$$
  \begin{aligned}
  \overline{Z_B(\Sigma)}
  & \in
  Z_B(\partial_{in} \Sigma)^{*} \otimes Z_B(\partial_{out} \Sigma)
   \\
   & \simeq
   Z_B(\partial \Sigma)
  \end{aligned}
  \,.
$$

Stated differently: the vector space $Z_B(\partial \Sigma)$ is the space of all "potential correlators" of $Z_B$ and $\overline{Z_B(\Sigma)}$ is the particular one chosen by the given model.

If $Z_B$ is really a [[CFT]] one calls a subspace $Bl_B(\Sigma) \subset Z(\partial\Sigma)$ of elements that respect conformal invariance in a certain way the space of _[[conformal block]]s_ and calls the assignment $\Sigma \mapsto Bl_B(\Sigma)$ the [[modular functor]] of the model.

Notice that by looking at all "potential correlators" this way we are suddenly assigning vector spaces in codimension 0 (on $\Sigma$), even though the axioms of an [[FQFT]] a priori only mention vector spaces (of states) assigned in codimension 1. Given all these spaces of "[[conformal block]]s", the (re)construction of $Z_B$ consists of choosing inside each $Bl_B(\Sigma)$ the actual correlator $\overline{Z_B(\Sigma)}$ (this way of looking at [[TQFT]]s $B$ is actually the way in which [Atiyah]() originally formuated the axioms of [[FQFT]]).

But since we are dealing now with vector spaces assigned to $n$-dimensional $\Sigma$, we can ask the following question:

is there an $(n+1)$-dimensional [[extended TQFT]] $A$ such that

1. there is an [[isomorphism]]

   $$
     Z_A(\Sigma) \simeq Z_B(\partial \Sigma) = Bl_B(\Sigma)
   $$

1. such that whenever $\hat \Sigma$ cobounds $\Sigma$ the linear map

$$
  Z_A : \mathbb{C} = Z_A(\hat \Sigma)
 \stackrel{Z_A(\hat \Sigma)}{\to}
  Z_A(\partial \hat \Sigma)
  \simeq
  Bl_B(\Sigma)
$$

sends $1 \in \mathbb{C}$ to $\overline{Z_B(\Sigma)}$.

If so, we say that $A$ is a **holographic dual** to $B$.

Notice that $Z_A(\Sigma)$ is the space of **[[state]]s** of $A$ over $\Sigma$, while $Bl_B(\Sigma)$ is the space of possible **[[correlator]]s** of $B$ over $\Sigma$. Under holography, the states of $A$ are identified with the correlators of $B$.

## Examples
 {#Examples}

### Poisson holography
 {#PoissonHolography}

One of the key statements of the holographic principle is that
[[field (physics)|fields]] of a [[bulk field theory]] correspond to [[sources]] in its [[boundary field theory]].

One set-up where this can be made a formal [[theorem]] is for [[2d Chern-Simons theory]] which is a [[non-perturbative field theory|non-perturbative]] [[Poisson sigma-model]]. This theorem is discussed at

* _[off-shell Poisson bracket -- boundary field theory interpretation](http://ncatlab.org/nlab/show/off-shell+Poisson+bracket#BoundaryFieldTheoryInterpretation)_.

* _[motivic quantization -- Poisson holography](motivic+quantization#PoissonHolography)_.

### Holography of higher Chern-Simons/CFT-type
 {#3dCS-2dCFT}

See also _[[AdS3-CFT2 and CS-WZW correspondence]]_.

#### RT-3d TQFT / rational 2d CFT

The class of examples of "Chern-Simons-type holography" we mention now has fairly completely and rigorously been understood. It is in turn a special and comparatively simple (but far from trivial) case of the historically earliest class of examples: [ordinary Chern-Simons theory dual to a 2d WZW model](#OrdinaryCSWZWModel) below.

For more see at _[[AdS3-CFT2 and CS-WZW correspondence]]_.

Given any [[modular tensor category]] $C$ the [[Reshetikhin-Turaev construction]] produces a 3-dimensional [[TQFT]] $Z_C$. Its space of states over a 2-dimensional surface can be identified (after some work) with a space of [[conformal block]]s for a [[WZW-model]]-liked $2d$ CFT. The [[FRS formalism]] provides a way to show that the states of $Z_C$ provides [[correlators]] that solve the [[sewing constraints]].

#### Ordinary Chern-Simons theory / WZW-model
 {#OrdinaryCSWZWModel}

For a given [[Lie group]] $G$, ordinary 3-dimensional $G$-[[Chern-Simons theory]] for a group $G$ is holographically dual to the 2-dimensional [[WZW-model]] describing the [[string]] propagating on $G$.

Here is a list with aspects of this correspondence:

1. At the level of [[action functionals]] the relation is directly seen by observing that on a 3-d [[manifold with boundary]] the [[Chern-Simons theory]] action is not gauge invariant, but has a boundary term depending on the gauge transformation. Since the gauge transformation is a function on the 2d boundary with values in $G$, this boundary term is like an action functional for a [[sigma-model]] with [[target space]] $G$, and indeed it is that (subject to some fine-tuning) of the $G$-[[WZW model]].

   A random source reviewing this is for instance ([Arcioni-Blau-Loughlin, p. 6](#ArcioniBlauLoughlin)).

1. More abstractly, at least for simply connected compact $G$, the action functionals are also related by [[transgression]] of [[moduli stacks]] as discussed at _[[schreiber:infinity-Chern-Simons theory]]_. The action functional of $G$-Chern-Simons theory is induced by the morphism

   $$
     \mathbf{c}_{conn} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}
   $$

   from the [[smooth infinity-groupoid|smooth]] [[moduli stack]] of $G$-[[connection on a bundle|bundles with connection]] to the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|moduli 3-stack]] of [[circle n-bundle with connection|circle 3-bundles with connection]] (discussed in detail at [[differential string structure]] ) in that for $\Sigma_3$ a compact 3d-dimensional surface the Chern-Simons action is the composite

   $$
     \exp(i S_{CS}(-)) : [\Sigma_3, \mathbf{B} G_{conn}] \stackrel{[\Sigma_3, \mathbf{c}_{conn}]}{\to} [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_3}(-))}{\to} U(1)
     \,,
   $$

   where the last morphism is given by [[fiber integration in ordinary differential cohomology]].

   Topological term in the WZW-model (the [[B-field]] [[background gauge field]]) is similarly the term appearing in [[codimension]] 2. This is discussed at _[Chern-Simons theory -- Geometric quantization -- In higher codimension](Chern-Simons%20theory#GeometricQuantHigher)_.

1. At the level of matching [[space of states]] of CS-theory with the [[partition function]] of the WZW model this is a computation obtained from the [[geometric quantization]] of the CS-action, originally due to ([Witten](#Witten)). A review is in ([Gawedzki, section 5](#Gawedzki)).

1. If one accepts  that the [[quantization]] of the $G$-Chern-Simons [[action functional]] yields the [[TQFT]] given by the [[Reshetikhin-Turaev construction]] applied to the [[modular tensor category]] of $G$-[[loop group]] representations, then a detailed construction of the correspondence CS-TQFT/WZW-CFT is what the [[FFRS-formalism]] achieves. See there for more details.

   More comments on the holographic interpretation of this formalism are in ([Kapustin-Saulina](#KapustinSaulina), [Fuchs-Schweigert-Valentino](#FuchsSchweigertValentino)).

#### Poisson $\sigma$-model / quantum mechanics

Ordinary [[quantum mechanics]] induced by [[quantization]] of a [[Poisson manifold]] -- which may be regarded as a 1-dimensional QFT -- is holographically dual to the 2-dimensional [[Poisson sigma-model]]
(implicitly observed by ([Kontsevich](#Kontsevich)) made explicit by ([CattaneoFelder](#CattaneoFelder)).

(Notice the [[Poisson sigma-model]] is the $(n = 2)$-case of the [[AKSZ sigma-model]] which is indeed an example of a [[schreiber:infinity-Chern-Simons theory]], as discussed there.)

#### A-model / quantum mechanics

Similarly the [[A-model]] on certain [[D-brane]]s gives a holographic description of ordinary [[quantum mechanics]]. ([Witten](#WittenAModel)).

See

* [[quantization via the A-model]]

Notice that the A-model arises from the [[Poisson sigma-model]], as discussed there.

#### Higher dimensional Chern-Simons theory / Self-dual higher gauge theory
 {#HigherDimCSAndSelfDualQFT}

##### Idea and examples

Generally, [[higher dimensional Chern-Simons theory]]
in dimension $4k+3$ (for $k \in \mathbb{N}$) is holographically related to [[self-dual higher gauge theory]] in dimension $4k+2$ (at least in the abelian case).

* $(k=0)$: ordinary 3-dimensional [[Chern-Simons theory]] is related to a [[string]] [[sigma-model]] on its boundary;

* $(k=1)$: 7-dimensional Chern-Simons theory is related to a [[fivebrane]] model on its boundary;

* $(k=2)$: 11-dimensional Chern-Simons theory is related to a parts of a [[type II string theory]] on its boundary (or that of the space-filling 9-[[brane]], if one wishes) ([BelovMoore](#BelovMoore)).

##### Some details

We indicate why [[higher dimensional Chern-Simons theory]] is -- if holographically related to anything -- holographically related to [[self-dual higher gauge theory]].

The [[phase space]] of [[higher dimensional Chern-Simons theory]] in [[dimension]] $4k+3$ on $\Sigma \times \mathbb{R}$ can be identified with the space of [[curvature|flat]] $2k+1$-forms on $\Sigma$. The [[presymplectic form]] on this space is given by the pairing

$$
  (\delta B_1, \delta B_2) \mapsto
  \int_\Sigma \delta B_1 \wedge \delta B_2
  \,.
$$

The [[geometric quantization]] of the theory requires that we choose a polarization of the [[complexification]] of this space (split the space of forms into "coordinates" and their "canonical momenta").

One way to achieve this is to choose a [[conformal structure]] on $\Sigma$. The corresponding [[Hodge star operator]]

$$
  \star : \Omega^{2k+1}(\Sigma) \to \Omega^{2k+1}(\Sigma)
$$

provides the polarization by splitting into self-dual and anti-self-dual forms:

notice that (by the formulas at [[Hodge star operator]]) we have on mid-dimensional forms

$$
  \star \star B = (-1)^{(2k+1)(4k+3)} B
  = - B
  \,.
$$

Therefore it provides a [[complex structure]] on $\Omega^{2k+1}(\Sigma) \otimes \mathbb{C}$.

We see that the symplectic structure on the space of forms can equivalently be rewritten as

$$
  \begin{aligned}
    \int_X B_1 \wedge B_2
    & = - \int_X B_1 \wedge \star \star B_2
  \end{aligned}
  \,.
$$

Here on the right now the [[Hodge star operator|Hodge inner product]] of $B_1$ with $\star B_2$ appears, which is invariant under applying the Hodge star to both arguments.

We then decompose $\Omega^{2k+1}(\Sigma)$ into the $\pm i$-[[eigenspace]]s of $\star$: say $B \in \Omega^{2k+1}(\Sigma)$ is  _imaginary self-dual_ if

$$
  \star B = i B
$$

and _imaginary anti-self-dual_ if

$$
  \star B = - i B
  \,.
$$

Then for imaginary self-dual $B_1$ and $B_2$ we find that the symplectic pairing is

$$
  \begin{aligned}
    (B_1, B_2)
    &=
    -i \int_X B_1 \wedge \star B_2
    \\
    & =
    -i \int_X (\star B_1) \wedge \star (\star B_2)
    \\
    & =
    +i \int_X B_1 \wedge \star B_2
  \end{aligned}
  \,.
$$

Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into [[Lagrangian subspace]]s.

Therefore a [[state]] of higher Chern-Simons theory on $\Sigma$ may locally be thought of as a function of the self-dual forms on $\Sigma$. Under holography this is (therefore) identified with the [[correlator]] of a [[self-dual higher gauge theory]] on $\Sigma$.

### Holography of AdS gravity/CFT-type

#### Type II on $AdS_5 \times S^5$ and $d = 4$ super Yang-Mills
 {#SYMAds5}

Conjecturally, [[type II string theory]] on a [[anti-de Sitter space]] background is holographically dual to [[super Yang-Mills theory]] on the asymptotic boundary.

See [[AdS/CFT correspondence]].

#### M-theory on $AdS_7 \times S^4$ and 6d $(2,0)$-SCFT on M5 branes
 {#6dAdS7}

[[11-dimensional supergravity|M-theory]] on $AdS_7 \times S^4$ is supposed to have as holographic boundary the [[6d (2,0)-superconformal QFT]]. See there for references.

#### M-theory on $AdS_4 \times S^7/\mathbb{Z}_k$ and Chern-Simons on M2 branes

See [[ABJM theory]].

## Related concepts

* [[holographic entanglement entropy]]

* [[holographic principle of higher category theory]]

* [[firewall problem]]

## References

### General

The idea of the holographic principle in [[quantum gravity]] originates with:

* {#tHooft93} [[Gerard 't Hooft]], _Dimensional Reduction in Quantum Gravity_, talk at *[Conference on Highlights of Particle and Condensed Matter Physics (SALAMFEST)](https://inspirehep.net/conferences/968293)* (1993) [[gr-qc/9310026](http://arxiv.org/abs/gr-qc/9310026), [spire:36137](https://inspirehep.net/literature/36137)]

* {#Susskind94} [[Leonard Susskind]], *The World as a hologram*, J. Math. Phys. **36** (1995) 6377-6396 [[hep-th/9409089](http://arxiv.org/abs/hep-th/9409089), [doi:10.1063/1.531249](https://doi.org/10.1063/1.531249)]

Early suggestion of holography in the form of [[QCD]] [[flux tubes]] being described by [[strings]] in a higher dimensional [[spacetime]] (later discussed as *[[AdS-QCD duality]]*):

* [[Alexander Polyakov]], *String Theory and Quark Confinement*, Nucl. Phys. Proc. Suppl. **68** (1998) 1-8 [[arXiv:hep-th/9711002](https://arxiv.org/abs/hep-th/9711002), doi:10.1016/S0920-5632(98)00135-2]

* [[Alexander Polyakov]], *The wall of the cave*, Int. J. Mod. Phys. A **14** (1999) 645-658 [[arXiv:hep-th/9809057](https://arxiv.org/abs/hep-th/9809057), [doi:10.1142/S0217751X99000324](https://doi.org/10.1142/S0217751X99000324)]

Review:

* R. Bousso, _The holographic principle_, Rev. Mod. Phys. __74__ (2002) 825-874 [[MR2003m:83048](http://www.ams.org/mathscinet-getitem?mr=1925130), [doi](http://link.aps.org/doi/10.1103/RevModPhys.74.825), [arXiv:hep-th/0203101](http://arxiv.org/abs/hep-th/0203101)]

See also

* [Oxford Holography Group](http://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/oxford_holography_group/holography_seminar/group_members.html), _[Background material for holography](http://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/oxford_holography_group/holography_seminar/material.html)_

### AdS/CFT

See the references at [[AdS/CFT correspondence]].

### Chern-Simons / CFT
 {#ReferencesCS-CFT}

#### On the level of action functionals

The identification of the [[space of quantum states]] of 3d $G$-[[Chern-Simons theory]] with the space of [[conformal blocks]] of the [[WZW model]] on $G$ is due to

* {#Witten89} [[Edward Witten]], *Quantum Field Theory and the Jones Polynomial*, Commun. Math. Phys. 121 (3) (1989) 351-399. [[euclid:cmp/1104178138](http://projecteuclid.org/euclid.cmp/1104178138), MR0990772]

A review of the standard holographic relation between 3d $G$-[[Chern-Simons theory]] and the [[WZW model]] on $G$ is for instance around p. 30 of

* {#Gawedzki99} [[Krzysztof Gawędzki]], _Conformal field theory: a case study_ ([arXiv:hep-th/9904145](http://arxiv.org/abs/hep-th/9904145))

Discussion of how [[gauge transformations]] of the [[action functional]] of [[Chern-Simons theory]] reproduce overe [[boundaries]] the action functional of the [[WZW model]] are for instance on p. 6 of

* Giovanni Arcioni, [[Matthias Blau]], Martin O'Loughlin, _On the boundary dynamics of Chern-Simons gravity_ ([arXiv:0210089](http://arxiv.org/abs/hep-th/0210089))
 {#ArcioniBlauLoughlin}

(And many other references. )

#### Matching of spaces of states to conformal blocks

The observation that the [[space of states]] in the [[geometric quantization]] of 3d [[Chern-Simons theory]] matches with the [[partition function]] of the [[WZW model]] is originally due to

* [[Edward Witten]] _Quantum Field Theory and the Jones Polynomial_ Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 ([project EUCLID](http://projecteuclid.org/euclid.cmp/1104178138))
 {#Witten}

A review is in section 5 of

* [[Krzysztof Gawedzki]], _Conformal field theory: a case study_  ([arXiv:hep-th/9904145](http://arxiv.org/abs/hep-th/9904145))
 {#Gawedzki}

#### Reshetikhin-Turaev 3d TQFT and rational 2d CFT
 {#RTAnd2dCFT}

Using the hypothesized relation between $G$-Chern-Simons [[TQFT]] to that given by the [[Reshetikhin-Turaev construction]] applied to the [[modular tensor category]] of $G$-[[loop group]] representations, a detailed discussion of the relation CS/WZW in given by the [[FFRS formalism]]. See there for more details

One article that contains a survey of much of the story is

* Jens Fjelstad, Jürgen Fuchs, [[Ingo Runkel]], [[Christoph Schweigert]], _Uniqueness of open/closed rational CFT with given algebra of open states_ ([arXiv:hep-th/0612306](http://arxiv.org/abs/hep-th/0612306)) .

The isomorphism between the RT-theory modular functor and the CFT conformal blocks is also discussed in

* {#AndersenUeno} [[Jørgen Andersen]], [[Kenji Ueno]], _Construction of the Reshetikhin-Turaev TQFT from conformal field theory_ ([arXiv:1110.5027](http://arxiv.org/abs/1110.5027))

Amplification of how the [[FRS formalism]] is inevitable once one adopts holography and [[QFT with defects]] is in

* [[Anton Kapustin]], [[Natalia Saulina]], _Surface operators in 3d TFT and 2d Rational CFT_ in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ AMS, 2011
 {#KapustinSaulina}

More along these lines is in

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], _Bicategories for boundary conditions and for surface defects in 3-d TFT_ ([arXiv:1203.4568](http://arxiv.org/abs/1203.4568))
 {#FuchsSchweigertValentino}

#### Self-dual higher gauge fields and higher abelian Chern-Simons

The idea of describing [[self-dual higher gauge theory]] by abelian Chern-Simons theory in one dimension higher originates in

* [[Edward Witten]], _Five-brane effective action in M-Theory_, J. Geom. Phys. __22__ (1997), no. 2, 103–133, [hep-th/9610234](http://arxiv.org/abs/hep-th/9610234)

* [[Edward Witten]], _Duality relations among topological effects in string theory_, J. High Energy Phys. 2000, no. 5, Paper 31, 31 pp. [arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086), [doi](http://dx.doi.org/10.1088/1126-6708/2000/05/031)

More discussion of the general principle is in

* Dmitriy Belov, [[Greg Moore]], _Holographic action for the self-dual field_, [arXiv:hep-th/0605038](http://arxiv.org/abs/hep-th/0605038)
 {#BelovMoore}

A quick exposition of the basic idea is in

* [[Jacques Distler]], _Actions for self-dual gauge fields_ ([blog](http://golem.ph.utexas.edu/~distler/blog/archives/000809.html))

The application of this to the description of type II [[string theory]] in 10-dimensions to 11-dimensional Chern-Simons theory is in the followup

* Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv](http://arxiv.org/abs/hep-th/0611020))

#### Poisson $\sigma$-model/A-model and quantum mechanics

* [[Maxim Kontsevich]], ...
 {#Kontsevich}

* [[Alberto Cattaneo]], [[Giovanni Felder]], ...
 {#CattaneoFelder}

* [[Edward Witten]], ...
 {#WittenAModel}

#### 3d Chern-Simons theory / 2d CFT

3-dimensional [[Chern-Simons theory]] in the context of holography is discussed for instance in

* Victor O. Rivelles, _Holographic Principle and AdS/CFT Correspondence_ ([arXiv](http://arxiv.org/abs/hep-th/9912139))

#### Chern-Simons/CFT in AdS/CFT

In

* [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ JHEP 9812:012,1998 ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))
 {#Witten98}

it is argued that in the [[AdS/CFT correspondence]] it is in fact just the Chern-Simon terms inside the corresponding [[supergravity]] theories whose states control the [[conformal block]]s of the dual [[CFT]]. So the CS/CFT correspondence is a part (a crucial part) of the AdS/CFT correspondence, at least for $AdS_5/CFT_4$ and $AdS_7/CFT_6$.

### Black hole / CFT correspondence

For [[Kerr black holes]]:

* [[Monica Guica]], [[Thomas Hartman]], Wei Song, [[Andrew Strominger]], _The Kerr/CFT Correspondence_, Phys. Rev. D **80** (2009) 124008 [[arXiv:0809.4266](http://arxiv.org/abs/0809.4266), [doi:10.1103/PhysRevD.80.124008](https://doi.org/10.1103/PhysRevD.80.124008)]

* Alejandra Castro, Alexander Maloney, [[Andrew Strominger]], _Hidden Conformal Symmetry of the Kerr Black Hole_ ([arXiv:1004.0996](http://arxiv.org/abs/1004.0996))

### General abstract formulation

An identification of boundary conditions and [[QFT with defects|defects]] as [[natural transformation]]s between higher dimensional [[FQFT]]s is discussed in

* [[Chris Schommer-Pries]], _Topological defects and classifying local topological field theories in low dimension_ ([[SchommerPriesDefects.pdf:file]])
{#SchommerPries}

See [[holographic principle of higher category theory]] for more on this.

Further discussion of formalization in [[extended TQFT]] is in

* [[Dan Freed]], _[[4-3-2 8-7-6]]_, talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012

[[!redirects holographic duality]]

[[!redirects holography]]
[[!redirects holographic dual]]
[[!redirects holographic duals]]


holographic principle -- table

**[[holographic principle]] in [[quantum field theory]]**

| [[bulk field theory]] | [[boundary field theory]] |
|---|---|
| [[dimension]] $n+1$ | [[dimension]] $n$ |
| [[field (physics)|field]] | [[source field|source]] |
| [[wave function]] | [[correlation function]] |
| [[space of quantum states]] | [[conformal blocks]] |


holographic principle of higher category theory

+-- {: .rightHandSide}
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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## The general abstract principle

In [[higher category theory]] it is easy to verify that a (strict) $1$-[[transfor]]mation $\lambda$ between [[n-functor]]s $F_1,F_2 : C \to D$ between [[strict n-categories]] $C$ and $D$

$$
  \lambda : F_1 \Rightarrow F_2
$$

is determined uniquely by an $(n-1)$-functor

$$
  \eta : C_{(n-1)} \to D^{\Delta^1}
$$

on the strict $(n-1)$-category obtained from $C$ by discarding the [[n-morphism]]s.  (Of course, not every such $(n-1)$-functor determines such a transformation; the missing condition is "naturality" at the top level.)

Analogous statements hold for general (weak) [[n-categories]], although they are more complicated to formulate; see below.

As with various other easy facts about [[category theory]], these become interesting statements when realized in a concrete context where certain structures are modeled by $n$-[[functor categories]] for all $n$.

### Examples in low dimension

We spell out explicitly the $(n-1)$-functorial nature of transformation for low values of $n$.

* **$(n=1)$** -- A [[natural transformation]] $\eta$ between [[functor]]s $F_1,F_2 : C \to D$ between ordinary [[categories]] consists of components which are given by a [[function]]

  $$
    \eta : Obj(C) \to Mor(D)
  $$

  that sends [[object]]s of $C$ to [[morphism]]s in $D$

  $$
    \eta : x \mapsto ( F_1(x) \stackrel{\eta(x)}{\to}  F_2(x))
    \,.
  $$

  Saying that such a function extends to a functor $C \to Arr(D)$:

  $$
    \eta :
    \left(
      \array{
        x
        \\
        \downarrow^{\mathrlap{\gamma_1}}
        \\
        y
        \\
        \downarrow^{\mathrlap{\gamma_2}}
        \\
        z
      }
    \right)
    \;\;\;
    \mapsto
    \;\;\;
    \left(
      \array{
        F_1(x) &\stackrel{\eta(x)}{\to}& F_2(x)
        \\
        {}^{\mathllap{F_1(f)}}\downarrow
         &=&
        \downarrow^{\mathrlap{F_2(f)}}
        \\
        F_1(y) &\stackrel{\eta(y)}{\to}& F_2(y)
        \\
        {}^{\mathllap{F_1(g)}}\downarrow
          &=&
        \downarrow^{\mathrlap{F_2(g)}}
        \\
        F_1(z) &\underset{\eta(z)}{\to}& F_2(z)
      }
    \right)
    \,.
  $$

  is equivalent to saying that these components form a natural transformation.  Since there are no nontrivial [[2-morphism]]s in $D$---in other words, the forgetful functor $Arr(D) \to D\times D$ is faithful---such an extension to a functor is necessarily unique.

  So we may regard the component function of $\eta$ as a [[0-functor]]

  $$
    \eta : \mathbf{sk}_0 C = Obj(C) \to D^{\Delta[1]} = Arr(D)
  $$

  from the [[discrete category]] on the set of objects of $C$ to the [[arrow category]] of $D$.

* **$(n=2)$** A [[pseudonatural transformation]] $\eta$ between (strict, say, for ease of of notation) [[2-functor]]s $F_1,F_2 : C \to D$ between ([[strict 2-category|strict]], for simplicity) [[2-categories]] is in components a 1-[[functor]] that functorially assigns pseudonaturality squares:

  $$
    \eta :
    \left(
      \array{
        x
        \\
        \downarrow^{\mathrlap{\gamma_1}}
        \\
        y
        \\
        \downarrow^{\mathrlap{\gamma_2}}
        \\
        z
      }
    \right)
    \;\;\;
    \mapsto
    \;\;\;
    \left(
      \array{
        F_1(x) &\stackrel{\eta(x)}{\to}& F_2(x)
        \\
        {}^{\mathllap{F_1(f)}}\downarrow
         &\swArrow_{\eta(f)}&
        \downarrow^{\mathrlap{F_2(f)}}
        \\
        F_1(y) &\stackrel{\eta(y)}{\to}& F_2(y)
        \\
        {}^{\mathllap{F_1(g)}}\downarrow
          &\swArrow_{\eta(g)}&
        \downarrow^{\mathrlap{F_2(g)}}
        \\
        F_1(z) &\underset{\eta(z)}{\to}& F_2(z)
       }
    \right)
  $$

  We may regard this as a 1-functor

  $$
    \eta : \mathbf{sk}_1 C \to Arr(D)
  $$

  from the underlying 1-category of $C$ to the arrow category of $D$, whose objects are morphisms in $D$, whose morphisms are squares in $D$, and whose composition is [[pasting]] of such squares (see [[double category]] for details).

  Again, saying that this 1-functor extends to a 2-functor from $C$ to the arrow 2-category of $D$ says precisely that these components form a pseudonatural transformation, and any such extension is unique when it exists since the forgetful 2-functor $Arr(D)\to D\times D$ is locally faithful.

* {#SchematicsIn3d} **$(n=3)$** -- A transformation between 3-functors is in components a  [[2-functor]] that sends [[2-morphism]]s in $C$ to cyclinders in $D$. This is shown in the $(n=3)$-row of the following diagram

  The pseudonaturality condition on $\eta$, which is componentwise the equation

  and the fact that there are only identity 3-morphisms in $D$ implies that this already uniquely extends to a 2-functor

  $$
    \eta : C \to Arr(D)
    \,,
  $$

  where on the right we have the 2-category whose objects are morphisms in $D$, whose morphisms are squares in $D$ and whose 2-morphisms are cylinders bounded by these squares.

### Formalizations {#Formalizations}

For [[strict ∞-categories]] modeled as globular [[strict ∞-categories]] we have the following simple statement of the general principle.

+-- {: .un_Lemma}
###### Observation

For $C,D \in Str n Cat$ and $F_1, F_2 : C \to D$ two strict $n$-functors, [[transfor]]mations $\eta : F_1 \Rightarrow F_2$ which are in components given by $n$-functors

$$
  \eta : C \to D^{G_1}
$$

are entirely specified by their underlying $(n-1)$-functors

$$
  \eta : C_{n-1} \to D^{G_1}
  \,.
$$

=--

For weak $n$-categories analogous statements hold, but may have a less straightforward formulation. What is always true is that the transformation $\eta$ is specified by its values on $(n-1)$-morphisms (and below) and will be functorial in a weak sense on these, but these $(n-1)$-morphisms and below will usually not form an $(n-1)$-category themselves, since they will compose coherently only up to $n$-morphisms.

One way to bring the general case into the above simple form is to invoke models by [[semi-strict ∞-categories]]. By [[Simpson's conjecture]], every [[∞-category]] has a model in which everything is strict except possibly the [[identities]] and their unitalness [[coherence law]]s. This means that if $C$ is such a semistrict model of an $n$-category, then $C_{n-1}$ is an $(n-1)$-[[semicategory]] and the transformation

$$
  \eta : C_{n-1} \to D^{\Delta[1]}
$$

is an $n$-functor on that. (By naturalness we have that $\eta$ is guaranteed also to respect the weak identities in $C$ in some way, but that way is not so easy to formalize.)

More generally, for any algebraic notion of weak $n$-category, there is a corresponding algebraic "$(n-1)$-dimensional" structure containing only the operations on $(n-1)$-dimensional cells and below in the given notion of weak $n$-category.  This is not in general a notion of weak $(n-1)$-category, but it may suffice to formulate the above principle precisely.  If the starting notion of $n$-category had strict associativity and interchange, then the resulting $(n-1)$-dimensional structure will be a notion of $(n-1)$-semicategory.

## Application in functorial QFT
 {#ApplicationsInQFT}

For instance in [[FQFT]] one models $n$-dimensional [[topological quantum field theories]] as [[(∞,n)-functor]]s on a flavor of an [[(∞,n)-category of cobordisms]]

$$
  Z : Bord_{n}^S \to \mathcal{C}
$$

(where the superscript $S$ is to remind us that this may be [[cobordism]]s equipped with some extra [[stuff, structure, property|structure]]).

It follows that with $Z_1, Z_2$ two such $n$-dimensional QFTs, a transformation $B : Z_1 \Rightarrow Z_2$ does look in components itself like an QFT -- which is _twisted_ by $Z_1$ and $Z_2$ in some sense (see [below]()) -- , but in dimension $(n-1)$.

More specifically, if $\mathcal{C}$ is a [[symmetric monoidal (∞,n)-category]] with tensor unit $1$ there is the trivial FQFT $\mathbf{1}$ given by the constant $(\infty,n)$-functor $\mathbf{1} : Bord_n \to \mathcal{C}$.

One can see in examples that the transformations

$$
  B : Z \Rightarrow \mathbf{1}
$$

encode _boundary conditions_ on cobordisms with boundary for the theory $Z$. Conversely, this means that one discovers on the boundary of the $n$-dimensional QFT $Z$ the $(n-1)$-dimensional QFT $B$. Or rather, this is the case if instead of [[natural transformation]]s $\eta$ one uses [[canonical transformation]]s: those component maps $\eta : C_{n-1} \to D^{I}$ that are required to be natural only with respect to the invertible $(n-1)$-morphisms in $C$.

For the case of $n=2$ and 2-dimensional cobordisms without any extra structure, a detailed version of these statements are given in ([Schommer-Pries](#SchommerPries)). For $n=3$ and the holographic relation between [[Reshetikhin-Turaev model]] and rational 2d [[CFT]] in [[FFRS-formalism]] some remarks are in ([Schreiber](#Schreiber)).

In the study of [[quantum field theory]] and [[string theory]] such kinds of relations between $n$-dimensional QFTs and $(n-1)$-dimensional QFTs on their boundary have been called the _[[holographic principle]]_ . The degree to which this principle has been formalized and the degree to which this formalization has been verified varies greatly. Examples include

* The boundary theory of 3-dimensional [[Chern-Simons theory]] is the 2-dimensional [[WZW-model]].

  This is probably the oldest known holographic relation between QFTs.

* In the 2-d QFT called the [[Poisson sigma-model]] with target the [[Poisson Lie algebroid]] coming from a [[Poisson manifold]] the boundary 3-point function computes the [[deformation quantization]] of the classical system described by that Poisson manifold.

  (This was in fact used implicitly by [[Maxim Kontsevich]] to solve deformation quantization. The relation to the Poisson sigma-model was made explicit by Cattaneo and Felder.)

* The boundary theory of the 2-dimensional [[A-model]] QFT on a target space that is a complexification of a classical [[phase space]] is the 1-dimensional QFT (= [[quantum mechanics]]) that is the [[geometric quantization]] of this phase space.

  This is described at [[quantization via the A-model]].

* The [[AdS/CFT correspondence]] conjectures specifically a holographic relation between [[quantum gravity]] on a 10-dimensional asymptotically [[anti-de Sitter spacetime]] and a superconformal [[Yang-Mills theory]] on its asymptotic boundary. More generally, this specific construction is supposed to apply, with variations, in many different dimensions.

See also [[higher category theory and physics]].

## References
  {#References}

The discussion of transformations between 2d FQFTs and how these encode boundary 1-[[brane]]s and defect 1-[[bi-brane]]s is in

* [[Chris Schommer-Pries]], _Topological defects and classifying local topological field theories in low dimension_ ([[SchommerPriesDefects.pdf:file]])
{#SchommerPries}

from slide 65 on.

A formally  comparatively well understood case of QFT holography is the relation between 3-dimensional [[Chern-Simons theory]] and the 2-dimensional [[WZW-model]]. This is formalized by the [[Reshetikhin-Turaev model]] on the 3-dimensional side and the [[Fuchs-Runkel-Schweigert construction]] on the 2-dimensional side.

Remarks on how the relation between Reshitikhin-Turaev and FSR seem to have an interpretation in terms transformations between 3-functors are at

* [[Urs Schreiber]], _Towards 2-functorial CFT_ ([blog entry](http://golem.ph.utexas.edu/category/2007/08/dbranes_from_tin_cans_part_x.html))
{#Schreiber}

There is it discussed how the basic [[string diagram]] that in FSR formalism encodes a field insertion on, possibly, a defect line and encodes the disk amplitudes of the CFT is the string diagram Poincaré-dual to the cylinder in a 3-category of [[n-vector space|3-vector space]]s.

For references on the [[holographic principle]] in QFT, see there.


holographic Rényi entropy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A modification ([Dong 16](#Dong16)) of [[Rényi entropy]] which satisfy a [[holography|holographic]] area law analogous to [[holographic entanglement entropy]]

(...)

## References

Due to:

* {#Dong16} [[Xi Dong]], *The Gravity Dual of Renyi Entropy*, Nature Communications 7, 12472 (2016) ([arXiv:1601.06788](https://arxiv.org/abs/1601.06788), [doi:10.1038/ncomms12472]( https://doi.org/10.1038/ncomms12472))

* [[Xi Dong]], *Holographic Renyi Entropy at High Energy Density*, Phys. Rev. Lett. 122, 041602 (2019) ([arXiv:1811.04081](https://arxiv.org/abs/1811.04081))

Review:

* [[Pratik Rath]], *Aspects of Holography And Quantum Error Correction*, 2020 ([pdf](https://digitalassets.lib.berkeley.edu/etd/ucb/text/Rath_berkeley_0028E_19810.pdf), [[RathHolographyAndQEC.pdf:file]])

Relation to [[quantum error correcting codes]]

* [[Chris Akers]], [[Pratik Rath]], *Holographic Renyi entropy from quantum error correction*,  J. High Energ. Phys. 2019, 52 (2019) ([arXiv:1811.05171](https://arxiv.org/abs/1811.05171))

See also:

* [[Ning Bao]], Mudassir Moosa, Ibrahim Shehzad, *The holographic dual of Rényi relative entropy*,  J. High Energ. Phys. 2019, 99 (2019) ([arXiv:1904.08433](https://arxiv.org/abs/1904.08433))

[[!redirects holographic Rényi entropies]]

[[!redirects holographic Renyi entropy]]
[[!redirects holographic Renyi entropies]]


holographic Schwinger effect -- references

### Holographic Schwinger effect
{#HolographicSchwingerEffectReferences}

Interpretation in [[holographic QCD]] of the [[Schwinger effect]] of [[vacuum polarization]] as exhibited by the [[DBI-action]] on [[flavor branes]]:

Precursor computation in [[open string|open]] [[string theory]]:

* [[Constantin Bachas]], [[Massimo Porrati]], _Pair Creation of Open Strings in an Electric Field_, Phys. Lett. B296 (1992) 77-84 ([arXiv:hep-th/9209032](https://arxiv.org/abs/hep-th/9209032))

Relation to the [[DBI-action]] of a [[probe brane]] in [[AdS/CFT]]:

* {#SemenoffZarembo11} [[Gordon Semenoff]], [[Konstantin Zarembo]], _Holographic Schwinger Effect_, Phys. Rev. Lett. 107, 171601 (2011) ([arXiv:1109.2920](https://arxiv.org/abs/1109.2920),  [doi:10.1103/PhysRevLett.107.171601](https://doi.org/10.1103/PhysRevLett.107.171601))

* {#BolognesiKieferRabinovici12} S. Bolognesi, F. Kiefer, E. Rabinovici, _Comments on Critical Electric and Magnetic Fields from Holography_, J. High Energ. Phys. 2013, 174 (2013) ([arXiv:1210.4170](https://arxiv.org/abs/1210.4170))

* Yoshiki Sato, Kentaroh Yoshida, _Holographic description of the Schwinger effect in electric and magnetic fields_,  J. High Energ. Phys. 2013, 111 (2013) ([arXiv:1303.0112](https://arxiv.org/abs/1303.0112))

* Yoshiki Sato, Kentaroh Yoshida, _Holographic Schwinger effect in confining phase_, JHEP 09 (2013) 134 ([arXiv:1306.5512](https://arxiv.org/abs/1306.5512)

* Yoshiki Sato, Kentaroh Yoshida, _Universal aspects of holographic Schwinger effect in general backgrounds_, JHEP 12 (2013) 051 ([arXiv:1309.4629](https://arxiv.org/abs/1309.4629))

* Daisuke Kawai, Yoshiki Sato, Kentaroh Yoshida, _The Schwinger pair production rate in confining theories via holography_, Phys. Rev. D 89, 101901 (2014) ([arXiv:1312.4341](https://arxiv.org/abs/1312.4341))

* Yue Ding, Zi-qiang Zhang, _Holographic Schwinger effect in a soft wall AdS/QCD model_ ([arXiv:2009.06179](https://arxiv.org/abs/2009.06179))

Relation to [[DBI-action]] of [[flavor branes]] in [[holographic QCD]]:

* {#HashimotoOka13} [[Koji Hashimoto]], [[Takashi Oka]], _Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization_, JHEP 10 (2013) 116 ([arXiv:1307.7423](https://arxiv.org/abs/1307.7423))

* {#HashimotoOkaSonoda14a} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Magnetic instability in AdS/CFT: Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD_, J. High Energ. Phys. 2014, 85 (2014) ([arXiv:1403.6336](https://arxiv.org/abs/1403.6336))

* {#HashimotoKinoshitaMurataOka14} [[Koji Hashimoto]], Shunichiro Kinoshita, Keiju Murata, [[Takashi Oka]], _Electric Field Quench in AdS/CFT_, J. High Energ. Phys. 2014 ([arXiv:1407.0798](https://arxiv.org/abs/1407.0798))

* {#HashimotoOkaSonoda14b} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Electromagnetic instability in holographic QCD_,  J. High Energ. Phys. 2015, 1 (2015) ([arXiv:1412.4254](https://arxiv.org/abs/1412.4254))

See also:

* Xing Wu, _Notes on holographic Schwinger effect_, J. High Energ. Phys. 2015, 44 (2015)  ([arXiv:1507.03208](https://arxiv.org/abs/1507.03208), doi:10.1007/JHEP09(2015)044)

* Kazuo Ghoroku, Masafumi Ishihara, _Holographic Schwinger Effect and Chiral condensate in SYM Theory_, J. High Energ. Phys. 2016, 11 (2016) (doi:10.1007/JHEP09(2016)011)

* Le Zhang, De-Fu Hou, Jian Li, _Holographic Schwinger effect with chemical potential at finite temperature_, Eur. Phys. J. A54 (2018) no.6, 94 ([spire:1677949](http://inspirehep.net/record/1677949), [doi:10.1140/epja/i2018-12524-4](https://doi.org/10.1140/epja/i2018-12524-4))

* Wenhe Cai, Kang-le Li, Si-wen Li, _Electromagnetic instability and Schwinger effect in the Witten-Sakai-Sugimoto model with D0-D4 background_,  Eur. Phys. J. C 79, 904 (2019) ([doi:10.1140/epjc/s10052-019-7404-1](https://doi.org/10.1140/epjc/s10052-019-7404-1))

* Zhou-Run Zhu, De-fu Hou, Xun Chen, _Potential analysis of holographic Schwinger effect in the magnetized background_ ([arXiv:1912.05806](https://arxiv.org/abs/1912.05806))

* Zi-qiang Zhang, Xiangrong Zhu, De-fu Hou, _Effect of gluon condensate on holographic Schwinger effect_, Phys. Rev. D 101, 026017 (2020) ([arXiv:2001.02321](https://arxiv.org/abs/2001.02321))

Review:

* Daisuke Kawai, Yoshiki Sato, Kentaroh Yoshida, _A holographic description of the Schwinger effect in a confining gauge theory_, International Journal of Modern Physics A Vol. 30, No. 11, 1530026 (2015) ([arXiv:1504.00459](https://arxiv.org/abs/1504.00459))

* [[Akihiko Sonoda]], _Electromagnetic instability in AdS/CFT_, 2016 ([spire:1633963](http://inspirehep.net/record/1633963), [[Sonoda16.pdf:file]])


holography as Koszul duality

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## Idea

The suggestion that, at least after suitable [[topological twist|topological twisting]], [[AdS/CFT duality]] becomes the statement that the [[higher algebra|higher]] [[algebras of observables]] on both sides of the correspondence are [[Koszul duality|Koszul duals]] of each other.

## References

Review:

* [[Natalie Paquette]], _Koszul Duality  in Field Theory and Holography_, 2019 ([pdf](https://www.math.ksu.edu/research/m-center/paquette_koszulduality.pdf), [[PaquetteKoszulDuality.pdf:file]])

Precursor on  [[topologically twisted supergravity]]:

* [[Kevin Costello]], [[Si Li]], _Twisted supergravity and its quantization_ ([arXiv:1606.00365](https://arxiv.org/abs/1606.00365), [spire:1466640](http://inspirehep.net/record/1466640))

For [[M2-branes]]:

* [[Kevin Costello]], _Holography and Koszul duality: the example of the M2 brane_ ([arXiv:1705.02500](https://arxiv.org/abs/1705.02500), [spire:1598491](http://inspirehep.net/record/1598491))

For [[D2-D4 brane bound states]]:

* Nafiz Ishtiaque, Seyed Faroogh Moosavian, Yehao Zhou, _Topological Holography: The Example of The D2-D4 Brane System_ ([arXiv:1809.00372](https://arxiv.org/abs/1809.00372))

For [[AdS3/CFT2 duality]]:

* [[Kevin Costello]], [[Natalie Paquette]], _Twisted Supergravity and Koszul Duality: A case study in $AdS_3$_ ([arXiv:2001.02177](https://arxiv.org/abs/2001.02177))

See also claims of _topological holography_

* [[Kevin Costello]], [[Davide Gaiotto]], _Twisted Holography_ ([arXiv:1812.09257](https://arxiv.org/abs/1812.09257))

* Jihwan Oh, Yehao Zhou, _Feynman diagrams and $\Omega$-deformed M-theory_ ([arXiv:2002.07343](https://arxiv.org/abs/2002.07343))

Review with emphasis on [[QFT with defects]]:

* [[Natalie Paquette]], Brian R. Williams, *Koszul duality in quantum field theory* ([arXiv:2110.10257](https://arxiv.org/abs/2110.10257))

See also:

* [[Kevin Costello]], [[Natalie Paquette]], _On the associativity of one-loop corrections to the celestial OPE_ ([arXiv:2204.05301](https://arxiv.org/abs/2204.05301))

* [[Natalie Paquette]], _4d physics from 2d chiral correlators_,
talk at [[Strings 2022]] [[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)]


holomorph

One would like to embed an abstract [[group]] into a bigger group $K$ in which every [[automorphism]] of $G$ is obtained by restricting (to $G$) an [[inner automorphism]] of $K$ that fixes $G$ as a subset of $K$.  The holomorph is the universal (smallest) solution to this problem.

Each group $G$ embeds into the symmetric group $Sym(G)$ on the [[underlying set]] of $G$ by the left [[regular representation]] $g\mapsto l_g$ where $l_g(h) = g h$. The [[image]] is isomorphic to $G$ (that is, the left regular representation of a discrete group is [[faithful representation|faithful]]). The [[normalizer]] of the image of $G$ in $Sym(G)$ is called the __holomorph__.

The holomorph occurs very naturally as the group of arrows of the [[2-group]] (groupoid internal to $Groups$).


holomorphic block

[[!redirects holomorphic blocks]]

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## Idea

What are called _holomorphic blocks_ for [[super Yang-Mills theory]] are roughly the analogue of [[conformal blocks]] for [[2d conformal field theory]] (see [Beem-Dimofte-Pasquetti 12, section 2.2](#BeemDimoftePasquetti12)).

## References

* {#BeemDimoftePasquetti12} [[Christopher Beem]], [[Tudor Dimofte]], [[Sara Pasquetti]], _Holomorphic Blocks in Three Dimensions_ ([arXiv:1211.1986](http://arxiv.org/abs/1211.1986))

* Fabrizio Nieri, [[Sara Pasquetti]], _Factorisation and holomorphic blocks in 4d_ ([arXiv.1507.00261](http://arxiv.org/abs/1507.00261))


holomorphic Chern-Simons theory

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## Idea

The type of [[field (physics)|field]] [[theory (physics)|theory]] called _holomorphic Chern-Simons theory_ is a variant of [[Chern-Simons theory]] where instead of [[Lie algebra valued differential forms]] on a real odd-[[dimension|dimensional]] [[manifold]] the [[field (physics)|fields]] are [[holomorphic differential forms]] with values in a [[Lie algebra]] on a complex odd-dimensional [[complex manifold]], the [[action functional]] otherwise having roughly the same structure as for standard [[Chern-Simons theory]].

## Definition

See e.g. ([Khesin-Wendt 08, section III 3.3](#KhesinWendt08))

## Properties

### Relation to $\beta$-$\gamma$-systems

Holomorphic CS may be understood in terms of a nonabelian version of the _[[beta-gamma system]]_ ([Costello 07, section 5.3](#Costello07), [Gwilliam, section 6.1.3](#Gwilliam)).

## Related concepts

* [[semi-holomorphic 4d Chern-Simons theory]]

* [[Chern-Simons theory]]

* [[Chern-Simons theory with complex gauge group]]

* [[beta-gamma system]]

* [[analytically continued Chern-Simons theory]]

* [[Spin Chern-Simons theory]]

## References

Reviews include

* {#KhesinWendt08} Boris Khesin, Robert Wendt, section III 3.3 of _The Holomorphic Chern–Simons Action Functional_ in _The Geometry of infinite-dimensional groups_, Springer 2008 ([[KhesinWendt08.pdf:file]])

Discussion in terms of [[factorization algebras of observables]] is in

* {#Costello07} [[Kevin Costello]], section 5.3 of _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv:0706.1533](http://arxiv.org/abs/0706.1533))

* {#Gwilliam} [[Owen Gwilliam]], section 6.1.3 of _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))


holomorphic de Rham complex

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## Idea

On a [[complex analytic space]] $\Sigma$, the _holomorphic de Rham complex_ is the restriction of the [[de Rham complex]] to [[holomorphic differential forms]] $\Omega^p$

$$
  \Omega^\bullet_\Sigma
  \coloneqq
  \left(
  \mathcal{O}_\Sigma \stackrel{\partial}{\longrightarrow} \Omega^1_\Sigma \stackrel{\partial}{\longrightarrow} \Omega^2_\Sigma \longrightarrow \cdots
  \right)
  \,.
$$

Notice here that since $\Omega^k$ denotes the holomorphic forms, this is the [[kernel]] of the [[Dolbeault operator]] $\bar \partial$ and hence the [[de Rham differential]] $\mathbf{d} = \partial + \bar \partial$ restricts to the holomorphic component $\partial$ as above.

Often considered is a version of this with singularities ("logarithmic de Rham complex") where one consideres instead meromorphic differential forms which are holomorphic in the bulk with logarithmic singularities towards compactification boundaries of $X$.

## Properties

### Relation to complex cohomology
 {#RelationToComplexCohomology}

For $\Sigma$ a [[complex manifold]] then the [[ordinary cohomology]] of $\Sigma$
with [[coefficients]] in the [[complex numbers]] is [[natural isomorphism|naturally isomorphic]] to the [[hypercohomology|hyper]]-[[abelian sheaf cohomology]] with coefficients in the holomorphic de Rham complex:

$$
  H^k(\Sigma,\mathbb{C})
  \simeq
  H^k(\Sigma, \Omega_\Sigma^\bullet)
$$

(e.g. [Voisin 02, theorem 8.1](#Voisin02)).

### Filtering and Relation to Hodge filtation

The holomorphic de Rham complex is naturally [[filtered object|filtered]] by degree with the $p$th filtering stage being

$$
  F^p \Omega^\bullet_X
  \coloneqq
  (0 \to \cdots \to \Omega^p_X \stackrel{\partial}{\longrightarrow}
   \Omega^{p+1} \stackrel{\partial}{\longrightarrow} \cdots)
  \,.
$$

Notice that here $\Omega^p$ is still regarded as sitting in degree $-p$, one just replaces by 0 in the holomorphic de Rham complex the groups of differential forms of degree less than $p$.

With this, the _[[Hodge filtration]]_ on $H^\bullet(X,\mathbb{C})$ is defined to be the [[filtration]] with $p$th stage the [[image]] of the [[hypercohomology|hyper]]-[[abelian sheaf cohomology]] with coefficients in the $p$th filtering stage of the holomorphic de Rham complex inside that with coefficients the full de Rham complex:

$$
  F^p H^k(X,\mathbb{C})
  \coloneqq
  im
  \left(
    H^k(X, F^p \Omega^\bullet_X)
    \to
    H^k(X, \Omega^\bullet_X)
  \right)
$$

(e.g. [Voisin 02, def. 8.2](#Voisin02)).

If $X$ happens to be a [[Kähler manifold]] then the [[Frölicher spectral sequence]]  shows that this coincides with the more traditional definition via [[harmonic differential forms]] (e.g. [Voisin 02, remark 8.29](#Voisin02)).

## Related concepts

* [[differential form with logarithmic singularities]]

* [[Dolbeault cohomology]]

* [[Hodge filtration]]

* [[Hodge cohomology]]

* [[Hodge-filtered differential cohomology]]

## References

* {#Voisin02} [[Claire Voisin]], section 8.2 of _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* {#Beilinson85} [[Alexander Beilinson]], _Higher regulators and values of L-functions_, J. Soviet Math. 30 (1985), 2036-2070 (reviewed in [Esnault-Viehweg 88](#EsnaultViehweg88)) ([doi](http://dx.doi.org/10.1007/BF02105861))

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], section 2.5 of: *Deligne-Beilinson cohomology*, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_, Perspectives in Mathematics **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]

On holomorphic de Rham cohomology of [[Stein manifolds]] (where it coincides with ordinary [[de Rham cohomology]], see [there](Stein+manifold#HolomorphicDeRhamCoincidesWithDeRhamOnSteinMfds)):

* [[Jean-Pierre Serre]]: *Quelques problemes globaux relatifs aux varietes de Stein*,  Colloque sur les fonctions de plusieurs variables (1953) [[doi:10.1007/978-3-642-39816-2_23](https://doi.org/10.1007/978-3-642-39816-2_23)]

* {#Serre54} [[Jean-Pierre Serre]]: §2.1 in: *Cohomologie et fonctions de variables complexes*, Séminaire Bourbaki 2 71 (1954) [[numdam:SB_1951-1954__2__213_0](http://www.numdam.org/item/SB_1951-1954__2__213_0)]

[[!redirects holomorphic de Rham complexes]]

[[!redirects holomorphic de Rham cohomology]]


holomorphic function

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## Definitions

A [[function]] $f$ between [[complex manifolds]] is __holomorphic__ if it is complex-[[differentiable function|differentiable]], or equivalently complex-[[analytic functions|analytic]].  One may also see definitions referring to the intermediate notions of [[continuously differentiable function|continuous differentiability]] or [[infinitely differentiable function|infinite differentiability]].

The theorem that every continuously complex-differentiable function is analytic is soft and due to [[Augustin Cauchy]]; the theorem that every complex-differentiable function is continuously differentiable is hard and due essentially to [[Édouard Goursat]] (which may be seen as filling in a gap in Cauchy's proof).  See [[Cauchy integral formula]] and [[Goursat theorem]].

On infinite-dimensional manifolds, we have several notions of holomorphic function; see [Wikipeda](https://en.wikipedia.org/wiki/Infinite-dimensional_holomorphy).

## Related concepts

* [[analytic function]]

* [[meromorphic function]]

* [[domain of holomorphy]]

* [[biholomorphic function]]

* [[Dolbeault complex]]

## References

* {#Voisin02} [[Claire Voisin]], section 1 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

In relation to the [[homotopy theory|homotopy-theoretic]] [[Oka principle]]:

* {#Forstneric11} [[Franc Forstnerič]], _Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis_, Springer 2011 ([doi:10.1007/978-3-642-22250-4](https://link.springer.com/book/10.1007/978-3-642-22250-4))

[[!redirects holomorphic function]]
[[!redirects holomorphic functions]]
[[!redirects holomorphic map]]
[[!redirects holomorphic maps]]


holomorphic line 2-bundle

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{:toc}

## Idea

Over a [[site]] of [[complex analytic spaces]], where the [[multiplicative group]] $\mathbb{G}_m$ classifies non-vanishing [[holomorphic functions]] and $\mathbf{B}\mathbb{G}_m$ classifies [[holomorphic line bundles]], then a _holomorphic line 2-bundle_ is a $\mathbb{G}_m$-[[principal 2-bundle]], modulated by maps to $\mathbf{B}^2 \mathbb{G}_m$.

This means that the [[moduli stack]] of holomorphic line 2-bundles on a [[complex analytic space]] or more generally on a [[complex analytic ∞-groupoid]] $X$  is the [[Brauer stack]]  $\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]$ (the line 2-bundle itself is the [[associated ∞-bundle]] to the $\mathbf{B}\mathbb{G}_m$-[[principal ∞-bundle]] which is the [[homotopy fiber]] of a given map $X \to \mathbf{B}^2 \mathbb{G}_m$). In particular [[equivalence classes]] of holomorphic line 2-bundles form the elements of the [[bigger Brauer group]] of $X$ (the [[Brauer group]] proper if they are torsion).

Discussion in terms of [[bundle gerbes]] includes ([Chatterjee 98](#Chatterjee98),[Brylinski 00](#Brylinski00) [Mathai-Stevenson 02, section 7](#MathaiStevenson02)).

## Properties

### Dixmier-Douady class

The [[Dixmier-Douady class]] of holomorphic line 2-bundles, hence the higher analog of the [[first Chern class]], is given by the [[connecting homomorphism]] on degee 2 of the [[long exact sequence in cohomology]] which is induced by the [[exponential exact sequence]] in [[complex analytic geometry]]:

$$
  DD\;\colon\;  H^2(-,\mathbb{G}_m) \longrightarrow H^3(-,\mathbb{Z})
  \,.
$$

### Relation to higher twistor transforms
 {#RelationToHigherTwistorTransforms}

Holomorphic line 2-bundles appear in the higher degree analogs of [[twistor transforms]]. See ([Chatterjee 98](#Chatterjee98)) and see [twistor -- References -- Application to self-dual 2-forms](twistor%20space#ReferencesApplicationToSelfDual2FormField)

## Related concepts

* [[higher complex analytic geometry]]

* [[holomorphic line n-bundle]]

* [[algebraic line 2-bundle]]

## References

### General

Discussion in relation to [[Beilinson regulators]] is in

* {#Brylinski94} [[Jean-Luc Brylinski]], _Holomorphic gerbes and the Beilinson regulator_, Astérisque 226 (1994): 145-174 ([[Brylinski94.pdf:file]])

Early discussion in terms of [[bundle gerbes]] includes

* {#Chatterjee98} [[David Chatterjee]], section 5 of _On Gerbs_, 1998  ([pdf](http://people.maths.ox.ac.uk/hitchin/hitchinstudents/chatterjee.pdf))

Discussion with an eye towards of holomorphic [[twisted K-theory]] is in

* {#MathaiStevenson02} [[Varghese Mathai]], [[Danny Stevenson]], _Chern character in twisted K-theory: equivariant and holomorphic cases_, Commun.Math.Phys. 236 (2003) 161-186 ([arXiv:hep-th/0201010v5](http://lanl.arxiv.org/abs/hep-th/0201010v5))

An equivariant example arising from more algebro-geometric origin is in

* [[Giovanni Felder]], [[Andre Henriques]], Carlo A. Rossi, [[Chenchang Zhu]], _A gerbe for the elliptic gamma function_, Duke Math. J. 141(2008), 1-74 [arXiv:math/0601337](http://arxiv.org/abs/math/0601337) ([doi](http://dx.doi.org/10.1215/S0012-7094-08-14111-0))

Discussion connecting explicitly to the holomorphic [[Brauer group]] includes

* {#BenBassat08} [[Oren Ben-Bassat]], _Gerbes and the Holomorphic Brauer Group of Complex Tori_, Journal of Noncommutative Geometry, Volume 6, Issue 3 (2012) 407-455 ([arXiv:0811.2746](http://arxiv.org/abs/0811.2746))

* Edoardo Ballico, [[Oren Ben-Bassat]], _Meromorphic Line Bundles and Holomorphic Gerbes_, Math. Res. Lett. 18 (2011), 6, 1-14 ([arXiv:1101.2216](http://arxiv.org/abs/1101.2216))

See also

* {#BenBassat11} [[Oren Ben-Bassat]], _Equivariant Gerbes on Complex Tori_ ([arXiv:1102.2312](http://arxiv.org/abs/1102.2312))

### Basic line 2-bundles on reductive groups
 {#ReferencesOnReductiveGroup}

The existence of the "basic" 2-line bundle (see at [[Chern-Simons line 3-bundle]]) on a complex [[reductive group]] (such as $SL(n,\mathbb{C})$) is mentioned in

* [[Jean-Luc Brylinski]], around theorem 5.4.10 (p. 226-227) of _Loop spaces and characteristic classes_, Birkhäuser

The actual construction appears in

* {#Brylinski00} [[Jean-Luc Brylinski]], _Gerbes on complex reductive Lie groups_ ([arXiv:math/0002158](http://arxiv.org/abs/math/0002158))

[[!redirects holomorphic line 2-bundles]]

[[!redirects holomorphic bundle gerbe]]
[[!redirects holomorphic bundle gerbes]]

[[!redirects holomorphic 2-line bundle]]
[[!redirects holomorphic 2-line bundles]]


holomorphic line n-bundle

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## Idea

The concept of [[line n-bundle]] in [[complex analytic geometry]], classified by maps into the $n$-fold [[delooping]] $\mathbf{B}^n \mathbb{G}_m$ of the [[multiplicative group]].

* for $n = 0$ these are [[holomorphic functions]] classified by the [[group of units]];

* for $n= 1$ these are [[holomorphic line bundles]], classified by [[Picard group]], modulated by [[Picard stack]];

* for $n =2$ these are [[holomorphic line 2-bundles]], classified by [[Brauer group]], modulated by [[Brauer stack]]

## Properties
 {#Properties}

(...)

## Related concepts

* [[algebraic line n-bundle]]

* [[complex analytic ∞-stack]]

## References

* {#Grothendieck64} [[Alexander Grothendieck]], _Le groupe de Brauer : II. Théories cohomologiques_. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. ([Numdam](http://www.numdam.org/item?id=SB_1964-1966__9__287_0))

[[!redirects holomorphic line n-bundles]]