+-- {: .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 ([Gaw&#281;dzki 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 * [[AdS/CFT correspondence]] * [[holographic entanglement entropy]] * [[holographic principle of higher category theory]] * [[asymptotic symmetry]] * [[bulk-boundary correspondence]] * [[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) &lbrack;[gr-qc/9310026](http://arxiv.org/abs/gr-qc/9310026), [spire:36137](https://inspirehep.net/literature/36137)&rbrack; * {#Susskind94} [[Leonard Susskind]], *The World as a hologram*, J. Math. Phys. **36** (1995) 6377-6396 &lbrack;[hep-th/9409089](http://arxiv.org/abs/hep-th/9409089), [doi:10.1063/1.531249](https://doi.org/10.1063/1.531249)&rbrack; 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 &lbrack;[arXiv:hep-th/9711002](https://arxiv.org/abs/hep-th/9711002), <a href="https://doi.org/10.1016/S0920-5632(98)00135-2">doi:10.1016/S0920-5632(98)00135-2</a>&rbrack; * [[Alexander Polyakov]], *The wall of the cave*, Int. J. Mod. Phys. A **14** (1999) 645-658 &lbrack;[arXiv:hep-th/9809057](https://arxiv.org/abs/hep-th/9809057), [doi:10.1142/S0217751X99000324](https://doi.org/10.1142/S0217751X99000324)&rbrack; Review: * R. Bousso, _The holographic principle_, Rev. Mod. Phys. __74__ (2002) 825-874 &lbrack;[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)&rbrack; * Nele Callebaut: *Introduction to holography* &lbrack;[arXiv:2603.05186](https://arxiv.org/abs/2603.05186)&rbrack; 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. &lbrack;[euclid:cmp/1104178138](http://projecteuclid.org/euclid.cmp/1104178138), MR0990772&rbrack; 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&#8211;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, J&#252;rgen 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&#8211;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 &lbrack;[arXiv:0809.4266](http://arxiv.org/abs/0809.4266), [doi:10.1103/PhysRevD.80.124008](https://doi.org/10.1103/PhysRevD.80.124008)&rbrack; * 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]]