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\section*{holographic principle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{some_details}{Some details}\dotfill \pageref*{some_details} \linebreak
\noindent\hyperlink{MoreDetails}{More details}\dotfill \pageref*{MoreDetails} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{PoissonHolography}{Poisson holography}\dotfill \pageref*{PoissonHolography} \linebreak
\noindent\hyperlink{3dCS-2dCFT}{Holography of higher Chern-Simons/CFT-type}\dotfill \pageref*{3dCS-2dCFT} \linebreak
\noindent\hyperlink{rt3d_tqft__rational_2d_cft}{RT-3d TQFT / rational 2d CFT}\dotfill \pageref*{rt3d_tqft__rational_2d_cft} \linebreak
\noindent\hyperlink{OrdinaryCSWZWModel}{Ordinary Chern-Simons theory / WZW-model}\dotfill \pageref*{OrdinaryCSWZWModel} \linebreak
\noindent\hyperlink{poisson_model__quantum_mechanics}{Poisson $\sigma$-model / quantum mechanics}\dotfill \pageref*{poisson_model__quantum_mechanics} \linebreak
\noindent\hyperlink{amodel__quantum_mechanics}{A-model / quantum mechanics}\dotfill \pageref*{amodel__quantum_mechanics} \linebreak
\noindent\hyperlink{HigherDimCSAndSelfDualQFT}{Higher dimensional Chern-Simons theory / Self-dual higher gauge theory}\dotfill \pageref*{HigherDimCSAndSelfDualQFT} \linebreak
\noindent\hyperlink{idea_and_examples}{Idea and examples}\dotfill \pageref*{idea_and_examples} \linebreak
\noindent\hyperlink{some_details_2}{Some details}\dotfill \pageref*{some_details_2} \linebreak
\noindent\hyperlink{holography_of_ads_gravitycfttype}{Holography of AdS gravity/CFT-type}\dotfill \pageref*{holography_of_ads_gravitycfttype} \linebreak
\noindent\hyperlink{SYMAds5}{Type II on $AdS_5 \times S^5$ and $d = 4$ super Yang-Mills}\dotfill \pageref*{SYMAds5} \linebreak
\noindent\hyperlink{6dAdS7}{M-theory on $AdS_7 \times S^4$ and 6d $(2,0)$-SCFT on M5 branes}\dotfill \pageref*{6dAdS7} \linebreak
\noindent\hyperlink{mtheory_on__and_chernsimons_on_m2_branes}{M-theory on $AdS_4 \times S^7/\mathbb{Z}_k$ and Chern-Simons on M2 branes}\dotfill \pageref*{mtheory_on__and_chernsimons_on_m2_branes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{adscft}{AdS/CFT}\dotfill \pageref*{adscft} \linebreak
\noindent\hyperlink{ReferencesCS-CFT}{Chern-Simons / CFT}\dotfill \pageref*{ReferencesCS-CFT} \linebreak
\noindent\hyperlink{on_the_level_of_action_functionals}{On the level of action functionals}\dotfill \pageref*{on_the_level_of_action_functionals} \linebreak
\noindent\hyperlink{matching_of_spaces_of_states_to_conformal_blocks}{Matching of spaces of states to conformal blocks}\dotfill \pageref*{matching_of_spaces_of_states_to_conformal_blocks} \linebreak
\noindent\hyperlink{RTAnd2dCFT}{Reshetikhin-Turaev 3d TQFT and rational 2d CFT}\dotfill \pageref*{RTAnd2dCFT} \linebreak
\noindent\hyperlink{selfdual_higher_gauge_fields_and_higher_abelian_chernsimons}{Self-dual higher gauge fields and higher abelian Chern-Simons}\dotfill \pageref*{selfdual_higher_gauge_fields_and_higher_abelian_chernsimons} \linebreak
\noindent\hyperlink{poisson_modelamodel_and_quantum_mechanics}{Poisson $\sigma$-model/A-model and quantum mechanics}\dotfill \pageref*{poisson_modelamodel_and_quantum_mechanics} \linebreak
\noindent\hyperlink{3d_chernsimons_theory__2d_cft}{3d Chern-Simons theory / 2d CFT}\dotfill \pageref*{3d_chernsimons_theory__2d_cft} \linebreak
\noindent\hyperlink{chernsimonscft_in_adscft}{Chern-Simons/CFT in AdS/CFT}\dotfill \pageref*{chernsimonscft_in_adscft} \linebreak
\noindent\hyperlink{black_hole__cft_correspondence}{Black hole / CFT correspondence}\dotfill \pageref*{black_hole__cft_correspondence} \linebreak
\noindent\hyperlink{general_abstract_formulation}{General abstract formulation}\dotfill \pageref*{general_abstract_formulation} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[quantum field theory]] what has come to be known as the \emph{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]]

\hypertarget{some_details}{}\subsubsection*{{Some details}}\label{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

\begin{displaymath}
Z(\Sigma) : 1 \to Z(\partial \Sigma)
  \,,
\end{displaymath}
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 \emph{holographic principle}, which however remained somewhat vague (\hyperlink{tHooft93}{t'Hooft 93}, \hyperlink{Susskind94}{Susskind 94}).

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

\begin{enumerate}%
\item 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 (\hyperlink{Witten89}{Witten 89}) and hence precedes the proposal of (\hyperlink{tHooft93}{t'Hooft 93}, \hyperlink{Susskind94}{Susskind 94}), but this relation was not recognized from this perspective earlier.)


\item 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]].



\end{enumerate}
\begin{remark}
\label{}\hypertarget{}{}
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 (\hyperlink{Witten98}{Witten98}) it is discussed how the \hyperlink{SYMAds5}{SYM/IIB duality} is carried by the Chern-Simons term $\int B_{NS} \wedge d B_{RR}$ in the [[type II string theory]] action, the \hyperlink{6dAdS7}{6d(2,0)/AdS7 duality} - is induced by the Chern-Simons term $\int C_3 \wedge d C_3 \wedge d C_3$ of the [[11-dimensional supergravity]] action.

\end{remark}
Below at \emph{\href{Examples}{Examples}} we list some systems for which something along these lines is known.

\hypertarget{MoreDetails}{}\subsubsection*{{More details}}\label{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 (\hyperlink{Gawedzki99}{Gawdzki 99}):

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 $\mathbb{C}$):

\begin{displaymath}
dim Z_B(X) \lt \infty
  \,.
\end{displaymath}
Then for $\Sigma : \partial_{in}{\Sigma} \to \partial_{out}{\Sigma}$ any [[cobordism]] of dimension $n$, the [[correlator]]

\begin{displaymath}
Z_B(\Sigma) : Z_B(\partial_{in} \Sigma) \to Z_B(\partial_{out} \Sigma)
\end{displaymath}
that $Z_B$ assigns may naturally be identified, under the [[closed monoidal category|closed monoidal structure]] of [[Vect]], as an element

\begin{displaymath}
\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}
  \,.
\end{displaymath}
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 \emph{[[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 \href{}{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

\begin{enumerate}%
\item there is an [[isomorphism]]

\begin{displaymath}
Z_A(\Sigma) \simeq Z_B(\partial \Sigma) = Bl_B(\Sigma)
\end{displaymath}

\item such that whenever $\hat \Sigma$ cobounds $\Sigma$ the linear map



\end{enumerate}
\begin{displaymath}
Z_A : \mathbb{C} = Z_A(\hat \Sigma)
 \stackrel{Z_A(\hat \Sigma)}{\to}
  Z_A(\partial \hat \Sigma)
  \simeq
  Bl_B(\Sigma)
\end{displaymath}
sends $1 \in \mathbb{C}$ to $\overline{Z_B(\Sigma)}$.

If so, we say that $A$ is a \textbf{holographic dual} to $B$.

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

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{PoissonHolography}{}\subsubsection*{{Poisson holography}}\label{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

\begin{itemize}%
\item \emph{\href{http://ncatlab.org/nlab/show/off-shell+Poisson+bracket#BoundaryFieldTheoryInterpretation}{off-shell Poisson bracket -- boundary field theory interpretation}}.


\item \emph{\href{motivic+quantization#PoissonHolography}{motivic quantization -- Poisson holography}}.



\end{itemize}
\hypertarget{3dCS-2dCFT}{}\subsubsection*{{Holography of higher Chern-Simons/CFT-type}}\label{3dCS-2dCFT}

See also \emph{[[AdS3-CFT2 and CS-WZW correspondence]]}.

\hypertarget{rt3d_tqft__rational_2d_cft}{}\paragraph*{{RT-3d TQFT / rational 2d CFT}}\label{rt3d_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: \hyperlink{OrdinaryCSWZWModel}{ordinary Chern-Simons theory dual to a 2d WZW model} below.

For more see at \emph{[[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]].

\hypertarget{OrdinaryCSWZWModel}{}\paragraph*{{Ordinary Chern-Simons theory / WZW-model}}\label{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:

\begin{enumerate}%
\item 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 (\hyperlink{ArcioniBlauLoughlin}{Arcioni-Blau-Loughlin, p. 6}).


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

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

\begin{displaymath}
\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)
  \,,
\end{displaymath}
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 \emph{\href{Chern-Simons%20theory#GeometricQuantHigher}{Chern-Simons theory -- Geometric quantization -- In higher codimension}}.


\item 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 (\hyperlink{Witten}{Witten}). A review is in (\hyperlink{Gawedzki}{Gawedzki, section 5}).


\item 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 (\hyperlink{KapustinSaulina}{Kapustin-Saulina}, \hyperlink{FuchsSchweigertValentino}{Fuchs-Schweigert-Valentino}).



\end{enumerate}
\hypertarget{poisson_model__quantum_mechanics}{}\paragraph*{{Poisson $\sigma$-model / quantum mechanics}}\label{poisson_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 (\hyperlink{Kontsevich}{Kontsevich}) made explicit by (\hyperlink{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.)

\hypertarget{amodel__quantum_mechanics}{}\paragraph*{{A-model / quantum mechanics}}\label{amodel__quantum_mechanics}

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

See

\begin{itemize}%
\item [[quantization via the A-model]]

\end{itemize}
Notice that the A-model arises from the [[Poisson sigma-model]], as discussed there.

\hypertarget{HigherDimCSAndSelfDualQFT}{}\paragraph*{{Higher dimensional Chern-Simons theory / Self-dual higher gauge theory}}\label{HigherDimCSAndSelfDualQFT}

\hypertarget{idea_and_examples}{}\paragraph*{{Idea and examples}}\label{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).

\begin{itemize}%
\item $(k=0)$: ordinary 3-dimensional [[Chern-Simons theory]] is related to a [[string]] [[sigma-model]] on its boundary;


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


\item $(k=2)$: 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) (\hyperlink{BelovMoore}{BelovMoore}).



\end{itemize}
\hypertarget{some_details_2}{}\paragraph*{{Some details}}\label{some_details_2}

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

\begin{displaymath}
(\delta B_1, \delta B_2) \mapsto
  \int_\Sigma \delta B_1 \wedge \delta B_2
  \,.
\end{displaymath}
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]]

\begin{displaymath}
\star : \Omega^{2k+1}(\Sigma) \to \Omega^{2k+1}(\Sigma)
\end{displaymath}
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

\begin{displaymath}
\star \star B = (-1)^{(2k+1)(4k+3)} B
  = - B
  \,.
\end{displaymath}
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{displaymath}
\begin{aligned}
    \int_X B_1 \wedge B_2
    & = - \int_X B_1 \wedge \star \star B_2 
  \end{aligned}
  \,.
\end{displaymath}
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 \emph{imaginary self-dual} if

\begin{displaymath}
\star B = i B
\end{displaymath}
and \emph{imaginary anti-self-dual} if

\begin{displaymath}
\star B = - i B
  \,.
\end{displaymath}
Then for imaginary self-dual $B_1$ and $B_2$ we find that the symplectic pairing is

\begin{displaymath}
\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} 
  \,.
\end{displaymath}
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 $\Sigma$.

\hypertarget{holography_of_ads_gravitycfttype}{}\subsubsection*{{Holography of AdS gravity/CFT-type}}\label{holography_of_ads_gravitycfttype}

\hypertarget{SYMAds5}{}\paragraph*{{Type II on $AdS_5 \times S^5$ and $d = 4$ super Yang-Mills}}\label{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]].

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

\hypertarget{mtheory_on__and_chernsimons_on_m2_branes}{}\paragraph*{{M-theory on $AdS_4 \times S^7/\mathbb{Z}_k$ and Chern-Simons on M2 branes}}\label{mtheory_on__and_chernsimons_on_m2_branes}

See [[ABJM theory]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[holographic entanglement entropy]]


\item [[holographic principle of higher category theory]]


\item [[firewall problem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The idea of the holographic principle originates in

\begin{itemize}%
\item [[Gerard `t Hooft]], \emph{Dimensional Reduction in Quantum Gravity} (\href{http://arxiv.org/abs/gr-qc/9310026}{gr-qc/9310026}).

\end{itemize}
\begin{itemize}%
\item [[Leonard Susskind]], \emph{The World as a hologram} , J. Math. Phys. 36 (1995) 6377--6396, (\href{http://arxiv.org/abs/hep-th/9409089}{hep-th/9409089})

\end{itemize}
A review is

\begin{itemize}%
\item R. Bousso, \emph{The holographic principle}, Rev. Mod. Phys. \textbf{74} (2002) 825-874, (\href{http://www.ams.org/mathscinet-getitem?mr=1925130}{MR2003m:83048}, \href{http://link.aps.org/doi/10.1103/RevModPhys.74.825}{doi}, \href{http://arxiv.org/abs/hep-th/0203101}{arXiv:hep-th/0203101})

\end{itemize}
See also

\begin{itemize}%
\item \href{http://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/oxford_holography_group/holography_seminar/group_members.html}{Oxford Holography Group}, \emph{\href{http://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/oxford_holography_group/holography_seminar/material.html}{Background materian for holography}}

\end{itemize}
\hypertarget{adscft}{}\subsubsection*{{AdS/CFT}}\label{adscft}

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

\hypertarget{ReferencesCS-CFT}{}\subsubsection*{{Chern-Simons / CFT}}\label{ReferencesCS-CFT}

\hypertarget{on_the_level_of_action_functionals}{}\paragraph*{{On the level of action functionals}}\label{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

\begin{itemize}%
\item [[Edward Witten]] \emph{Quantum Field Theory and the Jones Polynomial} Commun. Math. Phys. 121 (3) (1989) 351--399. MR0990772 (\href{http://projecteuclid.org/euclid.cmp/1104178138}{EUCLID})

\end{itemize}
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

\begin{itemize}%
\item [[Krzysztof Gawędzki]], \emph{Conformal field theory: a case study} (\href{http://arxiv.org/abs/hep-th/9904145}{arXiv:hep-th/9904145})

\end{itemize}
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

\begin{itemize}%
\item Giovanni Arcioni, [[Matthias Blau]], Martin O'Loughlin, \emph{On the boundary dynamics of Chern-Simons gravity} (\href{http://arxiv.org/abs/hep-th/0210089}{arXiv:0210089})

\end{itemize}
(And many other references. )

\hypertarget{matching_of_spaces_of_states_to_conformal_blocks}{}\paragraph*{{Matching of spaces of states to conformal blocks}}\label{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

\begin{itemize}%
\item [[Edward Witten]] \emph{Quantum Field Theory and the Jones Polynomial} Commun. Math. Phys. 121 (3) (1989) 351--399. MR0990772 (\href{http://projecteuclid.org/euclid.cmp/1104178138}{project EUCLID})

\end{itemize}
A review is in section 5 of

\begin{itemize}%
\item [[Krzysztof Gawedzki]], \emph{Conformal field theory: a case study} (\href{http://arxiv.org/abs/hep-th/9904145}{arXiv:hep-th/9904145})

\end{itemize}
\hypertarget{RTAnd2dCFT}{}\paragraph*{{Reshetikhin-Turaev 3d TQFT and rational 2d CFT}}\label{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

\begin{itemize}%
\item Jens Fjelstad, J\"u{}rgen Fuchs, [[Ingo Runkel]], [[Christoph Schweigert]], \emph{Uniqueness of open/closed rational CFT with given algebra of open states} (\href{http://arxiv.org/abs/hep-th/0612306}{arXiv:hep-th/0612306}) .

\end{itemize}
The isomorphism between the RT-theory modular functor and the CFT conformal blocks is also discussed in

\begin{itemize}%
\item [[Jørgen Andersen]], [[Kenji Ueno]], \emph{Construction of the Reshetikhin-Turaev TQFT from conformal field theory} (\href{http://arxiv.org/abs/1110.5027}{arXiv:1110.5027})

\end{itemize}
Amplification of how the [[FRS formalism]] is inevitable once one adopts holography and [[QFT with defects]] is in

\begin{itemize}%
\item [[Anton Kapustin]], [[Natalia Saulina]], \emph{Surface operators in 3d TFT and 2d Rational CFT} in [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} AMS, 2011

\end{itemize}
More along these lines is in

\begin{itemize}%
\item [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], \emph{Bicategories for boundary conditions and for surface defects in 3-d TFT} (\href{http://arxiv.org/abs/1203.4568}{arXiv:1203.4568})

\end{itemize}
\hypertarget{selfdual_higher_gauge_fields_and_higher_abelian_chernsimons}{}\paragraph*{{Self-dual higher gauge fields and higher abelian Chern-Simons}}\label{selfdual_higher_gauge_fields_and_higher_abelian_chernsimons}

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

\begin{itemize}%
\item [[Edward Witten]], \emph{Five-brane effective action in M-Theory}, J. Geom. Phys. \textbf{22} (1997), no. 2, 103--133, \href{http://arxiv.org/abs/hep-th/9610234}{hep-th/9610234}


\item [[Edward Witten]], \emph{Duality relations among topological effects in string theory}, J. High Energy Phys. 2000, no. 5, Paper 31, 31 pp. \href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086}, \href{http://dx.doi.org/10.1088/1126-6708/2000/05/031}{doi}



\end{itemize}
More discussion of the general principle is in

\begin{itemize}%
\item Dmitriy Belov, [[Greg Moore]], \emph{Holographic action for the self-dual field}, \href{http://arxiv.org/abs/hep-th/0605038}{arXiv:hep-th/0605038}

\end{itemize}
A quick exposition of the basic idea is in

\begin{itemize}%
\item [[Jacques Distler]], \emph{Actions for self-dual gauge fields} (\href{http://golem.ph.utexas.edu/~distler/blog/archives/000809.html}{blog})

\end{itemize}
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

\begin{itemize}%
\item Dmitriy Belov, [[Greg Moore]], \emph{Type II Actions from 11-Dimensional Chern-Simons Theories} (\href{http://arxiv.org/abs/hep-th/0611020}{arXiv})

\end{itemize}
\hypertarget{poisson_modelamodel_and_quantum_mechanics}{}\paragraph*{{Poisson $\sigma$-model/A-model and quantum mechanics}}\label{poisson_modelamodel_and_quantum_mechanics}

\begin{itemize}%
\item [[Maxim Kontsevich]], \ldots{}

\end{itemize}
\begin{itemize}%
\item [[Alberto Cattaneo]], [[Giovanni Felder]], \ldots{}

\end{itemize}
\begin{itemize}%
\item [[Edward Witten]], \ldots{}

\end{itemize}
\hypertarget{3d_chernsimons_theory__2d_cft}{}\paragraph*{{3d Chern-Simons theory / 2d CFT}}\label{3d_chernsimons_theory__2d_cft}

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

\begin{itemize}%
\item Victor O. Rivelles, \emph{Holographic Principle and AdS/CFT Correspondence} (\href{http://arxiv.org/abs/hep-th/9912139}{arXiv})

\end{itemize}
\hypertarget{chernsimonscft_in_adscft}{}\paragraph*{{Chern-Simons/CFT in AdS/CFT}}\label{chernsimonscft_in_adscft}

In

\begin{itemize}%
\item [[Edward Witten]], \emph{AdS/CFT Correspondence And Topological Field Theory} JHEP 9812:012,1998 (\href{http://arxiv.org/abs/hep-th/9812012}{arXiv:hep-th/9812012})

\end{itemize}
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 $AdS_7/CFT_6$.

\hypertarget{black_hole__cft_correspondence}{}\subsubsection*{{Black hole / CFT correspondence}}\label{black_hole__cft_correspondence}

\begin{itemize}%
\item Monica Guica, Thomas Hartman, Wei Song, [[Andrew Strominger]], \emph{The Kerr/CFT Correspondence} (\href{http://arxiv.org/abs/0809.4266}{arXiv:0809.4266})


\item Alejandra Castro, Alexander Maloney, [[Andrew Strominger]], \emph{Hidden Conformal Symmetry of the Kerr Black Hole} (\href{http://arxiv.org/abs/1004.0996}{arXiv:1004.0996})



\end{itemize}
\hypertarget{general_abstract_formulation}{}\subsubsection*{{General abstract formulation}}\label{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

\begin{itemize}%
\item [[Chris Schommer-Pries]], \emph{Topological defects and classifying local topological field theories in low dimension} ([[SchommerPriesDefects.pdf:file]])

\end{itemize}
See [[holographic principle of higher category theory]] for more on this.

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

\begin{itemize}%
\item [[Dan Freed]], \emph{[[4-3-2 8-7-6]]}, talk at \emph{\href{https://people.maths.ox.ac.uk/tillmann/ASPECTS.html}{ASPECTS of Topology}} Dec 2012

\end{itemize}
[[!redirects holographic duality]]

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



\end{document}