functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
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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 $n + 1$.
holographic principle in quantum field theory
bulk field theory | boundary field theory |
---|---|
dimension $n+1$ | dimension $n$ |
field | source |
wave function | correlation function |
space of quantum states | conformal blocks |
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
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, 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:
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 CFTs (conformal blocks) 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) and hence precedes the proposal of (t’Hooft 93, Susskind 94), but this relation was not recognized from this perspective earlier.)
Systems of quantum gravity in various dimensions as given by string theory on asymptotically anti de Sitter spacetimes have been checked not in total but in a multitude of special aspects in special cases to be dual to supersymmetric CFTs on their asymptotic boundary – this is called AdS/CFT correspondence.
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 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) it is discussed how the 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 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.
Below at Examples we list some systems for which something along these lines is known.
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 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):
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 $\mathbb{C}$):
Then for $\Sigma : \partial_{in}{\Sigma} \to \partial_{out}{\Sigma}$ any cobordism of dimension $n$, the correlator
that $Z_B$ assigns may naturally be identified, under the closed monoidal structure of Vect, as an element
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 blocks 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 blocks”, 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 TQFTs $B$ 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
there is an isomorphism
such that whenever $\hat \Sigma$ cobounds $\Sigma$ the linear map
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 states of $A$ over $\Sigma$, while $Bl_B(\Sigma)$ is the space of possible correlators of $B$ over $\Sigma$. Under holography, the states of $A$ are identified with the correlators of $B$.
One of the key statements of the holographic principle is that 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 Poisson sigma-model. This theorem is discussed at
See also AdS3-CFT2 and CS-WZW correspondence.
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 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 blocks 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.
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:
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).
More abstractly, at least for simply connected compact $G$, the action functionals are also related by transgression of moduli stacks as discussed at infinity-Chern-Simons theory. The action functional of $G$-Chern-Simons theory is induced by the morphism
from the smooth moduli stack of $G$-bundles with connection to the smooth moduli 3-stack of 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
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.
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). A review is in (Gawedzki, section 5).
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, Fuchs-Schweigert-Valentino).
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) made explicit by (CattaneoFelder).
(Notice the Poisson sigma-model is the $(n = 2)$-case of the AKSZ sigma-model which is indeed an example of a infinity-Chern-Simons theory, as discussed there.)
Similarly the A-model on certain D-branes gives a holographic description of ordinary quantum mechanics. (Witten).
See
Notice that the A-model arises from the Poisson sigma-model, as discussed there.
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).
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 flat $2k+1$-forms on $\Sigma$. The presymplectic form on this space is given by the pairing
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
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
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
Here on the right now the 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 $\pm i$-eigenspaces of $\star$: say $B \in \Omega^{2k+1}(\Sigma)$ is imaginary self-dual if
and imaginary anti-self-dual if
Then for imaginary self-dual $B_1$ and $B_2$ we find that the symplectic pairing is
Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into Lagrangian subspaces.
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 $\Sigma$.
Conjecturally, type II string theory on a anti-de Sitter space background is holographically dual to super Yang-Mills theory on the asymptotic boundary.
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.
See ABJM theory.
The idea of the holographic principle originates in
A review is
See also
See the references at AdS/CFT correspondence.
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
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
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
(And many other references. )
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
A review is in section 5 of
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
The isomorphism between the RT-theory modular functor and the CFT conformal blocks is also discussed in
Amplification of how the FRS formalism is inevitable once one adopts holography and QFT with defects is in
More along these lines is in
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
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, doi
More discussion of the general principle is in
A quick exposition of the basic idea is in
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
3-dimensional Chern-Simons theory in the context of holography is discussed for instance in
In
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 blocks 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 $AdS_7/CFT_6$.
Monica Guica, Thomas Hartman, Wei Song, Andrew Strominger, The Kerr/CFT Correspondence (arXiv:0809.4266)
Alejandra Castro, Alexander Maloney, Andrew Strominger, Hidden Conformal Symmetry of the Kerr Black Hole (arXiv:1004.0996)
An identification of boundary conditions and defects as natural transformations between higher dimensional FQFTs is discussed in
See holographic principle of higher category theory for more on this.
Further discussion of formalization in extended TQFT is in