# nLab AdS3-CFT2 and CS-WZW correspondence

Contents

### Context

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

One incarnation of the holographic principle in quantum field theory is the correspondence between 3d $G$-Chern-Simons theory as the bulk field theory and the 2d Wess-Zumino-Witten model on a suitable Lie group $G$ as the boundary field theory. This case stands out in that it was known and understood already before the holographic principle was formulated as such, motivated from bulk field theories of gravity. Notably the CS/WZW correspondence is an actual theorem instead of just a vague conjecture, as for much of the AdS-CFT correspondence.

Indeed, the natural equivalence between the space of quantum states of Chern-Simons theory on a surface $\Sigma$ and the space of conformal blocks of the WZW model on $\Sigma$ was understood in the seminal article (Witten 89) and subsequently discussed in much detail, see also at CS-theory – References. The explicit holographic correspondence between the wavefunctions of Chern-Simons theory and the correlators of the WZW model is reviewed for instance in (Gawędzki 99, around p. 30). For the case of abelian gauge group and with an eye towards generalization to self-dual higher gauge theory a review is in (Witten 96, section 2). (This correspondence is captured functorially by the notion of the modular functor of the 2d theory, see there for more.)

For instance the FRS formalism constructs all rational conformal field theories as full FQFTs holographically from the Reshetikhin-Turaev construction of the 3d Chern-Simons theory and fully classifies them this way.

Later the AdS-CFT correspondence came to be understood as a canonical or default implementation of the holographic principle. Here the bulk field theory is a theory of 3d quantum gravity which is very much like traditional Chern-Simons theory but may crucially differ from it, see at Chern-Simons gravity the comments on the non-perturbative regime. Instead some variant of CS3/WTW2 appears as one “sector” inside AdS3/CFT2, this is discussed in (Gukov-Martinec-Moore-Strominger 04).

But notice that also plain Chern-Simons theory is a string theory, but of topological strings. For more on this see at TCFT the section Worldsheet and effective background theories.

A general argument that in sectors of the AdS-CFT correspondence the conformal blocks on the CFT-side are given just by the higher dimensional Chern-Simons theory-sector inside the dual gravity theory is in (Witten98). This applies notably to the duality between 7-dimensional Chern-Simons theory and the conformal blocks in the 6d (2,0)-superconformal QFT on the M5-brane.

## References

### CS/WZW

The original article on the CS/WZW correspondence is

Reviews include

The relation of this to the AdS-CFT correspondence is discussed in

A general argument about the relation between AdS/CFT duality and infinity-Chern-Simons theory is in

An argument (via Chern-Simons gravity, but see the caveats there) that 3d quantum gravity with negative cosmological constant has as boundary field theory 2d Liouville theory is due to

• O. Coussaert, Marc Henneaux, P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961-2966 (arXiv:gr-qc/9506019)

Discussion of the Ising model 2d CFT as a boundary theory to a 3d TQFT based on the Turaev-Viro model, and the phenomenon of Kramers-Wannier duality, is discussed in

### $AdS_3$/$CFT_2$

Last revised on June 4, 2018 at 03:52:31. See the history of this page for a list of all contributions to it.