# CS/WZW correspondence The [[0089 Chern-Simons theory|Chern-Simons]]/[[0601 Weiss-Zumino-Witten models|Weiss-Zumino-Witten]] (CS/WZW) correspondence is a very important example of a duality between two theories living in different dimensions. Some variant of CS$_3$/WZW$_2$ appears as a sector inside the [[0073 AdS3-CFT2|AdS3/CFT2]] correspondence, but the nature of this correspondence is very different from [[0001 AdS-CFT|AdS/CFT]]. The relation between the two theories can be summarised by saying that the Hilbert space of CS theory is identified with the space of [[0031 Conformal block|conformal blocks]] of the WZW model. In this case, there is a path integral formulation of the 3D theory, and we can think of the vector space of conformal blocks as the space of boundary conditions for the 3d path integral. More generally, there is not necessarily a path integral formulation of any 3d theory. Nevertheless, it is sufficient to define a TQFT using [[0602 Moore-Seiberg construction|Moore-Seiberg data]], or more formally a modular tensor category. ## A natural choice of basis 3d handlebodies prepare a natural basis for the states in the Hilbert space $\Sigma$. These handlebodies come with Wilson lines along each solid cylinder section. ## Refs - original - [[1989#Witten]] - canonical quantisation - [[1989#Witten]] - [[1989#Elitzur, Moore, Schwimmer, Seiberg]] - CS$_3$/WZW$_2$ as a sector of AdS$_3$/CFT$_2$: - [[2004#Gukov, Martinec, Moore, Strominger (Mar, b)]] - modular tensor category approach - [[1991#Reshetikhin, Turaev]] - [[1994#Turaev (Book)]] - string net (alternative to Turaev-Viro) - [[2022#Bartlett]] ## Comments - 3d *quantum* gravity is very much like a CS but differs crucially from it - see page 4-6 in [[2007#Witten (Jun)]] - non-perturbatively very different ## Related dualities If we constrain the CS/WZW duality, we get a relation between 3D gravity and 2D Liouville. At finite $c$, this is realised by [[0596 Virasoro TQFT|Virasoro TQFT]]. The relation between 2D BF theory and 1D particle on a group is the dimensional reduced version of the relation between 3D Chern-Simons and 2D WZW. The constrained version is a duality between 2D JT gravity and 1D Schwarzian theory (see [[0655 JT-Schwarzian correspondence|JT-Schwarzian duality]]).