nLab AdS3-CFT2 and CS-WZW correspondence



Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Duality in string theory



One incarnation of the holographic principle in quantum field theory is the correspondence between 3d GG-Chern-Simons theory as the bulk field theory and the 2d Wess-Zumino-Witten model on a suitable Lie group GG 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 Σ\Sigma and the space of conformal blocks of the WZW model on Σ\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.



The original article on the CS/WZW correspondence is

More details worked out:


The relation of this CS 3/WZW 2CS_3/WZW_2-duality 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

3d Gravity and Chern-Simons theory

The original articles on 3d gravity, discussing its formulation as a Chern-Simons theory and discovering its holographic relation to a 2d CFT boundary field theory (well before AdS/CFT was conceived from string theory):

The corresponding non-perturbative quantization of 3-dimensional gravity, via quantization of 3d Chern-Simons theory:

  • Edward Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System Nucl. Phys. B311 (1988) 46. (web)

  • Herman Verlinde, Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space, Nuclear Physics B

    Volume 337, Issue 3, 25 June 1990, Pages 652-680 (doi:10.1016/0550-3213(90)90510-K)


Further developments:

AdS 3AdS_3/CFT 2CFT_2

Discussion of AdS/CFT correspondence for 3d gravity/2d CFT:

An exact correspondence of the symmetric orbifold CFT of Liouville theory with a string theory on AdS 3AdS_3 is claimed in:

based on

See also

Relation of AdS3/CFT2 to hyperbolic geometry and Arakelov geometry of algebraic curves:

In the context of holography as Koszul duality:

Generalization to boundary field theory:

See also:

  • Stefano Speziali, Spin 2 fluctuations in 1/4 BPS AdS3/CFT2 (arxiv:1910.14390)

  • Bruno Balthazar, Amit Giveon, David Kutasov, Emil J. Martinec, Asymptotically Free AdS3/CFT2 (arXiv:2109.00065)

AdS 3/CFT 2AdS_3/CFT_2 on D1/D5 branes

On AdS3-CFT2 for D1/D5 brane bound states and black hole entropy in string theory:

AdS 3/CFT 2AdS_3/CFT_2 on D2/D4-D6/D8 branes

On black\;D6-D8-brane bound states in massive type IIA string theory, with defect D2-D4-brane bound states inside them realizing AdS3-CFT2 as defect field theory “inside” AdS7-CFT6:

See also

Wilson lines computing holographic entropy in AdS 3/CFT 2AdS_3/CFT_2

Discussion of BTZ black hole entropy and more generally of holographic entanglement entropy in 3d quantum gravity/AdS3/CFT2 via Wilson line observables in Chern-Simons theory:

  • Martin Ammon, Alejandra Castro, Nabil Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110 (arXiv:1306.4338)

  • Jan de Boer, Juan I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS 3AdS_3, JHEP 1404:089, 2014 (arXiv:1306.4347)

  • Alejandra Castro, Stephane Detournay, Nabil Iqbal, Eric Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 (arXiv:1405.2792)

  • Mert Besken, Ashwin Hegde, Eliot Hijano, Per Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099 (arXiv:1603.07317)

  • Andreas Blommaert, Thomas G. Mertens, Henri Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (arXiv:1806.07765)

  • Ashwin Dushyantha Hegde, Role of Wilson Lines in 3D Quantum Gravity, 2019 (spire:1763572)

  • Xing Huang, Chen-Te Ma, Hongfei Shu, Quantum Correction of the Wilson Line and Entanglement Entropy in the AdS 3AdS_3 Chern-Simons Gravity Theory (arXiv:1911.03841)

  • Eric D'Hoker, Per Kraus, Gravitational Wilson lines in AdS 3AdS_3 (arXiv:1912.02750)

  • Marc Henneaux, Wout Merbis, Arash Ranjbar, Asymptotic dynamics of AdS 3AdS_3 gravity with two asymptotic regions (arXiv:1912.09465)

and similarly for 3d flat-space holography:

Discussion for 3d de Sitter spacetime:

  • Alejandra Castro, Philippe Sabella-Garnier, Claire Zukowski, Gravitational Wilson Lines in 3D de Sitter (arXiv:2001.09998)

Last revised on February 12, 2023 at 10:46:00. See the history of this page for a list of all contributions to it.