nLab AdS3-CFT2 and CS-WZW correspondence

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.

Related concepts

• p-adic AdS/CFT correspondence

• 3d quantum gravity

• BTZ black hole

• quantization of Chern-Simons theory/quantization of loop groups

• super 1-brane in 3d

• TT deformation

References

CS/WZW

The original article on the CS/WZW correspondence is

• Edward Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 3 (1989) 351399 [euclid:.cmp/1104178138, doi:10.1007/BF01217730, MR0990772]

More details worked out:

• Daniel C. Cabra, Gerardo L. Rossini, Explicit connection between conformal field theory and 2+1 Chern-Simons theory, Mod. Phys. Lett. A 12 (1997) 1687-1697 [arXiv:hep-th/9506054, doi:10.1142/S0217732397001722]

  (with motivation from Laughlin wavefunctions for anyons in condensed matter theory)

Reviews:

• Krzysztof Gawędzki, Conformal field theory: a case study, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) Conformal Field Theory – New Non-perturbative Methods In String And Field Theory, CRC Press (2000) [arXiv:hep-th/9904145, doi:10.1201/9780429502873]

• Edward Witten, section 2 of Five-Brane Effective Action In M-Theory J. Geom. Phys. 22: 103-133, 1997 (arXiv:hep-th/9610234)

The relation of this $CS_3/WZW_2$-duality to the AdS-CFT correspondence is discussed in

• Sergei Gukov, Emil Martinec, Gregory Moore, Andrew Strominger, Chern-Simons Gauge Theory and the $AdS_3/CFT_2$ Correspondence, in: Mikhail Shifman et al. (eds.) From fields to strings, vol. 2, 1606-1647, 2004 (arXiv:hep-th/0403225)

• Kristan Jensen, Chiral anomalies and AdS/CMT in two dimensions, JHEP 1101:109,2011 (arXiv:1012.4831)

• Per Kraus, Finn Larsen, Partition functions and elliptic genera from supergravity, JHEP 0701:002, 2007 (arXiv:hep-th/0607138)

• Per Kraus, Lectures on black holes and the $AdS_3/CFT_2$ correspondence, Lect. Notes Phys. 755: 193-247, 2008 (arXiv:hep-th/0609074)

• Ville Keranen, Chern-Simons interactions in AdS3 and the current conformal block (arXiv:1403.6881)

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

• Edward Witten, AdS/CFT Correspondence And Topological Field Theory JHEP 9812:012,1998 (arXiv:hep-th/9812012)

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

• Daniel Freed, Constantin Teleman, Topological dualities in the Ising model (arXiv:1806.00008)

Via factorization algebras:

• Owen Gwilliam, Eugene Rabinovich, Brian R. Williams, Factorization algebras and abelian CS/WZW-type correspondences, Pure and Applied Mathematics Quarterly 18 4 (2022) 1485–1553 [arXiv:2001.07888, doi:10.4310/PAMQ.2022.v18.n4.a7]

Formulation in homotopical AQFT:

• Marco Benini, Alastair Grant-Stuart, Alexander Schenkel, The linear CS/WZW bulk/boundary system in AQFT, Annales Henri Poincaré (2023) [arXiv:2302.06990]

3d Gravity and Chern-Simons theory

On 3-dimensional (quantum) gravity (general relativity) with cosmological constant, and its (non-)relation to Chern-Simons theory with non-compact gauge groups:

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):

• Stanley Deser, Roman Jackiw, Gerard 't Hooft, Three-dimensional Einstein gravity: Dynamics of flat space, Ann. Phys. 152 (1984) 220 (doi:10.1016/0003-4916(84)90085-X)

• Stanley Deser, Roman Jackiw, Three-dimensional cosmological gravity: Dynamics of constant curvature, Annals of Physics, Volume 153, Issue 2, 1 April 1984, Pages 405-416 (doi:10.1016/0003-4916(84)90025-3, spire:192694)

• Roman Jackiw, Lower dimensional gravity, Nuclear Physics B Volume 252, 1985, Pages 343-356 (doi:10.1016/0550-3213(85)90448-1, spire:204694)

• J. D. Brown, Marc Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity, Commun. Math. Phys. (1986) 104: 207 (doi:10.1007/BF01211590)

• A. Achucarro, Paul Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories, Phys. Lett. B180 (1986) 89 (doi:10.1016/0370-2693(86)90140-1, spire:21208)

• Steven Carlip, Inducing Liouville theory from topologically massive gravity, Nuclear Physics B Volume 362, Issues 1–2, 16 September 1991, Pages 111-124 (doi:10.1016/0550-3213(91)90558-F)

• 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)

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. B 311 (1988) 46. [doi:10.1016/0550-3213(88)90143-5

• Herman Verlinde, Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space, Nuclear Physics B 337 3 (1990) 652-680 [doi:10.1016/0550-3213(90)90510-K]

Review:

• Steven Carlip, Lectures in (2+1)-Dimensional Gravity, J. Korean Phys. Soc. 28: S447-S467, 1995 (arXiv:gr-qc/9503024)

• Steven Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge Monographs on Mathematical Physics (2003) (publisher)

• Steven Carlip, Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe, Living Rev. Rel. 8:1, 2005 (arXiv:gr-qc/0409039)

• Steven Carlip, My Research – (2+1)-Dimensional quantum gravity

• Laura Donnay, Asymptotic dynamics of three-dimensional gravity, Proceedings of Eleventh Modave Summer School in Mathematical Physics, POS 271 (2016) $[$arXiv:1602.09021, doi:10.22323/1.271.0001$]$

• Wout Merbis, Chern-Simons-like Theories of Gravity (arXiv:1411.6888)

• Chen-Te Ma, $AdS_3$ Einstein Gravity and Boundary Description: Pedagogical Review $[$arXiv:2310.04665$]$

Further developments:

• Edward Witten, Three-dimensional gravity revisited (2007) [arXiv:0706.3359]

• Paul Townsend, Massive 3d (super)gravity, slides, (pdf)

• Gaston Giribet, Black hole physics and AdS3/CFT2 correspondence, lectures at Croatian Black Hole School 2010

• Alan Garbarz, Gaston Giribet, Yerko Vásquez, Asymptotically AdS$_3$ solutions to topologically massive gravity at special values of the coupling constants, arxiv/0811.4464

• Rudranil Basu, Samir K Paul, Consistent 3D Quantum Gravity on Lens Spaces (arXiv:1109.0793)

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

• Viraj Meruliya, Sunil Mukhi, Palash Singh, Poincaré Series, 3d Gravity and Averages of Rational CFT (arXiv:2102.03136)

• Scott Collier, Lorenz Eberhardt, Mengyang Zhang, 3d gravity and Chern-Simons theory – references $[$arXiv:2304.13650$]$

See also:

• Nathan Benjamin, Scott Collier, Alexander Maloney, Pure Gravity and Conical Defects, Journal of High Energy Physics 2020 34 (2020) $[$arXiv:2004.14428, doi:10.1007/JHEP09(2020)034$]$

$AdS_3$/$CFT_2$

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

• Andrea Prinsloo, Vidas Regelskis, Alessandro Torrielli, Integrable open spin-chains in AdS3/CFT2 (arXiv:1505.06767)

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

• Lorenz Eberhardt, Matthias Gaberdiel, String theory on $AdS_3$ and the symmetric orbifold of Liouville theory (arXiv:1903.00421)

• Lorenz Eberhardt, Matthias Gaberdiel, Strings on $AdS_3 \times S^3 \times S^3 \times S^1$ (arXiv:1904.01585)

• Lorenz Eberhardt, Matthias Gaberdiel, Rajesh Gopakumar, Deriving the $AdS_3/CFT_2$ Correspondence (arXiv:1911.00378)

• Andrea Dei, Lorenz Eberhardt, Correlators of the symmetric product orbifold (arXiv:1911.08485)

based on

• Shouvik Datta, Lorenz Eberhardt, Matthias Gaberdiel, Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$ JHEP 1801 (2018) 146 (arXiv:1709.06393)

See also

• Lorenz Eberhardt, $AdS_3/CFT_2$ at higher genus (arXiv:2002.11729)

• Lorenz Eberhardt, A perturbative CFT dual for pure NS-NS $AdS_3$ strings, J. Phys. A: Math. Theor. 55 064001 [arXiv:2110.07535, arXiv:10.1088/1751-8121/ac47b2]

• Kiarash Naderi, DDF operators in the Hybrid Formalism [arXiv:2208.01617]

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

• Yuri Manin, Matilde Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5 (2002) 617-650 (arXiv:hep-th/0201036)

In the context of holography as Koszul duality:

• Kevin Costello, Natalie Paquette, Twisted Supergravity and Koszul Duality: A case study in $AdS_3$ (arXiv:2001.02177)

Generalization to boundary field theory:

• Sanjit Shashi, Quotient-AdS/BCFT: Holographic Boundary $CFT_2$ on $AdS_3$ Quotients (arXiv:2005.10244)

• Tadashi Takayanagi, Takahiro Uetoko, Chern-Simons Gravity Dual of BCFT (arXiv:2011.02513)

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)

Relating to random matrix theory:

• Gabriele Di Ubaldo, Eric Perlmutter, $AdS_3/RMT_2$ Duality [arXiv:2307.03707]

$AdS_3/CFT_2$ on D1/D5 branes

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

• Juan Maldacena, Section 4 of: The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998 (hep-th/9711200)

• Ofer Aharony, Steven Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz, Section 5 of: Large $N$ Field Theories, String Theory and Gravity, Phys. Rept. 323:183-386, 2000 (arXiv:hep-th/9905111)

• Gautam Mandal, A review of the D1/D5 system and five dimensional black hole from supergravity and brane viewpoint (arXiv:hep-th/0002184)

• Robbert Dijkgraaf, Juan Maldacena, Gregory Moore, Erik Verlinde, A Black Hole Farey Tail (arXiv:hep-th/0005003, spire:526744)

• E. Gava, A.B. Hammou, J.F. Morales, K.S.Narain, AdS/CFT correspondence and D1/D5 systems in theories with 16 supercharges, JHEP 0103:035, 2001 (arXiv:hep-th/0102043)

• Per Kraus, Section 4 of: Lectures on black holes and the $AdS_3/CFT_2$ correspondence, Lect. Notes Phys. 755: 193-247, 2008 (arXiv:hep-th/0609074)

$AdS_3/CFT_2$ 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:

• Giuseppe Dibitetto, Nicolò Petri, 6d surface defects from massive type IIA, JHEP 01 (2018) 039 (arxiv:1707.06154)

• Nicolò Petri, section 6.5 of: Supersymmetric objects in gauged supergravities (arxiv:1802.04733)

• Nicolò Petri, Surface defects in massive IIA, talk at Recent Trends in String Theory and Related Topics 2018 (pdf)

• Giuseppe Dibitetto, Nicolò Petri, $AdS_3$ vacua and surface defects in massive IIA (arxiv:1904.02455)

• Yolanda Lozano, Niall Macpherson, Carlos Nunez, Anayeli Ramirez, $1/4$ BPS $AdS_3/CFT_2$ (arxiv:1909.09636)

• Yolanda Lozano, Niall Macpherson, Carlos Nunez, Anayeli Ramirez, Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA (arxiv:1909.10510)

• Yolanda Lozano, Niall Macpherson, Carlos Nunez, Anayeli Ramirez, $AdS_3$ solutions in massive IIA, defect CFTs and T-duality (arxiv:1909.11669)

• Kostas Filippas, Non-integrability on $AdS_3$ supergravity (arxiv:1910.12981)

See also

• Andrea Legramandi, Niall Macpherson, $AdS_3$ solutions with $\mathcal{N}=(3,0)$ from $S^3 \timesS^3$ fibrations, (arXiv:1912.10509

Wilson lines computing holographic entropy in $AdS_3/CFT_2$

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 $AdS_3$, 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 $AdS_3$ Chern-Simons Gravity Theory (arXiv:1911.03841)

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

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

and similarly for 3d flat-space holography:

• Arjun Bagchi, Rudranil Basu, Daniel Grumiller, Max Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114, 111602 (2015) (arXiv 1410.4089)

• Rudranil Basu, Max Riegler, Wilson Lines and Holographic Entanglement Entropy in Galilean Conformal Field Theories, Phys. Rev. D 93, 045003 (2016) (arXiv:1511.08662)

• Wout Merbis, Max Riegler, Geometric actions and flat space holography (arXiv:1912.08207)

Discussion for 3d de Sitter spacetime:

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