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The AGT correspondence (AGT 09) is a relation between the partition function of $SU(2)^{n+3g-3}$-N=2 D=4 super Yang-Mills theory and Liouville theory on an $n$-punctured Riemann surface $C_{g,n}$ of genus $g$ (from which the super Yang-Mills theory is obtained by compactifying the worldvolume 6d (2,0)-supersymmetric QFT of two M5-branes, see at N=2 D=4 super Yang-Mills theory, the section Construction by compactification).
More generally, this construction yields something like a decomposition of the 6d (2,0)-superconformal QFT into a 2d SCFT “with values in 4d SYM field theory” (e.g. Tachikawa 10, slide 25 (33 of 54)). Hence composition with any kind of suitable invariant of the 4d field theories yields an actual 2d SCFT, for instance taking the superconformal index in 4d yields a 2d TQFT (GPRR 10). In this picture of “4d-SYM field theory-valued 2d SCFT” one has the following correspondences:
the complex structure in 2d is the coupling constants and theta angles etc in the 4d super Yang-Mills theory;
the mapping class group (large conformal transformations) in 2d is the (generalized) S-duality of the 4d theory.
The original articles are
Davide Gaiotto, $N=2$ dualities (arXiv:0904.2715)
Luis F. Alday, Davide Gaiotto, Yuji Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories (arXiv:0906.3219)
The 2d TQFT obtained from this by forming the 4d index is discussed in
Brief surveys include
Yuji Tachikawa, M5-branes, 4d gauge theory and 2d CFT, 2010 (pdf)
Abhijit Gadde, $\mathcal{N}= 2$ Dualities and 2d TQFT 2012 (pdf)
Nikolay Bovev, New SCFTs from wrapped branes, 2013 (pdf)
A detailed review is in
See also
Alexander Belavin, M. A. Bershtein, B. L. Feigin, A. V. Litvinov, G. M. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, http://arxiv.org/abs/1111.2803
Volker Schomerus, Paulina Suchanek, Liouville’s imaginary shadow, arxiv/1210.1856
A.Mironov, A.Morozov, The power of Nekrasov functions, arxiv/0908.2190
D. Galakhov, A. Mironov, A. Morozov, S-duality as a beta-deformed Fourier transform, arxiv/1205.4998
A. Mironov, Spectral duality in integrable systems from AGT conjecture, arxiv/1204.0913
A. Belavin, V. Belavin, AGT conjecture and integrable structure of conformal field theory for $c=1$, Nucl.Phys.B850:199-213 (2011) arxiv/1102.0343
A. Belavin, V. Belavin, M. Bershtein, Instantons and 2d Superconformal field theory, arxiv/1106.4001
Kazunobu Maruyoshi, Quantum integrable systems, matrix models, and AGT correspondence, seminar slides
Giulio Bonelli, Alessandro Tanzini, Hitchin systems, N=2 gauge theories and W-gravity, arxiv/0909.4031
Giulio Bonelli, Kazunobu Maruyoshi, Alessandro Tanzini, Quantum Hitchin systems via beta-deformed matrix models, arxiv/1104.4016
Oscar Chacaltana, Jacques Distler, Tinkertoys for Gaiotto Duality, JHEP 1011:099,2010, (arXiv:1008.5203, [chacaltana:2010ks,MR3046557])
Satoshi Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator, arXiv/1408.4132
The AGT correspondence is treated with the help of a Riemann-Hilbert problem in
Last revised on January 3, 2018 at 02:45:47. See the history of this page for a list of all contributions to it.