Abstract
We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional \( \mathcal{N} = 2 \) SU(2)n+3g−3 gauge theories with generalized quiver diagrams. We obtain the generalized matrix models from the perturbative evaluation of the Liouville correlation functions and verify the consistency of the description with respect to degenerations of the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the \( \mathcal{N} = 2 \) gauge theory as the spectral curve of the generalized matrix model, thus providing a check of AGT correspondence at all genera.
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A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Self-dual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [SPIRES].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [SPIRES].
D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [SPIRES].
D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [SPIRES].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [SPIRES].
A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].
V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, JETP Lett. 90 (2009) 708 [arXiv:0911.0363] [SPIRES].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [SPIRES].
V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [SPIRES].
G. Giribet, On triality in N = 2 SCFT with N f = 4, JHEP 01 (2010) 097 [arXiv:0912.1930] [SPIRES].
R. Santachiara and A. Tanzini, Moore-Read fractional quantum Hall wavefunctions and SU(2) quiver gauge theories, Phys. Rev. D 82 (2010) 126006 [arXiv:1002.5017] [SPIRES].
L. Hadasz, Z. Jaskolski and P. Suchanek, Proving the AGT relation for N f = 0, 1, 2 antifundamentals, JHEP 06 (2010) 046 [arXiv:1004.1841] [SPIRES].
D. Nanopoulos and D. Xie, N = 2 generalized superconformal quiver gauge theory, arXiv:1006.3486 [SPIRES].
V. Alba and A. Morozov, Check of AGT relation for conformal blocks on sphere, Nucl. Phys. B 840 (2010) 441 [arXiv:0912.2535] [SPIRES].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [SPIRES].
D. Nanopoulos and D. Xie, Hitchin equation, singularity and N = 2 superconformal field theories, JHEP 03 (2010) 043 [arXiv:0911.1990] [SPIRES].
S. Kanno, Y. Matsuo, S. Shiba and Y. Tachikawa, N = 2 gauge theories and degenerate fields of Toda theory, Phys. Rev. D 81 (2010) 046004 [arXiv:0911.4787] [SPIRES].
M. Taki, On AGT conjecture for pure super Yang-Mills and W-algebra, JHEP 05 (2011) 038 [arXiv:0912.4789] [SPIRES].
D. Nanopoulos and D. Xie, Hitchin equation, irregular singularity and N = 2 asymptotical free theories, arXiv:1005.1350 [SPIRES].
S. Kanno, Y. Matsuo and S. Shiba, Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver, Phys. Rev. D 82 (2010) 066009 [arXiv:1007.0601] [SPIRES].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [SPIRES].
J.-F. Wu and Y. Zhou, From Liouville to Chern-Simons, alternative realization of Wilson loop operators in AGT duality, arXiv:0911.1922 [SPIRES].
N. Drukker, D. Gaiotto and J. Gomis, The virtue of defects in 4D gauge theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [SPIRES].
F. Passerini, Gauge theory Wilson loops and conformal Toda field theory, JHEP 03 (2010) 125 [arXiv:1003.1151] [SPIRES].
C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [SPIRES].
L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [SPIRES].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, arXiv:1006.0977 [SPIRES].
M. Taki, Surface operator, bubbling Calabi-Yau and AGT relation, arXiv:1007.2524 [SPIRES].
H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a surface operator, irregular conformal blocks and open topological string, arXiv:1008.0574 [SPIRES].
C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N) conformal blocks from N = 2 SU(N) gauge theories, JHEP 01 (2011) 045 [arXiv:1008.1412] [SPIRES].
J. Gomis and B. LeFloch, ’t Hooft operators in gauge theory from Toda CFT, arXiv:1008.4139 [SPIRES].
K. Hosomichi, S. Lee and J. Park, AGT on the S-duality wall, JHEP 12 (2010) 079 [arXiv:1009.0340] [SPIRES].
N. Wyllard, W-algebras and surface operators in N = 2 gauge theories, J. Phys. A 44 (2011) 155401 [arXiv:1011.0289] [SPIRES].
Y. Yamada, A quantum isomonodromy equation and its application to N = 2 SU(N) gauge theories, J. Phys. A 44 (2011) 055403 [arXiv:1011.0292] [SPIRES].
K. Maruyoshi and M. Taki, Deformed prepotential, quantum integrable system and Liouville field theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [SPIRES].
G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [SPIRES].
L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [SPIRES].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [SPIRES].
H. Itoyama, K. Maruyoshi and T. Oota, The quiver matrix model and 2d-4d conformal connection, Prog. Theor. Phys. 123 (2010) 957 [arXiv:0911.4244] [SPIRES].
R. Schiappa and N. Wyllard, An A r threesome: matrix models, 2d CFTs and 4d N = 2 gauge theories, arXiv:0911.5337 [SPIRES].
A. Mironov, A. Morozov and S. Shakirov, Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [SPIRES].
A. Mironov, A. Morozov and S. Shakirov, Conformal blocks as Dotsenko-Fateev integral discriminants, Int. J. Mod. Phys. A 25 (2010) 3173 [arXiv:1001.0563] [SPIRES].
H. Itoyama and T. Oota, Method of generating q-expansion coefficients for conformal block and N = 2 Nekrasov function by beta-deformed matrix model, Nucl. Phys. B 838 (2010) 298 [arXiv:1003.2929] [SPIRES].
M. Fujita, Y. Hatsuda and T.-S. Tai, Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model, JHEP 03 (2010) 046 [arXiv:0912.2988] [SPIRES].
P. Sulkowski, Matrix models for β-ensembles from Nekrasov partition functions, JHEP 04 (2010) 063 [arXiv:0912.5476] [SPIRES].
A. Mironov, A. Morozov and A. Morozov, Conformal blocks and generalized Selberg integrals, Nucl. Phys. B 843 (2011) 534 [arXiv:1003.5752] [SPIRES].
A. Morozov and S. Shakirov, The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP 08 (2010) 066 [arXiv:1004.2917] [SPIRES].
H. Awata and Y. Yamada, Five-dimensional AGT relation and the deformed beta-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [SPIRES].
A. Alexandrov, Matrix models for random partitions, arXiv:1005.5715 [SPIRES].
A. Morozov and S. Shakirov, From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators, arXiv:1007.4100 [SPIRES].
H. Itoyama, T. Oota and N. Yonezawa, Massive scaling limit of beta-deformed matrix model of Selberg type, Phys. Rev. D 82 (2010) 085031 [arXiv:1008.1861] [SPIRES].
A. Brini, M. Mariño and S. Stevan, The uses of the refined matrix model recursion, arXiv:1010.1210 [SPIRES].
A. Mironov, A. Morozov and S. Shakirov, On ‘Dotsenko-Fateev’ representation of the toric conformal blocks, J. Phys. A 44 (2011) 085401 [arXiv:1010.1734] [SPIRES].
A. Mironov, A. Morozov and S. Shakirov, Brezin-Gross-Witten model as ‘pure gauge’ limit of Selberg integrals, JHEP 03 (2011) 102 [arXiv:1011.3481] [SPIRES].
T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [SPIRES].
T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP 07 (2010) 081 [arXiv:1006.0828] [SPIRES].
K. Maruyoshi and F. Yagi, Seiberg-Witten curve via generalized matrix model, JHEP 01 (2011) 042 [arXiv:1009.5553] [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [SPIRES].
M.C.N. Cheng, R. Dijkgraaf and C. Vafa, Non-perturbative topological strings and conformal blocks, arXiv:1010.4573 [SPIRES].
M. Goulian and M. Li, Correlation functions in Liouville theory, Phys. Rev. Lett. 66 (1991) 2051 [SPIRES].
E.P. Verlinde and H.L. Verlinde, Multiloop calculations in covariant superstring theory, Phys. Lett. B 192 (1987) 95 [SPIRES].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 conformal field theories on Riemann surfaces, Commun. Math. Phys. 115 (1988) 649 [SPIRES].
E.D’Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [SPIRES].
J. Bagger and M. Goulian, Coulomb gas representation on higher genus surfaces, Nucl. Phys. B 330 (1990) 488 [SPIRES].
C. Vafa, Conformal theories and punctured surfaces, Phys. Lett. B 199 (1987) 195 [SPIRES].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [SPIRES].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [SPIRES].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Sommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [SPIRES].
A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: the case of SU(N), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [SPIRES].
A. Popolitov, On relation between Nekrasov functions and BS periods in pure SU(N) case, arXiv:1001.1407 [SPIRES].
N. Nekrasov and E. Witten, The omega deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [SPIRES].
J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, arXiv:1005.2846 [SPIRES].
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [SPIRES].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045 [SPIRES].
B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and non-commutative algebraic geometry, JHEP 03 (2009) 094 [arXiv:0809.3367] [SPIRES].
L. Chekhov, B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and quantum algebraic geometry, arXiv:0911.1664 [SPIRES].
L. Chekhov, Logarithmic potential beta-ensembles and Feynman graphs, arXiv:1009.5940 [SPIRES].
L.O. Chekhov, B. Eynard and O. Marchal, Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach, Theor. Math. Phys. 166 (2011) 141 [arXiv:1009.6007] [SPIRES].
H.M. Farkas and I. Kra, Riemann surfaces, Springer-Verlag (1991).
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Bonelli, G., Maruyoshi, K., Tanzini, A. et al. Generalized matrix models and AGT correspondence at all genera. J. High Energ. Phys. 2011, 55 (2011). https://doi.org/10.1007/JHEP07(2011)055
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DOI: https://doi.org/10.1007/JHEP07(2011)055