Abstract
We discuss the Penner type matrix model recently proposed by Dijkgraaf and Vafa for a possible explanation of the relation between four-dimensional gauge theory and Liouville theory by making use of the connection of the matrix model to two-dimensional CFT. We first consider the relation of gauge couplings defined in UV and IR regimes of N f = 4, \( \mathcal{N} \) = 2 supersymmetric gauge theory being related as q UV = ϑ2(q IR)4/ϑ3(q IR)4. We then use this relation to discuss the action of modular transformation on the matrix model and determine its spectral curve.
We also discuss the decoupling of massive flavors from the N f = 4 matrix model and derive matrix models describing asymptotically free \( \mathcal{N} \) = 2 gauge theories. We find that the Penner type matrix theory reproduces correctly the standard results of \( \mathcal{N} \) = 2 supersymmetric gauge theories.
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L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from Four-dimensional gauge theories, arXiv:0906.3219 [SPIRES].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].
N. Wyllard, AN−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. Marshakov, A. Mironov and A. Morozov, On combinatorial expansions of conformal blocks, arXiv:0907.3946 [SPIRES].
D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, arXiv:0908.2064 [SPIRES].
A. Mironov and A. Morozov, The power of Nekrasov functions, Phys. Lett. B 680 (2009) 188 [arXiv:0908.2190] [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.V. Nanopoulos and D. Xie, On crossing symmmetry and modular invariance in conformal field theory and S duality in Gauge theory, Phys. Rev. D 80 (2009) 105015 [arXiv:0908.4409] [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, arXiv:0909.0945 [SPIRES].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, arXiv:0909.1105 [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. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].
G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, arXiv:0909.4031 [SPIRES].
L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, arXiv:0909.4776 [SPIRES].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, arXiv:0910.2225 [SPIRES].
H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, arXiv:0910.4431 [SPIRES].
K. Papadodimas, Topological anti-topological fusion in four-dimensional superconformal field theories, arXiv:0910.4963 [SPIRES].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, arXiv:0910.5670 [SPIRES].
V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, arXiv:0911.0363 [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].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, arXiv:0911.2353 [SPIRES].
A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: the case of SU(N), arXiv:0911.2396 [SPIRES].
H. Itoyama, K. Maruyoshi and T. Oota, Notes on the Quiver matrix model and 2D-4D conformal connection, arXiv:0911.4244 [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].
D. Gaiotto, N=2 dualities, arXiv:0904.2715 [SPIRES].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [SPIRES].
D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, arXiv:0904.4466 [SPIRES].
Y. Tachikawa, Six-dimensional D N theory and four-dimensional SO-USp quivers, JHEP 07 (2009) 067 [arXiv:0905.4074] [SPIRES].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [SPIRES].
D. Nanopoulos and D. Xie, N=2 SU Quiver with USP ends or SU ends with antisymmetric matter, JHEP 08 (2009) 108 [arXiv:0907.1651] [SPIRES].
N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [SPIRES].
K. Maruyoshi, M. Taki, S. Terashima and F. Yagi, New Seiberg dualities from N = 2 dualities, JHEP 09 (2009) 086 [arXiv:0907.2625] [SPIRES].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, arXiv:0909.1327 [SPIRES].
Y. Zhou, A note on Wilson loop in N = 2 Quiver/M theory gravity duality, arXiv:0910.4234 [SPIRES].
K. Papadodimas, Topological anti-topological fusion in four-dimensional superconformal field theories, arXiv:0910.4963 [SPIRES].
D. Gaiotto, Surface operators in N = 2 4D Gauge theories, arXiv:0911.1316 [SPIRES].
D. Nanopoulos and D. Xie, Hitchin equation, singularity and N = 2 superconformal field theories, arXiv:0911.1990 [SPIRES].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 Gauge systems, arXiv:0909.2453 [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: discrete case, Phys. Lett. B 265 (1991) 99 [SPIRES].
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and S. Pakuliak, Conformal matrix models as an alternative to conventional multimatrix models, Nucl. Phys. B 404 (1993) 717 [hep-th/9208044] [SPIRES].
N. Seiberg and E. Witten, Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [SPIRES].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [SPIRES].
R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238] [SPIRES].
R. Gopakumar, N = 1 theories and a geometric master field, JHEP 05 (2003) 033 [hep-th/0211100] [SPIRES].
S.G. Naculich, H.J. Schnitzer and N. Wyllard, The N = 2 U(N) gauge theory prepotential and periods from a perturbative matrix model calculation, Nucl. Phys. B 651 (2003) 106 [hep-th/0211123] [SPIRES].
N. Dorey, V.V. Khoze and M.P. Mattis, On N = 2 supersymmetric QCD with 4 flavors, Nucl. Phys. B 492 (1997) 607 [hep-th/9611016] [SPIRES].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP 08 (2007) 058 [hep-th/0702187] [SPIRES].
S.G. Naculich, H.J. Schnitzer and N. Wyllard, Matrix model approach to the N = 2 U(N) gauge theory with matter in the fundamental representation, JHEP 01 (2003) 015 [hep-th/0211254] [SPIRES].
F. Cachazo, N. Seiberg and E. Witten, Chiral rings and phases of supersymmetric gauge theories, JHEP 04 (2003) 018 [hep-th/0303207] [SPIRES].
M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [SPIRES].
J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145 [hep-th/9510129] [SPIRES].
T. Eguchi and S.-K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, Mod. Phys. Lett. A 11 (1996) 131 [hep-th/9510183] [SPIRES].
A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [SPIRES].
P. Sulkowski, Matrix models for 2* theories, Phys. Rev. D 80 (2009) 086006 [arXiv:0904.3064] [SPIRES].
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Eguchi, T., Maruyoshi, K. Penner type matrix model and Seiberg-Witten theory. J. High Energ. Phys. 2010, 22 (2010). https://doi.org/10.1007/JHEP02(2010)022
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DOI: https://doi.org/10.1007/JHEP02(2010)022