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On AGT description of \( \mathcal{N} = 2 \) SCFT with \(N_{f } = 4\). (English) Zbl 1269.81127

Summary: We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of \( \mathcal{N} = 2 \) SCFT. We focus our attention on the \(SU(2)\) theory with \(N _{f } = 4\) flavor symmetry, whose partition function, according to AGT, is given by the Liouville four-point function on the sphere. The gauge theory with \(N _{f } = 4\) is known to exhibit \(SO(8)\) symmetry. We explain how the Weyl symmetry transformations of \(SO(8)\) flavor symmetry are realized in the Liouville theory picture. This is associated to functional properties of the Liouville four-point function that are a priori unexpected. In turn, this can be thought of as a non-trivial consistency check of AGT conjecture. We also make some comments on elementary surface operators and WZW theory.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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References:

[1] D. Gaiotto, N=2 dualities, arXiv:0904.2715 [SPIRES]. · Zbl 1397.81362
[2] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, arXiv:0906.3219 [SPIRES]. · Zbl 1185.81111
[3] N.A. Nekrasov, Seiberg-Witten Prepotential from Instanton Counting, Adv. Theor. Math. Phys.7 (2004) 831 [hep-th/0206161] [SPIRES]. · Zbl 1056.81068
[4] V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, arXiv:0911.0363 [SPIRES].
[5] A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, arXiv:0910.5670 [SPIRES]. · Zbl 1272.81180
[6] A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett.B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].
[7] 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].
[8] A. Mironov and A. Morozov, The Power of Nekrasov Functions, Phys. Lett.B 680 (2009) 188 [arXiv:0908.2190] [SPIRES].
[9] G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, arXiv:0909.4031 [SPIRES]. · Zbl 1153.83352
[10] A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, arXiv:0908.2064 [SPIRES]. · Zbl 1196.81205
[11] A. Marshakov, A. Mironov and A. Morozov, On Combinatorial Expansions of Conformal Blocks, arXiv:0907.3946 [SPIRES]. · Zbl 1256.81101
[12] A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions, arXiv:0911.5721 [SPIRES]. · Zbl 1270.81139
[13] H. Itoyama, K. Maruyoshi and T. Oota, Notes on the Quiver Matrix Model and 2d-4d Conformal Connection, arXiv:0911.4244 [SPIRES]. · Zbl 1195.81103
[14] D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
[15] V.A. Fateev and A.V. Litvinov, On AGT conjecture, arXiv:0912.0504 [SPIRES]. · Zbl 1270.81203
[16] R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [SPIRES]. · Zbl 0999.81068
[17] T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory, arXiv:0911.4797 [SPIRES]. · Zbl 1270.81165
[18] S. Kanno, Y. Matsuo, S. Shiba and Y. Tachikawa, N=2 gauge theories and degenerate fields of Toda theory, arXiv:0911.4787 [SPIRES].
[19] L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, arXiv:0909.4776 [SPIRES].
[20] A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: the Case of SU(N), arXiv:0911.2396 [SPIRES]. · Zbl 1189.81237
[21] A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY Nf = 2NcQCD, JHEP11 (2009) 048 [arXiv:0909.3338] [SPIRES]. · doi:10.1088/1126-6708/2009/11/048
[22] A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys.B 825 (2010) 1 [arXiv:0908.2569] [SPIRES]. · Zbl 1196.81205 · doi:10.1016/j.nuclphysb.2009.09.011
[23] N. Wyllard, AN−1conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP11 (2009) 002 [arXiv:0907.2189] [SPIRES]. · doi:10.1088/1126-6708/2009/11/002
[24] R. Schiappa and N. Wyllard, An Arthreesome: Matrix models, 2D CFTs and 4D N = 2 gauge theories, arXiv:0911.5337 [SPIRES]. · Zbl 1312.81108
[25] A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, arXiv:0910.2225 [SPIRES]. · Zbl 1271.81157
[26] G. Bertoldi, S. Bolognesi, M. Matone, L. Mazzucato and Y. Nakayama, The Liouville geometry of N = 2 instantons and the moduli of punctured spheres, JHEP05 (2004) 075 [hep-th/0405117] [SPIRES]. · doi:10.1088/1126-6708/2004/05/075
[27] V. Pestun, Localization of the four-dimensional N = 4 SYM to a two- sphere and 1/8 BPS Wilson loops, arXiv:0906.0638 [SPIRES]. · Zbl 1397.81393
[28] 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]. · Zbl 1020.81911 · doi:10.1016/0550-3213(94)90214-3
[29] N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl.102 (1990) 319 [SPIRES]. · Zbl 0790.53059 · doi:10.1143/PTPS.102.319
[30] J. Teschner, Liouville theory revisited, Class. Quant. Grav.18 (2001) R153 [hep-th/0104158] [SPIRES]. · Zbl 1022.81047 · doi:10.1088/0264-9381/18/23/201
[31] Y. Nakayama, Liouville field theory: A decade after the revolution, Int. J. Mod. Phys.A 19 (2004) 2771 [hep-th/0402009] [SPIRES]. · Zbl 1080.81056
[32] J. Teschner, A lecture on the Liouville vertex operators, Int. J. Mod. Phys.A 19S2 (2004) 436 [hep-th/0303150] [SPIRES]. · Zbl 1080.81060
[33] M. Goulian and M. Li, Correlation Functions In Liouville Theory, Phys. Rev. Lett.66 (1991) 2051. · doi:10.1103/PhysRevLett.66.2051
[34] H. Dorn and H.J. Otto, On correlation functions for noncritical strings with c <= 1 but d >= 1, Phys. Lett.B 291 (1992) 39 [hep-th/9206053] [SPIRES].
[35] P. Di Francesco and D. Kutasov, World Sheet and Space Time Physics in Two Dimensional (Super) String Theory, Nucl. Phys.B 375 (1992) 119 [hep-th/9109005] [SPIRES]. · doi:10.1016/0550-3213(92)90337-B
[36] V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys.B 240 (1984) 312 [SPIRES]. · doi:10.1016/0550-3213(84)90269-4
[37] V.S. Dotsenko and V.A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c ¡ 1, Nucl. Phys.B 251 (1985) 691 [SPIRES]. · doi:10.1016/S0550-3213(85)80004-3
[38] A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys.B 477 (1996) 577 [hep-th/9506136] [SPIRES]. · Zbl 0925.81301 · doi:10.1016/0550-3213(96)00351-3
[39] G. Giribet and L. Nicolás, Comment on three-point function in AdS3/CFT2, J. Math. Phys.50 (2009) 042304 [arXiv:0812.2732] [SPIRES]. · Zbl 1214.81246 · doi:10.1063/1.3119003
[40] H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys.B 429 (1994) 375 [hep-th/9403141] [SPIRES]. · Zbl 1020.81770 · doi:10.1016/0550-3213(94)00352-1
[41] J. Teschner, On the Liouville three point function, Phys. Lett.B 363 (1995) 65 [hep-th/9507109] [SPIRES].
[42] A. Pakman, Liouville theory without an action, Phys. Lett.B 642 (2006) 263 [hep-th/0601197] [SPIRES]. · Zbl 1248.81202
[43] V.A. Fateev and A.V. Litvinov, Coulomb integrals in Liouville theory and Liouville gravity, JETP Lett.84 (2007) 531 [SPIRES]. · doi:10.1134/S0021364006220012
[44] V.A. Fateev and A.V. Litvinov, Multipoint correlation functions in Liouville field theory and minimal Liouville gravity, Theor. Math. Phys.154 (2008) 454 [arXiv:0707.1664] [SPIRES]. · Zbl 1192.81297 · doi:10.1007/s11232-008-0038-3
[45] 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]. · Zbl 1269.81078
[46] N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, arXiv:0909.1105 [SPIRES]. · Zbl 1270.81134
[47] D. Gaiotto, Surface Operators in N = 2 4D Gauge Theories, arXiv:0911.1316 [SPIRES]. · Zbl 1397.81363
[48] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys.B 241 (1984) 333 [SPIRES]. · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[49] V. Fateev and A.B. Zamolodchikov, Operator algebra and correlation functions in the two-dimensional SU(2) × SU(2) chiral Wess-Zumino model, Sov. J. Nucl. Phys.43 (1987) 657.
[50] V.G. Knizhnik and A.B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys.B 247 (1984) 83 [SPIRES]. · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2
[51] J. Teschner, Crossing symmetry in the H3+WZNW model, Phys. Lett.B 521 (2001) 127 [hep-th/0108121] [SPIRES]. · Zbl 1020.81028
[52] E. Frenkel, Lectures on the Langlands program and conformal field theory, hep-th/0512172 [SPIRES]. · Zbl 1196.11091
[53] J.L. Petersen, J. Rasmussen and M. Yu, Hamiltonian reduction of SL(2) theories at the level of correlators, Nucl. Phys.B 457 (1995) 343 [hep-th/9506180] [SPIRES]. · doi:10.1016/0550-3213(95)00503-X
[54] G. Giribet, On spectral flow symmetry and Knizhnik-Zamolodchikov equation, Phys. Lett.B 628 (2005) 148 [hep-th/0508019] [SPIRES]. · Zbl 1247.81424
[55] G.E. Giribet, A note on Z2symmetries of the KZ equation, J. Math. Phys.48 (2007) 012304 [hep-th/0608104] [SPIRES]. · Zbl 1121.81109 · doi:10.1063/1.2424789
[56] A.V. Stoyanovsky, A relation between the Knizhnik-Zamolodchikov and Belavin-Polyakov-Zamolodchikov systems of partial differential equations, math-ph/0012013 [SPIRES].
[57] S. Ribault and J. Teschner, H3+WZNW correlators from Liouville theory, JHEP06 (2005) 014 [hep-th/0502048] [SPIRES]. · doi:10.1088/1126-6708/2005/06/014
[58] Y. Hikida and V. Schomerus, H3+WZNW model from Liouville field theory, JHEP10 (2007) 064 [arXiv:0706.1030] [SPIRES]. · doi:10.1088/1126-6708/2007/10/064
[59] S. Ribault, Knizhnik-Zamolodchikov equations and spectral flow in AdS3string theory, JHEP09 (2005) 045 [hep-th/0507114] [SPIRES]. · doi:10.1088/1126-6708/2005/09/045
[60] S. Ribault, A family of solvable non-rational conformal field theories, JHEP05 (2008) 073 [arXiv:0803.2099] [SPIRES]. · doi:10.1088/1126-6708/2008/05/073
[61] S. Ribault, On sl3 Knizhnik-Zamolodchikov equations and W3 null-vector equations, JHEP10 (2009) 002 [arXiv:0811.4587] [SPIRES]. · doi:10.1088/1126-6708/2009/10/002
[62] G. Giribet, Y. Nakayama and L. Nicolás, Langlands duality in Liouville-H3+WZNW correspondence, Int. J. Mod. Phys.A 24 (2009) 3137 [arXiv:0805.1254] [SPIRES]. · Zbl 1168.83321
[63] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, hep-th/0604151 [SPIRES]. · Zbl 1128.22013
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