Abstract
It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a differential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate field. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary field is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of n’th order Fuchsian differential equations are discussed.
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References
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] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, 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] [INSPIRE].
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, in Strings, branes and dualities. Proceedings, NATO Advanced Study Institute, Cargese, France, May 26 - June 14, 1997 [hep-th/9801061] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09) (2009) [arXiv:0908.4052] [INSPIRE].
A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
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] [INSPIRE].
K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
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] [INSPIRE].
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
F. Fucito, J.F. Morales, D.R. Pacifici and R. Poghossian, Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] [INSPIRE].
F. Fucito, J.F. Morales and D.R. Pacifici, Deformed Seiberg-Witten Curves for ADE Quivers, JHEP 01 (2013) 091 [arXiv:1210.3580] [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. 3. The Yang-Baxter relation, Commun. Math. Phys. 200 (1999) 297 [hep-th/9805008] [INSPIRE].
M. Piatek, Classical conformal blocks from TBA for the elliptic Calogero-Moser system, JHEP 06 (2011) 050 [arXiv:1102.5403] [INSPIRE].
K. Bulycheva, H.-Y. Chen, A. Gorsky and P. Koroteev, BPS States in Omega Background and Integrability, JHEP 10 (2012) 116 [arXiv:1207.0460] [INSPIRE].
M. Piatek, Classical torus conformal block, N = 2∗ twisted superpotential and the accessory parameter of Lamé equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].
S.K. Choi, C. Rim and H. Zhang, Irregular conformal block, spectral curve and flow equations, JHEP 03 (2016) 118 [arXiv:1510.09060] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].
F. Fucito, J.F. Morales, R. Poghossian and D. Ricci Pacifici, Exact results in \( \mathcal{N}=2 \) gauge theories, JHEP 10 (2013) 178 [arXiv:1307.6612] [INSPIRE].
M. Bershtein and O. Foda, AGT, Burge pairs and minimal models, JHEP 06 (2014) 177 [arXiv:1404.7075] [INSPIRE].
S.K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Non-perturbative studies of N = 2 conformal quiver gauge theories, Fortsch. Phys. 63 (2015) 259 [arXiv:1502.05581] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 12 (2002) 071 [hep-th/0211170] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Wilson loops and chiral correlators on squashed spheres, JHEP 11 (2015) 064 [arXiv:1507.05426] [INSPIRE].
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205 [Teor. Mat. Fiz. 65 (1985) 347] [INSPIRE].
V.A. Fateev and S.L. Lukyanov, The Models of Two-Dimensional Conformal Quantum Field Theory with Z(n) Symmetry, Int. J. Mod. Phys. A 3 (1988) 507 [INSPIRE].
A. Bilal and J.-L. Gervais, Systematic Approach to Conformal Systems with Extended Virasoro Symmetries, Phys. Lett. B 206 (1988) 412 [INSPIRE].
V.A. Fateev and A.V. Litvinov, On differential equation on four-point correlation function in the Conformal Toda Field Theory, JETP Lett. 81 (2005) 594 [Pisma Zh. Eksp. Teor. Fiz. 81 (2005) 728] [hep-th/0505120] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I., JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
L. Takhtajan and P. Zograf, Action for the Liouville equation as a generating function for the accessory parameters and as a potential for the Weil-Petersson metric on the Teichmüller space, Funkts Anal. Prilozh. 19 (1985) 67.
N. Nekrasov, A. Rosly and S. Shatashvili, Darboux coordinates, Yang-Yang functional and gauge theory, Nucl. Phys. Proc. Suppl. 216 (2011) 69 [arXiv:1103.3919] [INSPIRE].
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical Conformal Blocks and Painleve VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].
J.-E. Bourgine, Y. Mastuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, arXiv:1512.02492 [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, arXiv:1512.05388 [INSPIRE].
M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [INSPIRE].
R. Flume, F. Fucito, J.F. Morales and R. Poghossian, Matone’s relation in the presence of gravitational couplings, JHEP 04 (2004) 008 [hep-th/0403057] [INSPIRE].
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Poghossian, R. Deformed SW curve and the null vector decoupling equation in Toda field theory. J. High Energ. Phys. 2016, 70 (2016). https://doi.org/10.1007/JHEP04(2016)070
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DOI: https://doi.org/10.1007/JHEP04(2016)070