Abstract
We present a new systematic way to evaluate the classical limit of the Virasoro irregular conformal block for arbitrary rank n based on the irregular partition function. In addition, we prove that the classical irregular conformal block has the exponential form as suggested by A. Zamolodchikov and Al. Zamolodchikov for the regular case. We provide an explicit calculation for the rank 2 case in detail.
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C. Rim and H. Zhang, Classical Virasoro irregular conformal block, JHEP 07 (2015) 163 [arXiv:1504.07910] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052.
S.K. Choi, C. Rim and H. Zhang, Virasoro irregular conformal block and beta deformed random matrix model, Phys. Lett. B 742 (2015) 50 [arXiv:1411.4453] [INSPIRE].
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] [INSPIRE].
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical conformal blocks and Painlevé VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].
M. Piatek and A.R. Pietrykowski, Classical irregular block, \( \mathcal{N}=2 \) pure gauge theory and Mathieu equation, JHEP 12 (2014) 032 [arXiv:1407.0305] [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
H. Itoyama and T. Oota, Method of generating q-expansion coefficients for conformal block and N = 2 Nekrasov function by β-deformed matrix model, Nucl. Phys. B 838 (2010) 298 [arXiv:1003.2929] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [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].
T. Nishinaka and C. Rim, Matrix models for irregular conformal blocks and Argyres-Douglas theories, JHEP 10 (2012) 138 [arXiv:1207.4480] [INSPIRE].
S.-K. Choi and C. Rim, Parametric dependence of irregular conformal block, JHEP 04 (2014) 106 [arXiv:1312.5535] [INSPIRE].
T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, JHEP 12 (2012) 050 [arXiv:1203.1052] [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].
G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin systems via β-deformed matrix models, arXiv:1104.4016 [INSPIRE].
P.J. Forrester and S.O. Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008) 489 [arXiv:0710.3981].
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Rim, C., Zhang, H. Classical Virasoro irregular conformal block II. J. High Energ. Phys. 2015, 97 (2015). https://doi.org/10.1007/JHEP09(2015)097
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DOI: https://doi.org/10.1007/JHEP09(2015)097