Abstract
We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks.
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V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [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].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
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 [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP 05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
L. Chekhov and M. Mazzocco, Colliding holes in Riemann surfaces and quantum cluster algebras, arXiv:1509.07044 [INSPIRE].
L. Chekhov, M. Mazzocco and V. Rubtsov, Painlev́e monodromy manifolds, decorated character varieties and cluster algebras, Int. Math. Res. Not. 2017 (2017) 7639 [arXiv:1511.03851].
D. Gaiotto, Asymptotically free \( \mathcal{N} \) = 2 theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
L. Hadasz, On the fusion matrix of the N = 1 Neveu-Schwarz blocks, JHEP 12 (2007) 071 [arXiv:0707.3384] [INSPIRE].
L. Hadasz, Z. Jaskolski and M. Piatek, Analytic continuation formulae for the BPZ conformal block, Acta Phys. Polon. B 36 (2005) 845 [hep-th/0409258] [INSPIRE].
O. Lisovyy, H. Nagoya and J. Roussillon, Irregular conformal blocks and connection formulae for Painlev́e V functions, J. Math. Phys. 59 (2018) 091409 [arXiv:1806.08344] [INSPIRE].
H. Nagoya, Irregular conformal blocks, with an application to the fifth and fourth Painlev́e equations, J. Math. Phys. 56 (2015) 123505 [arXiv:1505.02398] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
M. Pawelkiewicz, V. Schomerus and P. Suchanek, The universal Racah-Wigner symbol for Uq(osp (1|2)), JHEP 04 (2014) 079 [arXiv:1307.6866] [INSPIRE].
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq (SL(2, ℝ)), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE].
J. Teschner, A Lecture on the Liouville Vertex Operators, Int. J. Mod. Phys. A 19 (2004) 436.
J. Teschner, Quantization of moduli spaces of flat connections and Liouville theory, arXiv:1405.0359 [INSPIRE].
I. Nidaiev and J. Teschner, On the relation between the modular double of Uq(SL(2, ℝ)) and the quantum Teichmueller theory, arXiv:1302.3454 [INSPIRE].
J.-P. Ramis, Confluence et résurgence, J. Fac. Sci. Univ. Tokyo (Sec. 1A Math.) 36 (1989) 703.
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
S. Ruijsenaars, A Generalized Hypergeometric Function Satisfying Four Analytic Difference Equations of Askey-Wilson Type, Commun. Math. Phys. 206 (1999) 639.
Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
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Lenells, J., Roussillon, J. Confluent conformal blocks of the second kind. J. High Energ. Phys. 2020, 133 (2020). https://doi.org/10.1007/JHEP06(2020)133
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DOI: https://doi.org/10.1007/JHEP06(2020)133