Abstract
Generic c = 1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlevé VI equation with respect to one of its integration constants. Based on this relation, we show that c = 1 fusion matrix essentially coincides with the connection coefficient relating tau function asymptotics at different critical points. Explicit formulas for both quantities are obtained by solving certain functional relations which follow from the tau function expansions. The final result does not involve integration and is given by a ratio of two products of Barnes G-functions with arguments expressed in terms of conformal dimensions/monodromy data. It turns out to be closely related to the volume of hyperbolic tetrahedron.
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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].
J. Baik, R. Buckingham and J. Difranco, Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function, Comm. Math. Phys. 280 (2008) 463 [arXiv:0704.3636].
E. Basor and C.A. Tracy, Asymptotics of a τ -function and Toeplitz determinants with singular generating functions, Int. J. Mod. Phys. A7 (1992) 93.
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].
P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. 90 (2005)167 [math/0308221].
P. Boalch, The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math. 596 (2006)183 [math/0406281].
P.P. Boalch, Regge and Okamoto symmetries, Communications in Mathematical Physics 276 (2007)117 [math/0603398].
Y. Cho and H. Kim, On the volume formula for hyperbolic tetrahedra, Discr. Comp. Geom. 22 (1999)347.
T. Claeys, A. Its and I. Krasovsky, Emergence of a singularity for Toeplitz determinants and Painleve V, Duke Math. J. 160 (2011) 207 [arXiv:1004.3696] [INSPIRE].
P. Deift, A. Its and I. Krasovsky, Asymptotics of the Airy-kernel determinant, Comm. Math. Phys. 278 (2008) 643 [math/0609451].
P. Deift, A. Its, I. Krasovsky and X. Zhou, The Widom-Dyson constant for the gap probability in random matrix theory, J. Comput. Appl. Math. 202 (2007) 26 [math/0601535].
V. Dotsenko and V. 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 [INSPIRE].
B. Dubrovin and M. Mazzocco, Monodromy of certain Painlevé VI transcendents and reflection groups, Inv. Math. 141 (2000) 55 [math/9806056].
F. Dyson, Fredholm determinants and inverse scattering problems, Commun. Math. Phys. 47 (1976)171 [INSPIRE].
T. Ehrhardt, Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel, Commun. Math. Phys. 262 (2006) 317 [math/0401205].
B. Eynard and S. Ribault, Lax matrix solution of c = 1 Conformal Field Theory, arXiv:1307.4865 [INSPIRE].
D. Galakhov, A. Mironov and A. Morozov, S-duality as a beta-deformed Fourier transform, JHEP 08 (2012) 067 [arXiv:1205.4998] [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP 10 (2012)038 [Erratum ibid. 1210 (2012) 183] [arXiv:1207.0787] [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A 46 (2013) 335203 [arXiv:1302.1832] [INSPIRE].
M. Jimbo Monodromy problem and the boundary condition for some Painlevé equations, Publ. RIMS Kyoto Univ. 18 (1982) 1137.
A. V. Kitaev Grothendieck’s dessins d’enfants their deformations and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations, Algebra i Analiz 17 (2005) 224 [nlin/0309078].
I.V. Krasovsky, Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle, Int. Math. Res. Not. 2004 (2004) 1249 [math/0401258].
O. Lisovyy and Y. Tykhyy, Algebraic solutions of the sixth Painlevé equation, arXiv:0809.4873.
O. Lisovyy, Dyson’s constant for the hypergeometric kernel, in New trends in quantum integrable systems, B. Feigin et al. eds., World Scientific, Singapore (2011), arXiv:0910.1914 [INSPIRE].
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical conformal blocks and Painlevé VI, arXiv:1309.4700 [INSPIRE].
G. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177.
J. Murakami M. Yano On the volume of a hyperbolic and spherical tetrahedron, Comm. Anal. Geom. 13 (2005) 379.
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].
N. Nemkov, S-duality as Fourier transform for arbitrary ϵ 1 , ϵ 2, arXiv:1307.0773 [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 U (q)(SL(2, \( \mathbb{R} \))), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE].
I. Runkel and G. Watts, A nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].
V. Schomerus, Rolling tachyons from Liouville theory, JHEP 11 (2003) 043 [hep-th/0306026] [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
J. Teschner, A lecture on the Liouville vertex operators, Int. J. Mod. Phys. A 19S2 (2004) 436 [hep-th/0303150] [INSPIRE].
J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, Adv. Theor. Math. Phys. 15 (2011) 471 [arXiv:1005.2846] [INSPIRE].
G. Vartanov and J. Teschner, Supersymmetric gauge theories, quantization of moduli spaces of flat connections and conformal field theory, arXiv:1302.3778 [INSPIRE].
AlB. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Zh. Eksp. Teor. Fiz. 90 (1986) 1808.
A.lB. Zamolodchikov, On the three-point function in minimal Liouville gravity, hep-th/0505063 [INSPIRE].
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ArXiv ePrint: 1308.4092
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Iorgov, N., Lisovyy, O. & Tykhyy, Y. Painlevé VI connection problem and monodromy of c = 1 conformal blocks. J. High Energ. Phys. 2013, 29 (2013). https://doi.org/10.1007/JHEP12(2013)029
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DOI: https://doi.org/10.1007/JHEP12(2013)029