Abstract
To a correlation function in a two-dimensional conformal field theory with the central charge c = 1, we associate a matrix differential equation Ψ′ = LΨ, where the Lax matrix L is a matrix square root of the energy-momentum tensor. Then local conformal symmetry implies that the differential equation is isomonodromic. This provides a justification for the recently observed relation between four-point conformal blocks and solutions of the Painlevé VI equation. This also provides a direct way to compute the three-point function of Runkel-Watts theory — the common c → 1 limit of Minimal Models and Liouville theory.
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References
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].
A. Bytsko and J. Teschner, The integrable structure of nonrational conformal field theory, arXiv:0902.4825 [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].
L. Chekhov, B. Eynard and S. Ribault, Seiberg-Witten equations and non-commutative spectral curves in Liouville theory, J. Math. Phys. 54 (2013) 022306 [arXiv:1209.3984] [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].
V. Schomerus, Rolling tachyons from Liouville theory, JHEP 11 (2003) 043 [hep-th/0306026] [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].
M. Bergere and B. Eynard, Determinantal formulae and loop equations, arXiv:0901.3273 [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].
N. Iorgov, O. Lisovyy and Y. Tykhyy, Painlevé VI connection problem and monodromy of c = 1 conformal blocks, JHEP 12 (2013) 029 [arXiv:1308.4092] [INSPIRE].
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].
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ArXiv ePrint: 1307.4865
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Eynard, B., Ribault, S. Lax matrix solution of c = 1 conformal field theory. J. High Energ. Phys. 2014, 59 (2014). https://doi.org/10.1007/JHEP02(2014)059
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DOI: https://doi.org/10.1007/JHEP02(2014)059