×

Classical conformal blocks and Painlevé VI. (English) Zbl 1333.81375

Summary: We study the classical \(c \to \infty\) limit of the Virasoro conformal blocks. We point out that the classical limit of the simplest nontrivial null-vector decoupling equation on a sphere leads to the Painlevé VI equation. This gives the explicit representation of generic four-point classical conformal block in terms of the regularized action evaluated on certain solution of the Painlevé VI equation. As a simple consequence, the monodromy problem of the Heun equation is related to the connection problem for the Painlevé VI.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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]. · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[2] N.A. Nekrasov, On the BPS/CFT correspondence, seminar at The String Theory Group, University of Amsterdam, Amsterdam, The Netherlands, 3 February 2004.
[3] N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2004) 831 [hep-th/0206161] [INSPIRE]. · Zbl 1056.81068 · doi:10.4310/ATMP.2003.v7.n5.a4
[4] A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, in From Fields to Strings: Circumnavigating Theoretical Physics, M. Shifman et al. eds., World Scientific Publishing Co. Pte. Ltd. (2005), chapter 17, pg. 581 [ISBN: 978-981-238-955-8, 978-981-4482-04-2] [hep-th/0302191] [INSPIRE]. · Zbl 1081.81103
[5] N.A. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Progr. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE]. · Zbl 1233.14029 · doi:10.1007/0-8176-4467-9_15
[6] 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]. · Zbl 1185.81111 · doi:10.1007/s11005-010-0369-5
[7] A.B. Zamolodchikov, Conformal symmetry in two-dimensions: recursion representation of conformal block, Theor. Math. Phys.53 (1987) 1088 [Teor. Mat. Fiz.73 (1987) 103].
[8] 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]. · Zbl 0925.81301 · doi:10.1016/0550-3213(96)00351-3
[9] O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP10 (2012) 038 [Erratum ibid.1210 (2012) 183] [arXiv:1207.0787] [INSPIRE]. · Zbl 1397.81307
[10] A.B. Zamolodchikov, Conformal symmetry in two dimensions: An explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys.96 (1984) 419 [INSPIRE]. · doi:10.1007/BF01214585
[11] 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]. · Zbl 1242.81119 · doi:10.1007/s11005-011-0503-z
[12] N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in XVIth International Congress on Mathematical Physics, Prague, Czech Republic, 3-8 August 2009, P. Exner ed., World Scientific Publishing Co. Pte. Ltd. (2010), chapter 6, pg. 265 [ISBN: #9789814304634] [arXiv:0908.4052] [INSPIRE]. · Zbl 1214.83049
[13] N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE]. · Zbl 1393.81033
[14] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. T. Roy. Soc.A 308 (1982) 523 [INSPIRE]. · Zbl 0509.14014
[15] A. Bilal, V.V. Fock and I.I. Kogan, On the origin of W -algebras, Nucl. Phys.B 359 (1991) 635 [INSPIRE]. · doi:10.1016/0550-3213(91)90075-9
[16] P. Boalch, Symplectic manifolds and isomonodromic deformations, Adv. Math.163 (2001) 137. · Zbl 1001.53059 · doi:10.1006/aima.2001.1998
[17] N.A. 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]. · doi:10.1016/j.nuclphysbps.2011.04.150
[18] L.A. Takhtadzhyan and P.G. 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 [Funct. Anal. Appl.19 (1986) 219] (English translation). · Zbl 0612.32018
[19] P.G. Zograf and L.A. Takhtadzhyan, On the Liouville equation, accessory parameters and the geometry of the Teichmüller space for the Riemann surfaces of genus 0, Mat. Sbornik132 (1987) 147 [Math. USSR-Sbornik60 (1988) 143] (English translation). · Zbl 0642.32010
[20] F. Klein, Neue Beiträge zur Riemann’schen Functionentheorie, Math. Ann.21 (1883) 201. · JFM 15.0351.01 · doi:10.1007/BF01442920
[21] H. Poincaré, Sur les groupes des équations linéaires, Acta Math.4 (1884) 201. · JFM 16.0252.01 · doi:10.1007/BF02418420
[22] H. Poincaré, Les fonctions fuchsiennes et l’équation Δu = eu, J. Math. Pure. Appl.5 (1898) 157.
[23] 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]. · Zbl 1442.81059 · doi:10.4310/ATMP.2011.v15.n2.a6
[24] S.L. Lukyanov, Critical values of the Yang-Yang functional in the quantum sine-Gordon model, Nucl. Phys.B 853 (2011) 475 [arXiv:1105.2836] [INSPIRE]. · Zbl 1229.35270 · doi:10.1016/j.nuclphysb.2011.07.028
[25] D. Guzzetti, Tabulation of Painlevé 6 transcendents, Nonlinearity25 (2012) 3235 [arXiv:1108.3401]. · Zbl 1267.34152 · doi:10.1088/0951-7715/25/12/3235
[26] L. Schlesinger, Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten, J. Reine Angew. Math.141 (1912) 96. · JFM 43.0385.01
[27] M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ -function, PhysicaD 2 (1981) 407.
[28] E.K. Sklyanin, Goryachev-Chaplygin top and the inverse scattering method, Zap. Nauchn. Semin.133 (1984) 236) [J. Sov. Math.31 (1985) 3417] (English translation) [INSPIRE]. · Zbl 0576.70002
[29] E.K. Sklyanin, Separation of variables. New trends, Prog. Theor. Phys. Suppl.118 (1995) 35 [solv-int/9504001] [INSPIRE]. · Zbl 0868.35002 · doi:10.1143/PTPS.118.35
[30] I. Krichever, Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations, hep-th/0112096 [INSPIRE]. · Zbl 1044.70010
[31] B. Dubrovin and M. Mazzocco, Canonical structure and symmetries of the Schlesinger equations, Commun. Math. Phys.271 (2007) 289 [math.DG/0311261]. · Zbl 1146.32005 · doi:10.1007/s00220-006-0165-3
[32] A.-K. Kashani-Poor and J. Troost, Transformations of Spherical Blocks, JHEP10 (2013) 009 [arXiv:1305.7408] [INSPIRE]. · Zbl 1342.83386 · doi:10.1007/JHEP10(2013)009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.