Abstract
A multiparametric family of 2D Toda \({\tau}\) -functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of S n . The coefficients \({{F^{c_{1}, . . . , c_{l}}_{d_{1}, . . . , d_{m}}}(\mu, \nu)}\) in their series expansion over products \({P_{\mu}P^{'}_{\nu}}\) of power sum symmetric functions in the two sets of Toda flow parameters and powers of the l + m auxiliary parameters are shown to enumerate \({|\mu|=|\nu|=n}\) fold branched covers of the Riemann sphere with specified ramification profiles \({ \mu}\) and \({\nu}\) at a pair of points, and two sets of additional branch points, satisfying certain additional conditions on their ramification profile lengths. The first group consists of l branch points, with ramification profile lengths fixed to be the numbers \({(n-c_{1}, . . . , n-c_{l})}\) ; the second consists of m further groups of “coloured” branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers \({(d_{1}, . . . , d_{m})}\). The latter are counted with signs determined by the parity of the total number of such branch points. The coefficients \({{F^{c_{1}, . . . , c_{l}}_{d_{1}, . . . , d_{m}}}(\mu, \nu)}\) are also shown to enumerate paths in the Cayley graph of the symmetric group S n generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type \({\mu}\) and ending in the class of type \({\nu}\), with the first l consecutive subsequences of \({(c_{1}, . . . , c_{l})}\) transpositions strictly monotonically increasing, and the subsequent subsequences of \({(d_{1}, . . . , d_{m})}\) transpositions weakly increasing.
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Alexandrov A., Mironov A., Morozov A., Natanzon S.: Integrability of Hurwitz partition functions. I. Summary. J. Phys. A 45, 045209 (2012)
Alexandrov A., Mironov A., Morozov A., Natanzon S.: On KP-integrable Hurwitz functions. JHEP 1411, 080 (2014)
Ambjørn, J., Chekhov, L.: The matrix model for dessins d’enfants. Ann. Inst. Henri Poincaré, Comb. Phys. Interact. 1, 337–361 (2014). See also arXiv:1404.4240
Brown T.W.: Complex matrix model duality. Phys. Rev. D 83, 085002 (2011)
Diaconis, P., Greene, C.: Applications of Murphy’s elements. Stanford Technical Report 335 (1989)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Sato, M. (ed.) Non-Linear Integrable Systems Classical and Quantum Theory, Proceedings of RIMS Symposium (1981). World Scientific, (1983)
Eynard B., Orantin N.: Topological recursion in enumerative geometry and random matrices. J. Phys. A 42, 293001 (2009)
Fulton, W., Harris, J.: Representation Theory, Graduate Texts in Mathematics, vol. 129, Chapt. 4, and Appendix A. Springer, New York, Berlin, Heidelberg (1991)
Frobenius, G.: Über die Charaktere der symmetrischen Gruppe. Sitzber. Pruess. Akad. Berlin, pp. 516–534 (1900)
Gross K.I., Richards D.S.: Special functions of matrix arguments. I: Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Am. Math. Soc. 301, 781–811 (1987)
Goulden I.P., Guay-Paquet M., Novak J.: Monotone Hurwitz numbers and the HCIZ integral. Ann. Math. Blaise Pascal 21(1), 71–89 (2014)
Goulden, I.P., Guay-Paquet, M., Novak J.: Toda equations and piecewise polynomiality for mixed double Hurwitz numbers. arXiv:1307.2137
Goulden I.P., Jackson D.M.: Transitive factorizations into transpositions and holomorphic mappings on the sphere. Proc. Am. Math. Soc. 125(1), 51–60 (1997)
Guay-Paquet, M., Harnad, J.: 2D Toda \({\tau}\) -functions as combinatorial generating functions. Lett. Math. Phys. (2015, in press). arXiv:1405.6303
Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957)
Itzykson, C., Zuber, J.-B.: The planar approximation. II. J. Math. Phys. 21, 411–21 (1980)
Harnad, J., Orlov, A.Yu.: Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions. J. Phys. A 39, 8783–8809 (2006). See also arXiv:math-ph/0512056
Harnad, J., Orlov, A.Yu.: Convolution symmetries of integrable hierarchies, matrix models and tau functions. In: Deift, P., Forrester, P. (eds.) Random Matrix Theory, Interacting Particle Systems, and Integrable Systems. MSRI Publications, vol. 65, pp. 247–275. Cambridge University Press (2014). See also arXiv:0901.0323
Kazarian, M., Zograf, P.: ‘Virasoro constraints and topological recursion for Grothendieck’s dessin counting. arXiv:1406.5976
Kharchev S., Marshakov A., Mironov A., Morozov A.: Generalized Kazakov–Migdal–Kontsevich model: group theory aspects. Int. J. Mod. Phys. A 10, 2015 (1995)
Jucys A.A.: Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5(1), 107–112 (1974)
Lando S.: Combinatorial facets of Hurwitz numbers. In: Koolen, J., Kwak, J.H., Xu, M.Y. (eds.) Applications of Group Theory to Combinatorics, pp. 109–132. Taylor and Francis Group, London (2008)
Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences, with appendix by D. Zagier, vol. 141. Springer, New York (2004)
Macdonald I.G.: Symmetric Functions and Hall Polynomials. Clarendon Press, Oxford (1995)
de Mello Koch, R., Ramgoolam, S.: From matrix models and quantum fields to Hurwitz space and the absolute Galois group. arXiv:1002.1634
Murphy G.E.: A new construction of Young’s seminormal representation of the symmetric groups. J. Algebra 69, 287–297 (1981)
Mironov A.D., Morozov A.Yu., Natanzon S.M.: Complete set of cut-and-join operators in the Hurwitz–Kontsevich theory. Theor. Math. Phys. 166, 1–22 (2011)
Okounkov, A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7, 447–453 (2000). See also arXiv:math/0004128
Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory, and completed cycles. Ann. Math. 163 517–560 (2006); The equivariant Gromov–Witten theory of P 1. ibid. 163 561–605 (2006)
Orlov, A.Yu.: New solvable matrix integrals. Int. J. Mod. Phys. A 19(suppl 02), 276–93 (2004)
Orlov, A.Yu., Shiota, T.: Schur function expansion for normal matrix model and associated discrete matrix models. Phys. Lett. A 343, 384–96 (2005)
Orlov A.Yu., Scherbin D.M.: Hypergeometric solutions of soliton equations. Theor. Math. Phys. 128, 906–926 (2001)
Pandharipande R.: The Toda equations and the Gromov–Witten theory of the Riemann sphere. Lett. Math. Phys. 53, 59–74 (2000)
Takasaki, K.: Initial value problem for the Toda lattice hierarchy. In: Group Representation and Systems of Differential Equations. Advanced Studies in Pure Mathematics, vol. 4, pp. 139–163 (1984)
Takebe T.: Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy I. Lett. Math. Phys. 21, 77–84 (1991)
Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Group Representation and Systems of Differential Equations. Advanced Studies in Pure Mathematics, vol. 4, pp. 1–95 (1984)
Zograf, P.: Enumeration of Grothendieck’s dessins and KP hierarchy. arXiv:1312.2538 (2013)
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Communicated by N. Reshetikhin
Work of JH supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec—Nature et technologies (FRQNT); work of AO supported by RFBR Grant 14-01-00860.
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Harnad, J., Orlov, A.Y. Hypergeometric \({\tau}\) -Functions, Hurwitz Numbers and Enumeration of Paths. Commun. Math. Phys. 338, 267–284 (2015). https://doi.org/10.1007/s00220-015-2329-5
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DOI: https://doi.org/10.1007/s00220-015-2329-5