Title: G-Hurwitz numbers, colored cut-and-join quations, and integrable hierarchy

Speaker: Hanxiong Zhang (张汉雄), Tsinghua University

Venue: Room 09 at Quan-Zhai (全斋), New location for BICMR, Peking University

Time: 2:30-4:30 pm, May 10, 2012 (Thursday)

Abstract: Hurwitz numbers are classical objects in enumerative geometry, which relate the geometry of Riemann surfaces to the representation theory of symmetric groups. In the late 1890's, Hurwitz considered the problem of counting topologically distinct, almost simple, ramified covers of the projective line. He translated this geometric problem to a purely combinatorial problem, namely, factorizing a permutation into transpositions, thus obtained a closed formula in terms of characters of symmetric groups. The generating series of Hurwitz numbers can be written in a quite neat form using symmetric functions. In this form, one is able to prove that it satisfies the cut-and-join equation, and it is a tau function of the 2-Toda hierarchy.

Inspired by the development of orbifold Gromov-Witten theory, one is naturally led to consider the problem of generalizing Hurwitz numbers to the orbifold setting. In this talk, we will give the definition of a G-branched cover for arbitrary finite group G, and then analyze its monodromy representation. It turns out that the monodromy data are contained in the conjugacy classes of the wreath product G_d. We will give a geometric definition of the double G-Hurwitz numbers, and then obtain an explicit formula via their algebraic definition. Finally, we will prove that the generating function of double G-Hurwitz numbers satisfies the various colored cut-and-join equations, and it is the product of several copies of tau functions of the 2 Toda hierarchy.