G-Hurwitz Numbers, Colored Cut-and-join Equations, And Integrable Hierarchy

Hurwitz numbers are classical objects in enumerative geometry, which relate thegeometry of Riemann surfaces to the representation theory of symmetric groups. In thelate1890’s, Hurwitz considered the problem of counting topologically distinct, almostsimple, ramified covers of P1. He translated this geometric problem to a purely combi-natorial problem, namely, factorizing a permutation into transpositions, thus obtained aclosed formula in terms of characters of symmetric groups. The generating series of Hur-witz numbers can be written in a quite neat form using symmetric functions. In this form,one is able to prove that it satisfies some interesting partial diferential equations. Thefirst one is the cut-and-join equation, first proposed by Goulden and Jackson. The sec-ond kind of partial diferential equations satisfied by the generating functions of Hurwitznumbers are the KP or2-Toda hierarchy, via the famous boson-fermion correspondence.In fact, Okounkov showed that the generating series of double Hurwitz numbers is a taufunction of the2-Toda hierarchy.Inspired by the development of orbifold Gromov-Witten theory, one is naturally ledto consider the problem of generalizing Hurwitz numbers to the orbifold setting. In thispaper, we will give the definition of a G-branched cover for arbitrary finite group G, andthen analyze its monodromy representation. It turns out that the monodromy data arecontained in the conjugacy classes of the wreath product G_d. We will give a geometricdefinition of double G-Hurwitz numberH_G·(μ~+, μ~-, b) and then obtain an explicit formulafor H_G·(μ~+, μ~-, b) via its algebraic definition. This generalizes the result of[17]. Usingsymmetric functions, we define the generating function of double G-Hurwitz numbers,H_G·(t; p~+, p~-). We will prove that H_G·(t; p~+, p~-). satisfies the various colored cut-and-joinequations, and it is the product of|G*|copies of τ functions of the2-Toda hierarchy.