Hodge integrals with one λ-class

Abstract

Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space. ELSV formula connects the Hodge integrals with Hurwitz numbers, and the generating function of Hurwitz numbers satisfies the cut-and-join equation. Therefore, it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula. In this paper, at first, we will review the method introduced in Goulden et al.’s paper to get the λg conjecture for Hodge integral. Through some variables transformation, the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation. By comparing the coefficients of the lowest degree term of both sides in this equation, we can get the λg conjecture. Then, in a similar way, we obtain our main result in this paper: a recursive formula for Hodge integral of type contains only one λg−1-class. We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.