On double Hurwitz numbers with completed cycles

In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection theory of some analogues of the moduli space of curves. The structure of these conjectural "intersection numbers" is discussed in detail.Comment: 31 page

Equivalence of ELSV and Bouchard-Mariño conjectures for r-spin Hurwitz numbers

We propose two conjectures on Hurwitz numbers with completed (r+1)-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard-Mariño conjecture for ordinary Hurwitz numbers. Our r-ELSV formula is an equality between a Hurwitz number and an integral over the space of r-spin structures, that is, the space of stable curves with an rth root of the canonical bundle. Our r-BM conjecture is the statement that n-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve x=−yr+logy in the sense of Chekhov, Eynard, and Orantin. We show that the r-ELSV formula and the r-BM conjecture are equivalent to each other and provide some evidence for both

Integrals of ψ-classes over double ramification cycles

A double ramification cycle, or DR-cycle, is a codimension $g$ cycle in the moduli space $\overline{\mathcal M}_{g,n}$ of stable curves. Roughly speaking, given a list of integers $(a_1,\ldots,a_n)$, the DR-cycle ${\rm DR}_g(a_1,\ldots,a_n) \subset\overline{\mathcal M}_{g,n}$ is the locus of curves $(C,x_1,\ldots,x_n)$ such that the divisor $\sum a_ix_i$ is principal. We compute the intersection numbers of DR-cycles with all monomials in $\psi$-classes