On double Hurwitz numbers with completed cycles

TL;DR

This paper explores generalized double Hurwitz numbers with completed cycles, providing geometric interpretations, a cut-and-join operator, polynomiality properties, and proposing a conjectural formula linking them to intersection theory.

Contribution

It introduces a geometric framework and a cut-and-join operator for completed cycles, and proposes a conjectural formula relating these Hurwitz numbers to intersection theory.

Findings

Established a geometric interpretation of completed cycle Hurwitz numbers

Derived a cut-and-join operator for these numbers

Proved a piecewise polynomiality property

Abstract

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.