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 generalized 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