Hurwitz numbers with completed cycles and Gromov--Witten theory relative to at most three points (2311.06316v1)

Abstract: Hurwitz numbers with completed cycles are standard Hurwitz numbers with simple branch points replaced by completed cycles. In fact, simple branch points correspond to completed $2$-cycles. Okounkov and Pandharipande have established the remarkable GW/H correspondence, saying that the stationary sectors of the Gromov--Witten theory relative to $r$ points equal Hurwitz numbers with $r$ branch points besides the completed cycles. However, from the viewpoint of computation, known results for Hurwitz numbers (standard or with completed cycles) are mainly for $r\leq 2$. It is hard to obtain explicit formulas and then discuss the structural properties for the cases $r>2$. In this paper, we obtain explicit formulas for the case $r=3$ and uncover a number of structural properties of these Hurwitz numbers. For instance, we discover a piecewise polynomiality with respect to the orders of the completed cycles in addition to the parts of the profiles of branch points as usual, we show that certain hook-shape Hurwitz numbers are building blocks of all our Hurwitz numbers, and we prove an analogue of the celebrated $\lambda_g$-conjecture.