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

2023/11/10 by Ricky X. F. Chen, Chen, Ricky X. F., Zhen-Ran Wang +1
Mathematics · #05E10 #14H10 #20B30 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2311.06316

openalex publication_date 2023/11/10 · openalex created_date 2023/11/15 · openalex updated_date 2026/07/28

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≤ 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 λ_g-conjecture.

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