Hodge integrals and Hurwitz numbers via virtual localization

Abstract

Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula expressing Hurwitz numbers (counting covers of the projective line with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebro-geometric definition of a "relative space".

13 pages, 1 figure

References in corpus (3)

• Stable maps and branch divisors
• Relative Gromov-Witten Invariants
• Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

Cited by in corpus (12)

• Hurwitz numbers and intersections on moduli spaces of curves
• An algebro-geometric proof of Witten's conjecture
• Orbifold Hurwitz numbers and Eynard-Orantin invariants
• On double Hurwitz numbers in genus 0
• Geometry of meromorphic functions and intersections on moduli spaces of curves
• The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line
• The Hurwitz Enumeration Problem of Branched Covers and Hodge Integrals
• Loop equations and a proof of Zvonkine's $qr$-ELSV formula
• Nonlinear PDEs for Fredholm determinants arising from string equations
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• Combinatorics of binomial decompositions of the simplest Hodge integrals
• Tropical Hurwitz Spaces