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The Gromov–Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals
Author(s) -
Goulden I. P.,
Jackson D. M.,
Vakil R.
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/83.3.563
Subject(s) - mathematics , genus , moduli space , symplectic geometry , ansatz , hurwitz polynomial , pure mathematics , conjecture , algebraic number , generating function , series (stratigraphy) , recursion (computer science) , combinatorics , algebra over a field , mathematical physics , mathematical analysis , polynomial , paleontology , botany , algorithm , biology
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov‐Witten potential F of a point, the generating series for descendent integrals on the moduli space of curves, is a central object of study in Gromov‐Witten theory. We define a slightly enriched Gromov‐Witten potential G (including integrals involving one ‘λ‐class’), and show that, after a non‐trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber. We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented proves all such recursions. As examples we present surprisingly simple new recursions in genus 0 to 3. Similar recursions should exist for all genera. As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self‐contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics. 2000 Mathematical Subject Classification : primary 14H10, 81T40; secondary 05C30, 58D29.