The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals
| dc.creator | Goulden, Ian | |
| dc.creator | Jackson, David | |
| dc.creator | Vakil, Ravi | |
| dc.date | 1999-10-01 | |
| dc.date.accessioned | 2026-07-07T05:31:00Z | |
| dc.date.available | 2026-07-07T05:31:00Z | |
| dc.description | Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permuations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for Hodge integrals on the moduli space of curves, has been a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one "lambda-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 should prove "all" such recursions. Furthermore, there are many more recursions than previously suspected from geometry (and indeed they should exist in all genera); as examples we present surprisingly simple new recursions in genus up to 3 that are geometrically mysterious. 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. | |
| dc.description | 21 pages | |
| dc.identifier | https://arxiv.org/abs/math/9910004 | |
| dc.identifier | http://arxiv.org/abs/math/9910004 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79187 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | 14H10, 81T40 | |
| dc.title | The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals | |
| dc.type | text |