Towards the geometry of double Hurwitz numbers

By Ian P. Goulden, David M. Jackson, Ravi Vakil. 2003

169 citations recorded by the source corpus.

Abstract Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞ , and the branching over 0 and ∞ specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande's proof of Witten's conjecture. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case m = 1 (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber's intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves M g . For example, three formulae we give there will imply Faber's intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4f227f8f6cdecd6847f39d2f20c2125013e11b8c

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Citations in the corpus

Integrals of ψ-classes over double ramification cycles (2012, background): “,an) coincides, up to a simple factor, with the formula for one-part double Hurwitz numbers found in [2] (first equality of Theorem 3.”
Abelian Hurwitz-Hodge integrals (2008, background): “1-point Series For μ = (d ) consisting of a single part, the entire generating series for double Hurwitz numbers was computed in [13] as ∑ g≥0 t (−1)Hg(ν, (d )) = r! d r−1 |Aut(ν)| ∏ k≥1 ( sin(kt/2) kt/2 )mk(ν)−δk,1 , (20)”
Double Hurwitz numbers via the infinite wedge (2010, methodology): “In [GJV05], the following formula for one part double Hurwitz numbers, i.”
Chamber behavior of double Hurwitz numbers in genus 0 (2008, background): “ved. The only example considered in [Sh1] is n= 2, in which case, after some eﬀort, one gets the correct answer. A diﬀerent ELSV-type formula for the one-part double Hurwitz numbers is conjectured in [GJVa]. It is proved there for the case g = 0. The answer look as follows: h0 (d);ν = (n−1)!d Z M 0,”
THE MODULI SPACE OF CURVES AND GROMOV-WITTEN THEORY (2006, background): “licitly for W, and hence prove Faber’s intersection number conjecture. Because genus 0double Hurwitz number H0 α,β are only currently well-understood when one of the partition has at most 3parts (see [GJV2]), this program is not yet complete. However, it indeed yields: 7.11. Theorem [GJV3]. — Faber’”
Tropical Hurwitz numbers (2008, methodology): “We point the attention of the reader to a technical detail: in this section we wish to adopt the definition of Hurwitz numbers in [6], which differs from the classical one used so far in that the preimages of 0 and ∞ are marked.”