Hodge Integrals and Hurwitz Numbers via Virtual Localization

By T. Graber, Ravi Vakil. Compositio Mathematica, 2000

104 citations recorded by the source corpus.

We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. Our proof uses virtual localization on the moduli space of stable maps. We describe how the proof could be simplified by the proper algebro-geometric definition of a 'relative space'. Such a space has recently been defined by J. Li.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b7f9fb3e7493de17680f752dc396c0f2d0e55e21

How researchers see this paper

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In Geometry and Topology

1. R. Pandharipande
2. Marcos Mariño
3. M. Mulase
4. A. Okounkov
5. V. Bouchard
6. Naizhen Zhang
7. Ian P. Goulden
8. Ronald van Luijk

In Mathematical Physics

1. R. Pandharipande
2. A. Okounkov
3. M. Mulase
4. Marcos Mariño
5. Naizhen Zhang
6. Ronald van Luijk
7. V. Bouchard
8. B. Eynard

Citations in the corpus

• An algebro-geometric proof of Witten's conjecture (2006, background): “Our proof is based on the following, now well-known, properties of these numbers: • the ELSV formula [4, 5], see also [8], [13], relating Hurwitz numbers to the intersection theory on moduli spaces; • the relationship between Hurwitz numbers and integrable hierarchies conjectured by Pandharipande [1”
• K3 surfaces with Picard number one and infinitely many rational points (2005, methodology): “This result, the most technical in this paper, is established by computing the genus 0, one-pointed invariants via localization, together with the use of some beautiful Hodge integral computations from [Faber and Pandharipande 2000; Ekedahl et al. 2001; Graber and Vakil 2003a].”
• The Equivariant Gromov-Witten theory of P**1 (2002, methodology): “The operator formalism for Hodge integrals relies crucially upon a formula due to Ekedahl, Lando, Shapiro, and Vainstein (see [6] [7] [11] and also [21]) expressing basic Hurwitz numbers as Hodge integrals.”
• Hurwitz numbers, matrix models and enumerative geometry (2007, background): “Hurwitz numbers turn out to be closely related to topological gravity in two dimensions [GJV, OP], to Hodge integrals [ELSV, FP, GV ], and to the Gromov–Witten theory of P1 [OP1].”
• Towards the geometry of double Hurwitz numbers (2003, methodology): “We next prove an analogue of the genus expansion ansatz of Itzykson and Zuber for intersection numbers on the moduli space of curves (Itzykson-Zuber, 1992, (5.32)). The Itzykson-Zuber Ansatz was proved by Eguchi-Yamada-Yang (1995) and later by Goulden-Jackson-Vakil (2001, Theorem 3.”
• Higher genus Gromov–Witten invariants as genus zero invariants of symmetric products (2003, unknown): “Let Γt be the category whose objects are objects η of exp(Υt), together with an order two isomorphism σ : T (η) ∼= T (η), commuting with the multiplicity map m : T (η) → Z≥1.”
• Polynomial recursion formula for linear Hodge integrals (2009, methodology): “The celebrated Ekedahl-Lando-Shapiro-Vainshtein formula [6, 16, 29] relates Hurwitz numbers and linear Hodge integrals on the Deligne-Mumford moduli stack Mg,` consisting of stable algebraic curves of genus g with ` distinct nonsingular marked points subject to the stability condition 2g − 2 + ` > 0”
• Abelian Hurwitz-Hodge integrals (2008, background): “Since r must be an integer, Mg,γ(P1[a],µ) is empty unless the parity condition d = ∑ni=1 γi (mod a) holds.”