Toda equations for Hurwitz numbers

By A. Okounkov. 2000

301 citations recorded by the source corpus.

We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a tau-function for the Toda lattice hierarchy of Ueno and Takasaki.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f5638c72d37eb031e2e1d2f9f3010e32369df3fe

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Marcos Mariño
2. R. Pandharipande
3. Paul D. Johnson
4. B. Eynard
5. M. Mulase
6. A. Okounkov
7. V. Bouchard
8. Dimitri Zvonkine

In Statistical and Nonlinear Physics

1. Marcos Mariño
2. B. Eynard
3. V. Bouchard
4. R. Pandharipande
5. Paul D. Johnson
6. A. Okounkov
7. Ian P. Goulden
8. J. Harnad

Citations in the corpus

• Equivariant Gromov-Witten theory of one dimensional stacks (2009, background): “hort detour illustrating the connection to integrable systems: it shows that wreath Hurwitz numbers satisfy multiple commuting copies of the 2-Toda hierarchy. This result is an easy generalization of [Oko00], and should be viewed as a gentle inroduction to the 2-Toda hierarchy. 4.1 Hurwitz Numbers a”
• Infinite wedge and random partitions (1999, background): “These numbers, which are certain rather complicated polynomials in Bernoulli numbers, can be identifled with volumes of certain moduli spaces and are important in ergodic theory [14]. For the coverings of the sphere, see [ 28 ].”
• Fermionic representation for basic hypergeometric functions related to Schur polynomials (2000, background): “Let us note the paper [ 60 ] which turns to be different example of the tau function (3.3.1).”
• Abelian Hurwitz-Hodge integrals (2008, background): “The latter formalism naturally connects Hurwitz numbers to integrable systems [23; 26; 27].”
• Chamber behavior of double Hurwitz numbers in genus 0 (2008, unknown): “ extends previous computations carried out in [Ku] for n= 1,2. Much less is known for double Hurwitz numbers with arbitrary µand ν. In fact, there are only two general results. First, it is proved in [Ok] that the exponent of the generating function for the double Hurwitz numbers is a τ-function for”
• Algebra of differential operators associated with Young diagrams (2012, background): “−1,∆k−1,∆k >, (see, e.g., [2]). The Hurwitz numbers appear in different framewroks: strings and QCD [8], mirror symmetry [7], theory of singularities [1], matrix models [16, 20], integrable systems [26, 14, 25], Yang-Mills theory [6, 14] and the theory of moduli of curves [13, 12, 20] and other br”
• Polynomiality of monotone Hurwitz numbers in higher genera (2012, background): “The double Hurwitz numbers were first studied by Okounkov [16], who addressed a conjecture of Pandharipande [17] in Gromov–Witten theory by proving that a certain generating function for these numbers is a solution of the 2-Toda lattice hierarchy from the theory of integrable systems.”
• Towards the geometry of double Hurwitz numbers (2003, background): “3] for some of the subsequent connections, for example through the Gopakumar-Vafa conjecture).”
• Weighted Hurwitz Numbers and Topological Recursion (2018, background): “There has also been important progress in relating Hurwitz numbers to other classes of enumerative geometric invariants [2, 21,26,33,40,55,65,68,69,72] and matrix models [1, 6–9,17,21,25,61,64,68,69].”
• The Equivariant Gromov-Witten theory of P**1 (2002, background): “The Toda conjecture was further studied in [24] [19] [8] [9] and, for the stationary sector, proved in [22].”
• The KP hierarchy, branched covers, and triangulations (2008, background): “This result was first proved by Okounkov [Ok], using a different method, and then more recently by Orlov [Or], and Kazarian [Ka].”
• The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers (2009, methodology): “In recent years these equations have played an essential role in the study of Hurwitz numbers [24, 25, 34, 40, 41, 42].”
• 2D Toda τ-Functions as Combinatorial Generating Functions (2015, unknown): “Several examples of this construction are provided in Section 3, starting with the generating function for the double Hurwitz numbers first studied by Okounkov [26].”
• Hurwitz numbers, matrix models and enumerative geometry (2007, methodology): “(3.43) P(z) = −z + O (z 2 ). Then ¯ must satisfy”