Toda equations for Hurwitz numbers

2000/04/19 by Andrei Okounkov, Andreĭ Okounkov, Okounkov, Andrei · 6 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math-ph #math.AG #math.CO #math.MP #math.RT

paper · pdf · doi:10.48550/arxiv.math/0004128

10 pages

arxiv created 2000/04/19 · openalex publication_date 2000/04/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider ramified coverings of P¹ 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.

Cited by

• Random Young diagrams and Jacobi Unitary Ensemble
• Tropicalising hypergeometric τ-functions
• Open Hurwitz numbers and the mKP hierarchy
• Infinite wedge and random partitions
• Path operators and (q,t)-tau functions
• Admissible covers and stable maps
• Mirror symmetry and C^×

Related

• Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structure of Hurwitz spaces and combinatorial results)
• Open Hurwitz numbers and the mKP hierarchy
• From Toda Hierarchy to KP Hierarchy
• Generating Functions for Hurwitz-Hodge Integrals
• Hirota Quadratic Equations for the Extended Toda Hierarchy