Explicit closed algebraic formulas for Orlov–Scherbin $n$-point functions

Bychkov, Boris; Dunin-Barkowski, Petr; Kazarian, Maxim; Shadrin, Sergey

doi:10.5802/jep.202

Explicit closed algebraic formulas for Orlov–Scherbin

n

-point functions
[Formules algébriques closes explicites pour les fonctions à

n

points d’Orlov-Scherbin]

Bychkov, Boris ¹ ; Dunin-Barkowski, Petr ² ; Kazarian, Maxim ³ ; Shadrin, Sergey ⁴

¹ Faculty of Mathematics, National Research University Higher School of Economics Usacheva 6, 119048 Moscow, Russia and Center of Integrable Systems, P.G. Demidov Yaroslavl State University Sovetskaya 14, 150003, Yaroslavl, Russia and current affiliation: Department of Mathematics, University of Haifa Mount Carmel, 3488838, Haifa, Israel
² Faculty of Mathematics, National Research University Higher School of Economics Usacheva 6, 119048 Moscow, Russia and HSE–Skoltech International Laboratory of Representation Theory and Mathematical Physics, Skoltech Nobelya 1, 143026, Moscow, Russia and NRC “Kurchatov Institute” – ITEP 117218 Moscow, Russia
³ Faculty of Mathematics, National Research University Higher School of Economics Usacheva 6, 119048 Moscow, Russia and Center for Advanced Studies, Skoltech Nobelya 1, 143026, Moscow, Russia
⁴ Korteweg-de Vries Institute for Mathematics, University of Amsterdam Postbus 94248, 1090 GE Amsterdam, The Netherlands

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1121-1158.

Résumé
Abstract

Nous présentons une nouvelle formule explicite en termes de sommes sur les graphes pour les fonctions de corrélation à $n$ points des nombres de Hurwitz doubles pondérés formels généraux provenant des fonctions tau de Kadomtsev-Petviashvili de type hypergéométrique (également connues sous le nom de fonctions de partition d’Orlov-Scherbin). Nous utilisons notamment le changement de variables suggéré par la courbe spectrale associée, et notre formule s’avère être une expression polynomiale dans un certain petit ensemble de fonctions formelles définies sur la courbe spectrale.

We derive a new explicit formula in terms of sums over graphs for the $n$ -point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev–Petviashvili tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.

Reçu le : 2021-09-20
Accepté le : 2022-07-04
Publié le : 2022-07-13

DOI : 10.5802/jep.202

Classification : 05A15, 37K10, 14H30, 14N10, 37K30
Keywords: Hurwitz numbers, KP tau functions, Fock space
Mot clés : Nombres de Hurwitz, tau fonction KP, espace de Fock

Affiliations des auteurs :

Bychkov, Boris ¹ ; Dunin-Barkowski, Petr ² ; Kazarian, Maxim ³ ; Shadrin, Sergey ⁴

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     title = {Explicit closed algebraic formulas for {Orlov{\textendash}Scherbin} $n$-point functions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Bychkov, Boris; Dunin-Barkowski, Petr; Kazarian, Maxim; Shadrin, Sergey. Explicit closed algebraic formulas for Orlov–Scherbin $n$-point functions. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1121-1158. doi : 10.5802/jep.202. http://www.numdam.org/articles/10.5802/jep.202/

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