Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations

TL;DR

This paper explores the algebraic structure of Hurwitz partition functions, demonstrating their relation to WDVV equations and introducing new solutions via cut-and-join operators and their multiplication properties.

Contribution

It establishes the multiplication properties of cut-and-join operators and connects them to solutions of the WDVV equations in the context of Hurwitz partition functions.

Findings

Ordinary Hurwitz numbers yield trivial WDVV solutions with finite variables.

Generalized Hurwitz numbers provide non-trivial solutions with infinite variables.

A specific solution is linked to a subring generated by dilatation operators.

Abstract

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. The ordinary Hurwitz numbers for a given number of sheets in the covering provide trivial (sums of exponentials) solutions to the WDVV equations, with finite number of time-variables. The generalized Hurwitz numbers from arXiv:0904.4227 provide a non-trivial solution with infinite number of times. The simplest solution of this type is associated with a subring, generated by the dilatation operators tr X(d/dX).