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

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).

24 pages

References in corpus (4)

• Generation of Matrix Models by W-operators
• Algebra of differential operators associated with Young diagrams
• Cut-and-Join operator representation for Kontsevich-Witten tau-function
• Virasoro constraints for Kontsevich-Hurwitz partition function

Cited by in corpus (12)

• On KP-integrable Hurwitz functions
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Genus expansion of HOMFLY polynomials
• On genus expansion of superpolynomials
• Cut-and-join structure and integrability for spin Hurwitz numbers
• Checks of integrality properties in topological strings
• Spectral curves and $W$-representations of matrix models
• Hurwitz numbers from Feynman diagrams
• From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
• From $r$-Spin Intersection Numbers to Hodge Integrals
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• Twisted Cherednik spectrum as a $q,t$-deformation