From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators

Abstract

In this letter,we present our conjecture on the connection between the Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the $GL(\infty )$ group element. An important feature of this group element is its simplicity: this is a group element of the Virasoro subalgebra of $gl(\infty )$. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich--Witten tau-function.

13 pages

References in corpus (11)

• Instantons and Merons in Matrix Models
• Generation of Matrix Models by W-operators
• Free fermions and tau-functions
• Integrability of Hurwitz Partition Functions. I. Summary
• Matrix Models for Random Partitions
• Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
• Cut-and-Join operator representation for Kontsevich-Witten tau-function
• Virasoro constraints for Kontsevich-Hurwitz partition function
• CFT and topological recursion
• Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations
• Matrix models as conformal field theories: genus expansion

Cited by in corpus (8)

• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Genus expansion of matrix models and $\hslash$ expansion of KP hierarchy
• Virasoro constraints and polynomial recursion for the linear Hodge integrals
• From $r$-Spin Intersection Numbers to Hodge Integrals
• Connecting the Kontsevich-Witten and Hodge tau-functions by the $\widehat{GL(\infty )}$ operators
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• From Kontsevich-Witten to linear Hodge integrals via Virasoro operators