Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra

Abstract

In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich-Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac-Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.

arXiv comment65 pages; minor corrections

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Cited by in corpus (20)

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• Weighted Hurwitz numbers and topological recursion: an overview
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• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies
• Open intersection numbers and free fields
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Relations on ${\overline{\mathcal{M}}}_{g,n}$ and the negative $r$-spin Witten conjecture
• 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
• Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers
• On higher Brézin-Gross-Witten tau-functions
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
• 4D limit of melting crystal model and its integrable structure
• From Kontsevich-Witten to linear Hodge integrals via Virasoro operators
• Matrix model for the stationary sector of Gromov-Witten theory of ${\mathbf{P}}^1$
• The ordered exponential representation of GKM using the $W_{1+\infty }$ operator
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula