Virasoro constraints and polynomial recursion for the linear Hodge integrals

Abstract

The Hodge tau-function is a generating function for the linear Hodge integrals. It is also a tau-function of the KP hierarchy. In this paper, we first present the Virasoro constraints for the Hodge tau-function in the explicit form of the Virasoro equations. The expression of our Virasoro constraints is simply a linear combination of the Virasoro operators, where the coefficients are restored from a power series for the Lambert W function. Then, using this result, we deduce a simple version of the Virasoro constraints for the linear Hodge partition function, where the coefficients are restored from the Gamma function. Finally, we establish the equivalence relation between the Virasoro constraints and polynomial recursion formula for the linear Hodge integrals.

33 pages

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• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
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Cited by in corpus (5)

• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
• From Kontsevich-Witten to linear Hodge integrals via Virasoro operators
• A connection between the Kontsevich-Witten and Brezin-Gross-Witten tau-functions
• The ordered exponential representation of GKM using the $W_{1+\infty }$ operator