Correlation functions of the KdV hierarchy and applications to intersection numbers over M¯g,n

Abstract

We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formulae of a new type for the generating series of intersection numbers of ψ-classes as well as of mixed ψ-and κ-classes in full genera.Keywords. KdV hierarchy; tau-function; wave function; correlation function; intersection number. Contents Introduction and resultsThe famed Korteweg-de Vries (KdV) equationhas long been known to be integrable. It belongs to an infinite family of pairwise commuting nonlinear evolutionary PDEs called the KdV hierarchy. The hierarchy can be described in terms of isospectral deformations of the Lax operator( 1.2) Namely, the k-th equation of the KdV hierarchy, in the normalization of the present paper, reads( 1.4) Here, the independent variables t 0 , t 1 , t 2 , ... are called times. The symbol L 2k+1 2 + stands for the differential part of the pseudo-differential operator L 2k+1 2 , see e.g. the book [16] for details. The k = 1 equation of (1.3) coincides with (1.1). As customary in the literature, we shall identify t 0 with the spatial variable x.The notion of tau-function for the KdV hierarchy was introduced by the Kyoto school [50,27,15] during the 1970s-1980s. In 1991, E. Witten, in his study of two-dimensional quantum gravity [52], conjectured that the generating function of the intersection numbers of ψ-classes on the Deligne-Mumford moduli spaces M g,n of stable algebraic curves is a tau-function of the KdV hierarchy. Witten's conjecture was later proved by M. Kontsevich [34]; see [33,49,44] for several alternative proofs. Moreover the so-called "tau structures" of KdV-like hierarchies became one of the central subjects in the study of the deep relation between integrable hierarchies and Gromov-Witten invariants [22,20,21].

FZZT branes in JT gravity and topological gravity

“…This shift of coupling t k due to the insertion of the FZZT branes agrees with the known result in the literature 4 [20] (see e.g. [21] for the expression in our convention). In the theory of soliton equations the shift is generated by the action of the vertex operator of [22], which is viewed as the infinitesimal Bäcklund transformation for the KdV equation [23].…”

“…Let us consider the genus expansion of the BA function ψ(ξ) = ψ(ξ; t 0 ). It is known that the BA function is written as a ratio of tau-function τ (t) = e F (t) [20,21]:…”

JT gravity, KdV equations and macroscopic loop operators

“…where P (g) is a degree-2 polynomial of g. The = 0 case is computed in [30] with the famous result P 0 (g) = 1. The = 1 case has appeared in [36] with the result…”

“…As discussed in [36,63], one can integrate the equation for R in (2.49) once. By multiplying R to the first equation in (2.49) we find On the other hand, we can take the classical limit of (D.9) directly.…”

Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials

“…The hierarchy (2.2) is known to be tau-symmetric [7,21,25,38]. In the setting of [5,7], that means that, there exist a family of differential polynomials Ω k, of q can , indexed by two integers k, ∈ E + (satisfying certain natural non-degeneracy condition), such that for all m, k, ∈ E + ,…”