Introduction to Classical Integrable Systems

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].

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“…The identity (4.1) is a standard result; e.g. see the formula (3.12) in [18] or (11.24) in [3]. The others follow from (4.1).…”

Section: Multi-point Correlation Functions Of the Kdv Hierarchymentioning

confidence: 96%

“…[3], pag. 389), expressing the generating function of τ 0 τ j , can be easily obtained as an immediate corollary of Theorem 1.2: Corollary 1.5.…”

Section: )mentioning

confidence: 99%

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

Bertola

Dubrovin

Yang

2016

Physica D: Nonlinear Phenomena

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“…It is necessary to emphasize that the form ω (i) is not closed and is degenerate on the space of all the operators L. It becomes closed after restriction onto certain subvarieties. As we shall see below, only the forms ω (0) and ω (1) are non-degenerate on the corresponding subvariety. That allows to regard the total space of operators L as a Poisson manifold foliated by symplectic leaves of two types.…”

Section: The Hamiltonian Theory Of the Reduced Systemsmentioning

confidence: 99%

“…Lemma 2.2 implies that ψ i (p) has pole of order i at the marked point p + . Then the standard arguments (for details see [1]) show that ψ i is a meromorphic function on Γ having away of the marked point g poles γ 1 , . .…”

Section: Necessary Factsmentioning

confidence: 99%