Transformation Groups for Soliton Equations

Date, Etsurō; Kashiwara, Masaki; Miwa, Tetsuji; Jimbo, Michio

doi:10.1142/9789812812650_0032

Cited by 249 publications

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“…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.…”

Section: Introduction and Resultsmentioning

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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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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Section: Introduction and Resultsmentioning

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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“…Although specialists in integrable hierarchies are well aware of these facts, it is not an easy task to find their compact and readable exposition. We refer the reader to [15], [2], and [16] for a description of the relationship between integrable hierarchies and the geometry of semi-infinite Grassmannians. The KP hierarchy is a system of partial differential equations for an unknown function H. The KP equation (2.3) is the first equation in this system.…”

Section: Odds and Endsmentioning

confidence: 99%

An algebro-geometric proof of Witten’s conjecture

Kazarian

Lando

2007

J. Amer. Math. Soc.

126

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“…In the papers of E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Lie algebras and vertex operators were incorporated into the soliton theory. (See the series of seven notes [DJKM1], and the paper [DJKM2]). In this setting, the τ -function appears in very general way.…”

Section: The Tau-function Of the Schlesinger Systemmentioning

confidence: 99%

Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I

Katsnelson

Volok

2004

Current Trends in Operator Theory and Its Applications

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Abstract. In the present paper we discuss the general facts, concerning the Schlesinger system: the τ -function, the local factorization of solutions of Fuchsian equations and holomorphic deformations. We introduce the terminology "isoprincipal" for the deformations of Fuchsian equations with general (not necessarily non-resonant) matrix coefficients, corresponding to solutions of the Schlesinger system. Every isoprincipal deformation is isomonodromic. The converse is also true in the non-resonant case, but not in general.In the forthcoming sequel we shall give an explicit description of a class of rational solutions of the Schlesinger system, based on the techniques, developed here, and the realization theory for rational matrix functions. NOTATIONS.• C stands for the complex plane;• C stands for the extended complex plane (= the Riemann sphere): C = C ∪ ∞; • C n stands for the n -dimensional complex space; • in the coordinate notation, a point t ∈ C n is written as t = (t 1 , . . . , t n ) ; • C n * is the set of those points t = (t 1 , . . . , t n ) ∈ C n , whose coordinates t 1 , . . . , t n are pairwise different: C n * = C n \ 1≤i,j≤n, i =j {t : t i = t j } ; • M k stands for the set of all k × k matrices with complex entries; • [ . , . ] denotes the commutator: for A, B ∈ M k , [A, B] = AB − BA;• I stands for the unity matrix of the appropriate dimension; • R(M k ) stands for the set of all rational M k -valued functions R with detR(z) ≡ 0; • P(R) stands for the set of all poles of the function R, N (R) stands for the set of all poles of the function R −1 ; P(R) is said to be the pole set of the function R, N (R) is said to be the zero set of the function R.

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Transformation Groups for Soliton Equations

Cited by 249 publications

References 0 publications

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

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

An algebro-geometric proof of Witten’s conjecture

Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I

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