An algebro-geometric proof of Witten’s conjecture

Kazarian, Maxim; Lando, S. K.

doi:10.1090/s0894-0347-07-00566-8

Cited by 125 publications

(134 citation statements)

References 22 publications

(32 reference statements)

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“…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].…”

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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“…Okounkov and R. Pandharipande [2001] and M. Mirzakhani [2007a;2007b] (see also [Mulase and Safnuk 2006]) gave different approaches through the enumeration of branched coverings of ‫ސ‬ 1 and the Weil-Petersson volume, respectively. More recently, M. Kazarian and S. Lando [2005] obtained an algebro-geometric proof starting from the ELSV formula [Ekedahl et al 2001]. In this section, we show that the Virasoro constraints is encoded in the join-cut relation of the Hurwitz numbers as the first nontrivial relation among the system of relations obtained via the asymptotic analysis.…”

Section: Dvv-formula Through the Asymptotic Analysismentioning

confidence: 62%

“…More recently, M. Kazarian and S. Lando [2005] obtained an algebro-geometric proof by using the ELSV formula to relate the intersection indices of ψ-classes to Hurwitz numbers.…”

Section: Introductionmentioning

confidence: 99%

Virasoro constraints and Hurwitz numbers through asymptotic analysis

Kim¹,

Liu²

2009

Pacific J. Math.

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We present a new method, using asymptotic analysis, to obtain a system of recursion relations for integrals on moduli spaces of curves. By applying the asymptotic analysis on the join-cut relation of the ELSV formula, we derive the recursion relation of R. Dijkgraaf and E. and H. Verlinde, which is exactly the Virasoro constraints for topological gravity. This gives an alternative proof of Witten's Conjecture using asymptotic analysis.

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“…In the meantime, we also proved the following result, which explains why in order to prove the Witten-Kontsevich theorem, it suffices to prove that the generating function (1) satisfies the classical KdV equation (2), as was done in [13].…”

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confidence: 73%

The n-point functions for intersection numbers on moduli spaces of curves

Liu

2011

Advances in Theoretical and Mathematical Physics

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Abstract. Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of n-point functions in terms of summation over binary trees.

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An algebro-geometric proof of Witten’s conjecture

Abstract: We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2-sphere.Comment: 12 pages, no figure

Cited by 125 publications

References 22 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

Virasoro constraints and Hurwitz numbers through asymptotic analysis

The n-point functions for intersection numbers on moduli spaces of curves

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