Recursion Formulae of Higher Weil-Petersson Volumes

Liu, Kefeng; Xu, Hao

doi:10.1093/imrn/rnn148

Cited by 31 publications

(37 citation statements)

References 22 publications

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“…Finally, we show that the families of Virasoro algebras used in [23,27,29,30,32,33] for the study of the topological recursion also fit into our framework and have a natural geometrical interpretation (see Section 4.4). In particular, it's shown that a family of τ -functions satisfying KdV and Virasoro is equivalent to a family of actions of the Witt algebra.…”

Section: Arxiv:11100729v2 [Mathag] 7 Jul 2015mentioning

confidence: 87%

“…Recent results on intersection theory are based on the so-called topological recursion [23,27,29,30,32,33] which are formulae involving families of τ -functions depending on an infinite number of parameters such that the whole family lies entirely on the space of functions satisfying KdV and Virasoro constraints. One of these families already appeared in Kontsevich's work [25,Section 3.4].…”

Section: Universal Familymentioning

confidence: 99%

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Algebro-Geometric Solutions of the Generalized Virasoro Constraints

Martín

José²

2015

SIGMA

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Abstract. We will describe algebro-geometric solutions of the KdV hierarchy whose τ -functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten-Kontsevich τ -function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with Gl(n)-opers on the punctured disk.

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Section: Arxiv:11100729v2 [Mathag] 7 Jul 2015mentioning

confidence: 87%

Section: Universal Familymentioning

confidence: 99%

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

Martín

José²

2015

SIGMA

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“…There exist several methods for computing the integrals (7.1), including application of the Virasoro constraints [43,44,24,46,54,38], the quasi-triviality approach [22,57,28], as well as an interesting method in the original paper of [1]. We propose a yet different approach, based on Thm.…”

Section: Application To Higher Weil-petersson Volumesmentioning

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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“…For generalizations of (III) to the case of intersection numbers involving higher Mumford's κ-classes see [LX1] , [E] and [Ka].…”

Section: Referencesmentioning

confidence: 99%

Towards large genus asymptotics of intersection numbers on moduli spaces of curves

Mirzakhani

Zograf

2015

Geom. Funct. Anal.

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We explicitly compute the diverging factor in the large genus asymptotics of the Weil-Petersson volumes of the moduli spaces of n-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the WeilPetersson volumes in the inverse powers of the genus with coefficients that are polynomials in n. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.

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Recursion Formulae of Higher Weil-Petersson Volumes

Cited by 31 publications

References 22 publications

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

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

Towards large genus asymptotics of intersection numbers on moduli spaces of curves

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