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

Liu, Kefeng; Xu, Hao

doi:10.4310/atmp.2011.v15.n5.a1

Cited by 18 publications

(27 citation statements)

References 24 publications

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“…As we will see, the contribution from the reduced invariants are correction terms when the assumption of simply decorated stars is not satisfied. The formula (73) can be viewed as a combinatorial solution to the n-point function of [24] in genus one. It is interesting to find a higher genera analog of this formula, which may shed light on the computation of Gromov-Witten invariants in higher genera.…”

Section: Simply Decorated Starsmentioning

confidence: 99%

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

Hu¹

2014

Int Math Res Notices

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For local Calabi-Yau manifolds which are total spaces of vector bundle over algebraic GKM manifolds, we propose a formal definition of reduced Genus one Gromov-Witten invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced formula, whose global version in the compact cases is due to A. Zinger. As an application we prove the conjecture in a previous article on the genus one Gromov-Witten invariants of local Calabi-Yau manifolds which are total spaces of concave splitting vector bundles over projective spaces. In particular, we prove the mirror formulae for genus one Gromov-Witten invariants of K P 2 and K P 3 , conjectured by Klemm, Zaslow and Pandharipande. In the appendix we derive the modularity of genus one Gromov-Witten invariants for the local P 2 as a consequence of the results on Ramanujan's cubic transformation. Inspired by the localized standard versus reduced formula, we show that the ordinary genus one Gromov-Witten invariants of Calabi-Yau hypersurfaces in projective spaces can be computed by virtual localization and quantum hyperplane property after the contribution of a genus one vertex is replaced by a modified one.

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Section: Simply Decorated Starsmentioning

confidence: 99%

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

Hu¹

2014

Int Math Res Notices

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“…Theorem 2.1 can be restated as [14]. Suppose for fixed integers r > 0 and s ≥ 0, equation (2.4) holds for all integers m ≥ 2g + r + s − 3.…”

Section: Conventionsmentioning

confidence: 99%

New topological recursion relations

Liu

Pandharipande

2010

J. Algebraic Geom.

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Simple boundary expressions for the k th power of the cotangent line class ψ 1 on M g,1 are found for k ≥ 2g. The method is by virtual localization on the moduli space of maps to P 1 . As a consequence, nontrivial tautological classes in the kernel of the boundary push-forward mapare constructed. The geometry of genus g + 1 curves, then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by

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“…It would be interesting to prove directly the equivalence between the formula (1.36), the (explicit) integral/recursive formulae of "n-point functions" given by Okounkov [48], Liu-Xu [36,37], Brézin-Hikami [10,11], and Kontsevich's main identity [34]. We indicate the relation between these three as follows.…”

Section: Further Remarksmentioning

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

Cited by 18 publications

References 24 publications

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

New topological recursion relations

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

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