The Gromov-Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals

Abstract We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed from our formulae. As a second result, we identify the Laguerre partition function with only positive couplings and a special value of the parameter $$\alpha =-1/2$$ α = - 1 / 2 with the modified GUE partition function, which has recently been introduced in Dubrovin et al. (Hodge-GUE correspondence and the discrete KdV equation. arXiv:1612.02333) as a generating function for Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.

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On Hurwitz numbers and Hodge integrals

Comptes Rendus de l Académie des Sciences - Series I - Mathematics ◽

10.1016/s0764-4442(99)80435-2 ◽

1999 ◽

Vol 328 (12) ◽

pp. 1175-1180 ◽

Cited By ~ 17

Author(s):

Torsten Ekedahl ◽

Sergei Lando ◽

Michael Shapiro ◽

Alek Vainshtein

Keyword(s):

Hurwitz Numbers ◽

Hodge Integrals

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Journal of High Energy Physics ◽

10.1007/jhep12(2020)038 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Matrix Models ◽

Matrix Model ◽

Special Interest ◽

Hermitian Matrix ◽

Geometric Meaning ◽

Kp Hierarchy ◽

Hurwitz Numbers ◽

Kp Equations ◽

Genus Expansion ◽

Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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Generating functions for weighted Hurwitz numbers: enumeration of branched coverings

Tau Functions and their Applications ◽

10.1017/9781108610902.015 ◽

2021 ◽

pp. 369-387

Keyword(s):

Generating Functions ◽

Hurwitz Numbers ◽

Branched Coverings

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Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

Letters in Mathematical Physics ◽

10.1007/s11005-021-01396-z ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Massimo Gisonni ◽

Tamara Grava ◽

Giulio Ruzza

Keyword(s):

Generating Functions ◽

Combinatorial Interpretation ◽

Hurwitz Numbers ◽

Jacobi Ensemble ◽

Matrix Ensembles ◽

Wilson Polynomials ◽

Unitary Ensemble ◽

Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

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RECURSION FOR MASUR-VEECH VOLUMES OF MODULI SPACES OF QUADRATIC DIFFERENTIALS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000638 ◽

2021 ◽

pp. 1-6

Author(s):

Maxim Kazarian

Keyword(s):

Moduli Spaces ◽

Recursion Relation ◽

Quadratic Differentials ◽

Hodge Integrals

Abstract We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

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