Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants

AbstractWe describe doubleHurwitz numbers as intersection numbers on the moduli space of curves Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

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Pseudo-Anosov Theory

10.23943/princeton/9780691147949.003.0015 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Braid Groups ◽

Intersection Numbers ◽

Branched Covers ◽

Algebraic Integers ◽

Dehn Twists ◽

Mapping Classes ◽

Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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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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Transverse Hilbert schemes and completely integrable systems

Complex Manifolds ◽

10.1515/coma-2017-0015 ◽

2017 ◽

Vol 4 (1) ◽

pp. 263-272 ◽

Cited By ~ 1

Author(s):

Niccolò Lora Lamia Donin

Keyword(s):

Integrable Systems ◽

Hilbert Scheme ◽

Symplectic Form ◽

Initial Surface ◽

Hilbert Schemes ◽

Completely Integrable Systems ◽

Completely Integrable ◽

Symplectic Surface ◽

Complex Symplectic ◽

Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

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Intersection triangles and block intersection numbers of Steiner systems

Mathematische Zeitschrift ◽

10.1007/bf01418308 ◽

1974 ◽

Vol 139 (2) ◽

pp. 87-104 ◽

Cited By ~ 14

Author(s):

Benedict H. Gross

Keyword(s):

Steiner Systems ◽

Intersection Numbers ◽

Block Intersection

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MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

Modern Physics Letters A ◽

10.1142/s0217732398002904 ◽

1998 ◽

Vol 13 (34) ◽

pp. 2731-2742 ◽

Cited By ~ 1

Author(s):

YUTAKA MATSUO

Keyword(s):

Hilbert Scheme ◽

Matrix Theory ◽

Fock Space ◽

Homology Class ◽

Orthogonal Basis ◽

Inner Product ◽

Virasoro Symmetry ◽

Hilbert Scheme Of Points ◽

The Matrix ◽

Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

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