The intersection numbers of the p-spin curves from random matrix theory

This article investigates the intersection numbers of the moduli space of p-spin curves with the help of matrix models. The explicit integral representations that are derived for the generating functions of these intersection numbers exhibit p Stokes domains, labelled by a "spin"-component l taking values l = −1, 0, 1, 2, ..., p − 2. Earlier studies concerned integer values of p, but the present formalism allows one to extend our study to half-integer or negative values of p, which turn out to describe new types of punctures or marked points on the Riemann surface. They fall into two classes: Ramond (l = −1), absent for positive integer p, and Neveu-Schwarz (l = −1). The intersection numbers of both types are computed from the integral representation of the n-point correlation functions in a large N scaling limit. We also consider a supersymmetric extension of the random matrix formalism to show that it leads naturally to an additional logarithmic potential. Open boundaries on the surface, or admixtures of R and NS punctures, may be handled by this extension.

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“…For p = 3, similarly from the explicit expression for the two-point function [44], one does not find R-punctures. Indeed for p = 3, we have [44],…”

Section: Two Point Correlation Function U( 1 2 )mentioning

confidence: 61%

Punctures and p-Spin Curves from Matrix Models

Brézin

Hikami

2020

J Stat Phys

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“…This result can be obtained by applying equation (4.37) [7]. In this O(2N) model, we have the following condition, the same as for Riemann surfaces with spin j and s-marked points…”

Section: Gue X ∈ U(n)mentioning

confidence: 96%

“…The Airy function Ai(z) was used for the case of two marked points for p = 3 in [7]. We obtain thus the explicit expression for the intersection numbers…”

Section: Gue X ∈ U(n)mentioning

confidence: 99%

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Random matrix, singularities and open/close intersection numbers

Brézin

Hikami

2015

J. Phys. A: Math. Theor.

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The s-point correlation function of a Gaussian Hermitian random matrix theory, with an external source tuned to generate a multi-critical singularity, provides the intersection numbers of the moduli space for the p-th spin curves through a duality identity. For one marked point, the intersection numbers are expressed to all order in the genus by Bessel functions. The matrix models for the Lie algebras of O(N) and Sp(N) provide the intersection numbers of non-orientable surfaces. The Kontsevich-Penner model, and higher p-th Airy matrix model with a logarithmic potential, are investigated for the open intersection numbers, which describe the topological invariants of non-orientable surfaces with boundaries. String equations for open/closed Riemann surface are derived from the structure of the s-point correlation functions. The Gromov-Witten invariants of CP 1 model are evaluated for one marked point as an application of the present method.

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“…We then solve the non-compact topological quantum field theory coupled to gravity using a dispersionful (or quantum) KdV hierarchy in section 4. In section 5, we discuss the extent to which our solution relates to the approach of determining the correlators of a non-compact model through analytic continuation in the central charge (or level) of the conformal field theory [18,19]. We conclude in section 6 and comment on the conceptual implications of the duality for two-dimensional gravity.…”

Section: Introductionmentioning

confidence: 99%

A duality in two-dimensional gravity

Ashok

Troost

2019

J. High Energ. Phys.

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We demonstrate an equivalence between two integrable flows defined in a polynomial ring quotiented by an ideal generated by a polynomial. This duality of integrable systems allows us to systematically exploit the Korteweg-de Vries hierarchy and its tau-function to propose amplitudes for non-compact topological gravity on Riemann surfaces of arbitrary genus. We thus quantize topological gravity coupled to non-compact topological matter and demonstrate that this phase of topological gravity at N = 2 matter central charge larger than three is equivalent to the phase with matter of central charge smaller than three.

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The intersection numbers of the p-spin curves from random matrix theory

Cited by 10 publications

References 29 publications

Punctures and p-Spin Curves from Matrix Models

Punctures and p-Spin Curves from Matrix Models

Random matrix, singularities and open/close intersection numbers

A duality in two-dimensional gravity

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