Intersection theory from duality and replica

Abstract

Kontsevich's work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In this article we show that a duality between k-point functions on $N\times N$ matrices and N-point functions of $k\times k$ matrices, plus the replica method, familiar in the theory of disordered systems, allows one to recover Kontsevich's results on the intersection numbers, and to generalize them to other models. This provides an alternative and simple way to compute intersection numbers with one marked point, and leads also to some new results.

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• Sum rules for characters from character-preservation property of matrix models
• From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
• Absorption of closed strings by giant gravitons
• On an Airy matrix model with a logarithmic potential
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• Computing topological invariants with one and two-matrix models
• Scaling limits of correlations of characteristic polynomials for the Gaussian beta-ensemble with external source
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• The Harer--Zagier recursion for an irregular spectral curve
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• Averages of products and ratios of characteristic polynomials in polynomial ensembles
• Punctures and p-spin curves from matrix models II
• Superintegrability and Kontsevich-Hermitian relation
• Random Matrix, Singularities and Open/Close Intersection Numbers
• Punctures and p-spin curves from matrix models
• Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential
• Phase transitions for products of characteristic polynomials under Dyson Brownian motion