Intersection theory from duality and replica

TL;DR

This paper introduces a duality and replica method approach to compute intersection numbers of moduli spaces, providing a simpler alternative to Kontsevich's original techniques and extending results to new models.

Contribution

It presents a novel duality and replica framework that generalizes Kontsevich's intersection number calculations to broader matrix models.

Findings

Recovered Kontsevich's intersection numbers using duality and replica methods

Extended intersection number computations to new matrix models

Provided a simpler alternative approach to existing methods

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.