Duality and replicas for a unitary matrix model

TL;DR

This paper explores a generalized Airy matrix model with a parameter p, revealing duality and replica methods that recover phases of a related unitary matrix model, linking it to intersection numbers of moduli space.

Contribution

It introduces a duality and replica approach to analyze a generalized Airy matrix model, connecting it to the phases of a unitary matrix model and intersection theory.

Findings

Recovered weak and strong coupling phases of the unitary model

Established new results for phase expansions

Linked the unitary model to intersection numbers

Abstract

In a generalized Airy matrix model, a power $p$ replaces the cubic term of the Airy model introduced by Kontsevich. The parameter $p$ corresponds to Witten's spin index in the theory of intersection numbers of moduli space of curves. A continuation in $p$ down to $p= -2$ yields a well studied unitary matrix model, which exhibits two different phases in the weak and strong coupling regions, with a third order critical point in-between. The application of duality and replica to the $p$-th Airy model allows one to recover both the weak and strong phases of the unitary model, and to establish some new results for these expansions. Therefore the unitary model is also indirectly a generating function for intersection numbers.