Matrix integrals and cluster algebras

Abstract

Cluster algebras were originated from the study of the canonical basis of quantum groups, as well as the investigation of the total positivity in semi-simple Lie theory. It is well known that the formal matrix integrals with cubic potential are the generating function of discrete surfaces for a given topology. In this presentation, we try to establish certain relationships between matrix integrals and geometric cluster algebras. We propose that these surfaces are also the geometric models of cluster algebras.