The loop equations for noncommutative geometries on quivers

Abstract

We define a path integral over Dirac operators that averages over noncommutative geometries on a fixed graph, as the title reveals, using quiver representations. We prove algebraic relations that are satisfied by the expectation value of the respective observables, computed in terms of integrals over unitary groups, with weights defined by the spectral action. These equations generalise the Makeenko-Migdal equations -- the constraints of lattice gauge theory -- from lattices to arbitrary graphs. As a perspective, our loop equations are combined with positivity conditions (on a matrix parametrised by composition of Wilson loops). On a simple quiver this combination known as `bootstrap' is fully worked out. The respective partition function boils down to an integral known as Gross-Witten-Wadia model; their solution confirms the solution bootstrapped by our loop equations.