Spin Calogero-Moser periodic chains and two dimensional Yang-Mills theory with corners

Abstract

The Quantum Calogero–Moser spin system is a superintegrable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of the corresponding simple Lie group. Here we describe its natural generalization known as quantum Calogero–Moser spin chain or $N$-spin Calogero–Moser system. In the first part of this paper we show that quantum Calogero–Moser spin chain is a quantum superintegrable systems. Then we show that the Euclidean multi-time propagator for this model can be written as a partition function of a two-dimensional Yang–Mills theory on a cylinder. Then we argue that the two-dimensional Yang–Mills theory withWilson loops with “outer ends” should be regarded as the theory on space times with non-removable corners. Partition functions of such theory satisfy non-stationary Calogero–Moser equations. In this paper the underlying Lie group $G$ is a compact connected, simply connected simple Lie group.