On the Characterization of Classical Dynamical Systems Using Supersymmetric Nonlinear $σ$-models

Abstract

We construct a two dimensional nonlinear $σ$-model that describes the Hamiltonian flow in the loop space of a classical dynamical system. This model is obtained by equivariantizing the standard N=1 supersymmetric nonlinear $σ$-model by the Hamiltonian flow. We use localization methods to evaluate the corresponding partition function for a general class of integrable systems, and find relations that can be viewed as generalizations of standard relations in classical Morse theory.