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

TL;DR

This paper develops a supersymmetric nonlinear sigma-model to analyze classical dynamical systems, using localization techniques to evaluate partition functions and generalize Morse theory relations.

Contribution

It introduces a novel supersymmetric sigma-model framework for classical dynamics and applies localization to derive new relations extending Morse theory.

Findings

Partition function evaluated for integrable systems

Relations generalizing classical Morse theory

Framework connects supersymmetry with classical dynamics

Abstract

We construct a two dimensional nonlinear $\sigma$-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 $\sigma$-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.