On the characterization of classical dynamical systems using supersymmetric nonlinear $\sigma$-models

We construct a two dimensional nonlinear \sigma-model that describes the Hamiltonian flow in the loop space of classical dynamical systems. This model is obtained from the standard N=1 supersymmetric nonlinear \sigma-model, by breaking its (1,1) supersymmetry with Hamiltonian flow. We use localization methods to evaluate the partition function for a general class of integrable Hamiltonians, determined by the condition that the action must exhibit a (1,0) supersymmetry. For these Hamiltonians we find relations that can be viewed as generalizations of the classical Morse equality. In particular, these Hamiltonians appear to saturate the lower bound in the Arnold conjecture. We also discuss the general case, and in particular point out that there appears to be some parallelism between dynamical supersymmetry breaking and the Arnold conjecture.