Topological Sigma-model, Hamiltonian Dynamics and Loop Space Lefschetz Number

TL;DR

This paper links topological quantum field theory, Hamiltonian dynamics, and loop space topology, showing that a topological sigma-model can compute invariants related to periodic solutions of Hamiltonian systems, generalizing classical fixed point theorems.

Contribution

It establishes a novel connection between Floer's instanton equations, quantum cohomology, and loop space Lefschetz numbers, extending classical Morse theory to infinite dimensions.

Findings

Floer's instanton equation relates to a functional Euler character in quantum cohomology.

Localization shows the functional Euler character matches de Rham cohomology for Kähler manifolds.

Results support the Arnold conjecture on periodic solutions in Hamiltonian systems.

Abstract

We use path integral methods and topological quantum field theory techniques to investigate a generic classical Hamiltonian system. In particular, we show that Floer's instanton equation is related to a functional Euler character in the quantum cohomology defined by the topological nonlinear $\sigma$--model. This relation is an infinite dimensional analog of the relation between Poincar\'e--Hopf and Gauss--Bonnet--Chern formul\ae$~$ in classical Morse theory, and can also be viewed as a loop space generalization of the Lefschetz fixed point theorem. By applying localization techniques to path integrals we then show that for a K\"ahler manifold our functional Euler character coincides with the Euler character determined by the finite dimensional de Rham cohomology of the phase space. Our results are consistent with the Arnold conjecture which estimates periodic solutions to classical Hamilton's equations in terms of de Rham cohomology of the phase space.