The supersymmetric sigma model, topological quantum mechanics and knot invariants

Abstract

A field theoretical discussion of the complex constructed by Floer in order to prove the Arnold conjecture concerning a Morse theory for the fixed point set of diffeomorphisms of a symplectic manifold is offered. The approach is modelled on the supersymmetric non-linear sigma model in (1 + 1)-space-time dimensions. The associated topological quantum mechanics are proved to be related with the Lefschetz formula of fixed points of holomorphic maps on complex manifolds. Expectation values of vertex operators provide a quantum mechanical version of the Witten-Jones invariants of knot theory. Stochastic quantization of the topological quantum mechanics is performed in the setting of the supersymmetric sigma model. Expectation values of significant operators in topological quantum mechanics are, non-trivially, obtained as the average, large stochastic time behaviour, of similar observables in the supersymmetric sigma model. Applications of the supersymmetric sigma model to the physics of liquid crystal materials are suggested. A topological theory of spin for relativistic particles moving in curved space-time arises from topological quantum mechanics.