Stochastic and topological gauge theories

Abstract

We show that the form of the Langevin equations for the stochastic quantization of a Yang-Mills theory in n-dimensional space is dictated by the geometrical structure. The arbitrary Lie algebra zero form, initially introduced by Zwanziger to induce a drift force along gauge orbits, is now defined as an additional gauge field component which permits stochastic time-dependent gauge transformations. The action of the functional integral representation of the Langevin equation is the same as that of an ( n+1)-dimensional topological quantum field theory. Covariance between the stochastic time and the physical space directions is made possible. As an application, the action which generates Donaldson invariants in four-dimensional space is shown to be identical to the supersymmetric action describing the stochastic quantization of the three-dimensional Chern-Simons theory, which generates Jones invariants.