Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family μ P of measures on a space of functions on the two-torus, parametrized by a polynomial P (the Wess-Zumino-Landau-Ginzburg model). The second is a family μ G of measures on a space G of maps from P 1 to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family μ M,G of measures on the product of a space of connections on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of g-valued three-forms on M. We show that these measures are positive, and that the measures μ G are Borel probability measures. As an application we show that formulas arising from expectations in the measures μ G reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures μ M,SU(2), where M is a homology threesphere, will yield the Casson invariant of M.