Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

Abstract

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 $$\mu_P^{s,t}$$ 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 $$\mu_{\mathcal{G}}^{s,t}$$ of measures on a space $${\mathcal{G}}$$ of maps from $${\mathbb{P}}^1$$ to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family $$\mu_{M,G}^{s,t}$$ 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 $${\mathfrak{g}}$$ -valued three-forms on M. We show that these measures are positive, and that the measures $$\mu_{\mathcal{G}}^{s,t}$$ are Borel probability measures. As an application we show that formulas arising from expectations in the measures $$\mu_{\mathcal{G}}^{s,1}$$ reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures $$\mu_{M,SU(2)}^{s,t}$$ , where M is a homology three-sphere, will yield the Casson invariant of M.