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 μPs,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 μs,t of measures on a space of maps from 1 to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family μM,Gs,t of measures on the product of a space of connection s on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of -valued three-forms on M. We show that these measures are positive, and that the measures μs,t are Borel probability measures. As an application we show that formulas arising from expectations in the measures μs,1 reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures μM,SU(2)s,t, where M is a homology three-sphere, will yield the Casson invariant of M.
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