Measures on Banach Manifolds and Supersymmetric Quantum Field Theory
arXiv:math/0509104 · doi:10.1007/s00220-007-0359-3
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 of measures on a space of functions on the two-torus, parametrized by a polynomial (the Wess-Zumino-Landau-Ginzburg model). The second is a family $μ_\cG^{s,t}$ of measures on a space $\cG$ of maps from to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family of measures on the product of a space of connection s on the trivial principal bundle with structure group on a three-dimensional manifold with a space of $\fg$-valued three-forms on We show that these measures are positive, and that the measures $μ_\cG^{s,t}$ are Borel probability measures. As an application we show that formulas arising from expectations in the measures $μ_\cG^{s,1}$ reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures where is a homology three-sphere, will yield the Casson invariant of
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