April 6, 2006
Speaker: Jonathan Weitsman, UC Santa Cruz
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 $\mu_\cG^{s,t}$ of measures on a space $\cG$
of maps from $\P^1$ 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 $\fg$-valued three-forms on $M.$
We show that these measures are positive, and that the measures $\mu_\cG^{s,t}$ are Borel probability measures. As an application we
show that formulas arising from expectations in the measures
$\mu_\cG^{s,1}$ reproduce formulas discovered by Frenkel and Zhu in
the theory of vertex operator algebras. We conjecture that a similar
computation for the measures $\mu_{M,SU(2)}^{s,t}$ where $M$ is a
homology three-sphere, will yield the Casson invariant of $M.$
