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November 06, 2005
Alien or just smoking?
This is an abstract of the Colloquium in our Department. I wonder, if the speaker is an alien or just smoking?
| We show how to construct measures on Banach manifolds associated= to supersymmetric quantum field theories. 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 $M.$ |
Posted by Victor at November 6, 2005 03:18 PM