Pure Maths Colloquium: Volumes of classical supermanifolds
8 February 2016 16:00 in CM221
Volumes in supergeometry may exhibit unexpected features such as vanishing for nontrivial spaces. This is due to the nature of Berezin's integration, to which measure-theoretic arguments are not applicable. In 1970s, Berezin discovered that the total "Haar measure" of the unitary supergroup identically vanishes. Recently Witten conjectured that the same may be true to the "super Liouville" volumes of symplectic supermanifolds. In response to Witten's question, I provided a counterexample with a nice analytic expression for the volume. This formula has features typical for some other examples as well. Namely, it turns out that expressions for volumes for superanalogs of classical manifolds such as spheres and Stiefel manifolds can be obtained by an analytic continuation of the formulas for the classical case, with respect to parameters of "index type". I will explain this in the talk. I will also give a simple geometric explanation (and a generalization) of Berezin's result.
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