Manchester Geometry Seminar 2013/2014

Thursday 1 May 2014. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

On Volumes of Classical Supermanifolds. II

Theodore Voronov (University of Manchester)

"Volume" in the super case may show unexpected features, due to properties of the Berezin integration. For example, the invariant volume of the unitary supergroup $U(n|m)$ vanishes whenever $nm>0$, i.e., when it does not reduce to the ordinary group. This was discovered by Berezin in 1970s. Some time ago, Witten asked me a question as to whether the Liouville volume of compact symplectic supermanifolds should always be zero. The answer is negative: as a counterexample one can consider the complex projective superspace endowed with the analog of the classical Fubini--Study form. An interesting observation is that the explicit formula for the volume in this case can be obtained by analytic continuation of the formula for the ordinary (purely even) complex projective space. This is holds true for some other examples, and we shall discuss that in the talk. (Work in progress. I spoke earlier on this subject at the seminar, but this talk will be independent and no preliminary knowledge is assumed.)