Volumes of classical supermanifolds

This work started from a question by E. Witten, who asked me (in fall 2012), whether the volume of every compact (even) symplectic supermanifold with respect to the super analog of the Liouville measure should vanish (except for the case of ordinary symplectic manifolds). He had some arguments in favor of such a conjecture. I presented a counterexample, which is the volume of the complex projective superspace, the simplest nontrivial case being CP1|1 where the volume is 2π. The general formula for the volume of CPn|m of radius R (with respect to the super analog of the Fubini-Study metric) is R2(n-m)πn 2m / Γ(n-m+1).

To put Witten's question into context, it was long known (Berezin's theorem, 1970s) that the volume of the super analog of the unitary group, the unitary supergroup U(n|m), vanishes unless m=0 or n =0 (which is the case of an ordinary group, either U(n) or U(m)).

The above formula led me to the investigation of other formulas for "super" volumes. As it turns out, - an "experimental fact" - the volumes of various classical supermanifolds (upon some universal normalization) can be obtained by formulas that are analytic continuation of the formulas for the volumes of the corresponding ordinary manifolds. This looks as a mystery if one thinks about the formal algebraic nature of the Berezin integral in comparison with the ordinary measure theory. Also, this fact prompts for the investigation of a meaning of these "normalized volumes" in non-integer dimensions and in infinite dimension. There is another challenging connection with the "universal Lie algebra" program of Vogel-Deligne (see the works by R. Mkrtchyan, Mkrtchyan-Veselov and Mkrtchyan-Khudaverdian).

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There has been some new development in this area very recently, see a preprint by Stanford and Witten: arXiv:1907.03363. Among other things, they have deduced a formula for the Liouville volume of a symplectic supermanifold in topological terms (via the Chern class of the associated vector bundle, i.e. the normal bundle to the reduced submanifold). This prompts new very attractive paths for research.