On volumes of classical supermanifolds

Abstract: We consider the volumes of classical supermanifolds such as the supersphere, complex projective superspace, and Stiefel and Grassmann supermanifolds, with respect to the natural metrics or symplectic structures. We show that the formulas for the volumes, upon certain universal normalization, can be obtained by an analytic continuation from the formulas for the volumes of the corresponding ordinary manifolds. Volumes of nontrivial supermanifolds may identically vanish. In 1970s, Berezin discovered that the total Haar measure of the unitary supergroup $\un(n|m)$ vanishes unless $m=0$ or $n=0$, i.e., unless it reduces to the ordinary unitary group $\un(n)$ or $\un(m)$. Witten recently suggested that the (Liouville) volume of a compact even symplectic supermanifold should always be zero if it is not an ordinary manifold. Our calculations provide counterexamples to this conjecture. On the other hand, we give a simple explanation of Berezin's statement and generalize it to the Stiefel supermanifold $\st_{r|s}(\C{n|m})$. There are also possible connections with the recent works by Mkrtchyan and Veselov on `universal formulas' in Lie algebra theory.