On volumes of classical supermanifolds

Th. Th. Voronov

Sbornik. Mathematics, Tome 207 (2016) no. 11, pp. 1512-1536

Résumé

We consider the volumes of classical supermanifolds (such as the supersphere, complex projective superspace, Stiefel and Grassmann supermanifolds) with respect to natural metrics or symplectic structures. We show that the formulae for the volumes of these supermanifolds can be obtained from the formulae for the volumes of the corresponding ordinary manifolds (under some universal normalization of the volume) by analytic continuation with respect to parameters. The volumes of nontrivial supermanifolds may be identically equal to zero. In the 1970s Berezin showed that the total Haar measure of the unitary supergroup $\mathbf{U}(n|m)$ vanishes except in the cases $m=0$ and $n=0$, when the supergroup is the ordinary unitary group $\mathbf{U}(n)$ or $\mathbf{U}(m)$. Some time ago Witten conjectured that the Liouville volume of a compact even symplectic supermanifold is always equal to zero (except for ordinary manifolds). We give counterexamples to this conjecture, present a simple explanation of Berezin's theorem, and generalize this theorem to the Stiefel supermanifold $\mathbf{V}_{r|s}(\mathbf C^{n|m})$. We mention a connection with recent work of Mkrtchyan and Veselov on universal formulae in Lie algebra theory. Bibliography: 32 titles.

Détail

Keywords: supermanifolds, symplectic structure, Riemannian metric, Riemannian submersion, Berezin integral.
Mots-clés : volume