Superpfaffian

Abstract

Let V=V_0+V_1 be a real finite dimensional supervector space provided with a non-degenerate antisymmetric even bilinear form B. Let spo(V) be the Lie superalgebra of endomorphisms of V which preserve B. We consider spo(V) as a supermanifold. We show that a choice of an orientation of V_1 and of a square root i of -1 determines a very interesting generalized function on the supermanifold spo(V), the superPfaffian. When V=V_1, spo(V) is the orthogonal Lie algebra so(V_1) and the superPfaffian is the usual Pfaffian, a square root of the determinant. When V=V_0, spo(V) is the symplectic Lie algebra sp(V_0) and the superPfaffian is a constant multiple of the Fourier transform of one the two minimal nilpotent orbits in the dual of the Lie algebra sp(V_0), and is an analytic square root of the inverse of the determinant in the open subset of invertible elements of spo(V). Our opinion is that the superPfaffians (there are four of them, corresponding to the two orientations on V_1, and to the two square roots of -1) are fundamental objects. At least, they occur in the study of equivariant cohomology of supermanifolds, and in the study of the metaplectic representation of the metaplectic group with Lie algebra spo(V). In this article, we present the definition and some basic properties of the superPfaffians.