Spherical harmonics and integration in superspace: II

Abstract

The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007 J. Phys. A: Math. Theor. 40 7193) is further elaborated. A detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces, which also determines the irreducible pieces under the action of SO(m) x Sp(2n). In the second part of the paper, this decomposition is used to describe all possible integrations over the supersphere. It is then shown that only one possibility yields the orthogonality of spherical harmonics of different degrees. This is the so-called Pizzetti-integral of which it was shown in (De Bie and Sommen 2007 J. Phys. A: Math. Theor. 40 7193) that it leads to the Berezin integral.