Spherical harmonics and integration in superspace II

TL;DR

This paper advances the understanding of spherical harmonics in superspace by detailing their structure and identifying the unique integration method that preserves orthogonality, linking it to the Berezin integral.

Contribution

It provides a detailed decomposition of spherical harmonics in superspace and characterizes the unique integration over the supersphere that maintains orthogonality.

Findings

Decomposition of spherical harmonics into bosonic and fermionic parts.

Identification of the Pizzetti-integral as the unique orthogonality-preserving integration.

Connection of the Pizzetti-integral to the Berezin integral.

Abstract

The study of spherical harmonics in superspace, introduced in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212], 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 degree. This is the so-called Pizzetti-integral of which it was shown in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212] that it leads to the Berezin integral.