Conformal symmetries of the super Dirac operator

TL;DR

This paper explores the symmetries of the super Dirac operator, extending classical conformal symmetry concepts to supersymmetric settings and analyzing its kernel as a representation of these symmetries.

Contribution

It introduces the superalgebra of symmetries osp(m|2n) for the super Dirac operator and extends it to conformal symmetries, providing new insights into invariant operators in super geometries.

Findings

Superalgebra osp(m|2n) is essential for symmetries of the super Dirac operator.

Explicit realization of Howe dual pair osp(1|2) x osp(m|2n) is provided.

The kernel of the super Dirac operator is studied as a representation of symmetry algebras.

Abstract

In this paper, the Dirac operator, acting on super functions with values in super spinor space, is defined along the lines of the construction of generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of the superalgebra of symmetries osp(m|2n) is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a representation of both algebras. The construction also gives an explicit realization of the Howe dual pair osp(1|2) x osp(m|2n) < osp(m + 4n|2m + 2n). Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries.