Conformal symmetries of the super Dirac operator

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 $\mathfrak{osp}(m|2n)$ is a new and essential feature in this approach. This algebra of symmetries is extended to the algebra of conformal symmetries $\mathfrak{osp}(m+1,1|2n)$. 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 $\mathfrak{osp}(1|2)\times\mathfrak{osp}(m|2n)\subset \mathfrak{osp}(m+4n|2m+2n)$. Finally, the super Dirac operator gives insight into the open problem of classifying invariant first order differential operators in super parabolic geometries.