Intertwining differential operators for spinor-form representations of the conformal group

Abstract

New first-order conformally covariant differential operators P k on spinor- k-forms, i.e., tensor products of contravariant spinors with k-forms, in an arbitrary n-dimensional pseudo-Riemannian spin manifold, are introduced. This provides a series of generalizations of the Dirac operator ▿ ̷ , in analogy with the series of generalizations (introduced by the author in [1]) of the Maxwell operator and the conformally covariant Laplacian □ on functions. In particular, new intertwining operators for representations of SU(2, 2) and SO( p + 1, q + 1) are found. Related nonlinear covariant operators are also introduced, and mixed nonlinear covariant systems are obtained by coupling to the Yang-Mills-Higgs-Dirac system in dimension 4. The spinor-form bundle is isomorphic with E (3) = E ⊗ E ⊗ E, where E is the spin bundle, and the P k give a covariant operator on sections of E (3). This is generalized to a covariant operator on E (2 l + 1) . The relation of powers of these operators to higher-order covariant operators on lower spin bundles (analogous to the relation between ▿ ̷ and □ ) is discussed.