Spinors over a Cone, Dirac Operator, and Representations of Spin(4, 4)

Abstract

The representations of Spin(4, 2) e seem to be of particular physical interest since its quotient SO(4, 2) e is the conformal group of the spacetime. Kostant [7] has considered the Laplacian on a projective cone in R 8 and has shown that the kernel H of the Laplacian is an irreducible unitary representation of SO(4, 4) e . Moreover, it is known [9] that there is a Howe pair ( SO(2), SO(4, 2) e ) in SO(4, 4) e and that the weight spaces of SO(2) in H are irreducible representations of SO(4, 2) e . He conjectured that the analogous facts should hold for the Dirac operator acting on the spinor fields on S 3 × S 3 ⊆ R 8. We introduce a hitherto unknown embedding of the bundle of spinors on S 3 × S 3 in the bundle of spinors on R 8. This embedding makes it possible to obtain a particularly convenient expression for the Dirac operator on S 3 × S 3 in terms of spinors and vector fields on the underlying R 8 in particular the conformal invariance is manifest. We explicitly describe the kernel F of the restriction of the Dirac operator to even spinor fields and show that it is an irreducible representation of Spin(4, 4) e . We also show that it has an invariant sesquilinear scalar product (with is not, however, positive definite). The existence of this scalar product can also be established by showing that F ⊆ S + ⊗ H where S + the vector space of even spinors. We identify within Spin(4, 4) e a Howe pair consisting of SO(2) and Spin(4, 2) e and show that the weight spaces of SO(2) in F are irreducible representations of Spin(4, 2) e .