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

✍ Scribed by W. Biedrzycki

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 872 KB
• Volume: 113
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1993.1046

✦ Synopsis

The representations of (\operatorname{Spin}(4,2)), seem to be of particular physical interest since its quotient (S O(4,2)), 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 (S O(4,4){e}). Moreover, it is known [9] that there is a Howe pair ( (S O(2), S O(4,2)) ) in (S O(4,4)), and that the weight spaces of (S O(2)) in (H) are irreducible representations of (S O(4,2){r}). He conjectured that the analogous facts should hold for the Dirac operator acting on the spinor fields on (S^{3} \times S^{3} \subseteq R^{8}). We introduce a hitherto unknown embedding of the bundle of spinors on (S^{3} \times S^{3}) in the bundle of spinors on (R^{6}). This embedding makes it possible to obtain a particularly convenient expression for the Dirac operator on (S^{3} \times S^{3}) in terms of spinors and vector fields on the underlying (R^{x}); 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 (\operatorname{Spin}(4,4)).. 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 \subseteq S^{+} \otimes H) where (S^{\text {, }}), is the vector space of even spinors. We identify within (\operatorname{Spin}(4,4)), a Howe pair consisting of (S O(2)) and (\operatorname{Spin}(4,2)), and show that the weight spaces of (S O(2)) in (F) are irreducible representations of (\operatorname{Spin}(4,2)).. . 1993 Academic Press. Inc.