Eigenvalues of Dirac and Rarita-Schwinger operators. (English) Zbl 1080.53044

Hong, Doojin

Abłamowicz, Rafał (ed.), Clifford algebras. Applications to mathematics, physics, and engineering. Papers from the 6th international conference on Clifford algebras and their applications in mathematical physics, Cookeville, TN, USA, May 20–25, 2002. Boston, MA: Birkhäuser (ISBN 0-8176-3525-4/hbk). Progress in Mathematical Physics 34, 201-210 (2004).

Let \(S^n\) be the unit \(n\)-sphere, \(M=S^1\times S^{n-1}\) (\(n\) even) with the standard Lorentzian or Riemannian metric and nontrivial spin structure on \(S^1\), the standard Riemannian metric and the standard spin structure on \(S^{n-1}\). Spinors and twistors on \(M\) are considered in order to get the eigenvalues of the squares of the Dirac and Rarita-Schwinger operators on \(M\). The computations are based on results of T. Branson [J. Funct. Anal. 106, 314–328 (1992; Zbl 0778.58066); Proc. Am. Math. Soc. 126, 1031–1042 (1998; Zbl 0890.47030); J. Lie Theory 9, 491–506 (1999; Zbl 1012.22026)] and T. Branson and O. Hizaji [“Bochner-Weitzenböck formulas associated with the Rarita-Schwinger operator”, Int. J. Math. 13, No. 2, 137–182 (2002; Zbl 1109.53306), see also arXiv:hep-th/0110014].
For the entire collection see [Zbl 1052.15001].

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

53C27	Spin and Spin\({}^c\) geometry
53C30	Differential geometry of homogeneous manifolds
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

Dirac operator; Rarita-Schwinger operator; spinor; twistor; eigenvalue; bundle; Bochner-Weitzenböck formula