Spin-1/2 Fields in Riemannian Geometries

Abstract

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal _e n ^A’ _A to the boundary and a pair of independent spinor fields ψ^A and $\widetilde\psi ^{A'}$ . This chapter studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. If $\sqrt 2 _e n_A^{A'} \psi ^{\rm A} \mp \widetilde\psi ^{A'} \equiv \Phi ^{A'}$ is set to zero on a three-sphere bounding flat Euclidean four-space, the modes of the massless spin-1/2 field multiplying harmonics having positive eigenvalues for the intrinsic three-dimensional Dirac operator on S³ should vanish on S³. Remarkably, this coincides with the property of the classical boundary-value problem when spectral. boundary conditions are imposed on S³ in the massless case. Moreover, the boundary term in the action functional is proportional to the integral on the boundary of Φ^A’ _e n _AA’ ψ^A. The existence of self-adjoint extensions of the Dirac operator subject to supersymmetric boundary conditions is then proved. The global theory of the Dirac operator in compact Riemannian manifolds is finally described.