Complex General Relativity pp 93-110 | Cite as

# 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 *S*^{3} should vanish on *S*^{3}. Remarkably, this coincides with the property of the classical boundary-value problem when spectral. boundary conditions are imposed on *S*^{3} 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.

## Keywords

Dirac Operator Boundary Term Riemannian Geometry Symmetric Operator Spinor Field## Preview

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