Local Boundary Conditions for Dirac-type operators

TL;DR

This paper classifies all local smooth boundary conditions that produce self-adjoint Dirac-type operators on manifolds with boundary, providing a comprehensive linear-algebraic framework especially in low dimensions.

Contribution

It develops a linear-algebraic classification of local boundary conditions ensuring self-adjointness and regularity for Dirac operators, including explicit classifications in 3 and 4 dimensions.

Findings

Classified all local self-adjoint regular boundary conditions in low dimensions.

Provided a linear-algebraic translation of boundary condition criteria.

Extended analysis to transmission boundary conditions.

Abstract

We consider Dirac-type operators on manifolds with boundary, and set out to determine all local smooth boundary conditions that give rise to (strongly) regular self-adjoint operators. By combining the general theory of boundary value problems for Dirac operators as in [BB12] and pointwise considerations, for local smooth boundary conditions the question of being self-adjoint resp. regular is fully translated into linear-algebraic language at each boundary point. We analyse these conditions and classify them in low dimensions and ranks. In particular, we classify all local self-adjoint regular boundary conditions for Dirac spinors (four spinor components) in dimensions $3$ and $4$. With the same techniques we can also treat transmission boundary conditions.