Self-adjoint codimension 2 boundary conditions for Dirac operators

25.06.2020, 16:15  –  Online-Seminar
Forschungsseminar Differentialgeometrie

Bernd Ammann (Regensburg)

Let N be an oriented compact submanifold of codimension 2 in an oriented complete Riemannian manifold M. We assume that M\N is spin and carries a unitary line bundle L. We study the self-adjoint extensions of the Dirac operator acting on spinors on M\N, twisted by L. For the special case of M=S³ with the round metric a special boundary condition within this framework was described in recent work by Portman, Sok and Solovej. It is said to provide magnetic link invariants for the link N and is motivated by stability of matter. Our article reconstructs these boundary condition in a more geometric language, which allows us to extend the result to the general setting and to classify all self-adjoint extensions of the Dirac operator in terms of Lagrangian subspaces of a symplectic Hilbert space. Some associated regularity questions are still work in progress.

The motivation for our work are a better understanding of these link invariants, to prepar the ground for potential K-theoretic generalizations of them, and potential connections to the positive mass theorem for non-spin manifolds.

This project is in part a collaboration with Nadine Große and it is motivated by stimulating discussions with Boris Botvinnik and Nikolai Saveliev.

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