Sumati Surya (Raman Research Institute, Bangalore, India)
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.
Access data at:https://moodle2.uni-potsdam.de/course/view.php?id=24418