Conjugate linear perturbations of Dirac operators and Majorana fermions

Abstract

We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw-Rossi operator, which describes electronic excitations on topological superconductors. We also describe the low energy spectrum of these operators on complete surfaces, under mild hypotheses. CONTENTSIntroduction 1 1. Generalized Jackiw-Rossi theory on pseudo-Riemannian manifolds 3 2. The 2-dimensional Jackiw-Rossi theory 8 3. Generalized Jackiw-Rossi theory on complete surfaces 15 4. Applications 23 Appendix A. Bilinear pairings and conjugate linear maps 28 Appendix B. The Bogoliubov-de Gennes equation 31 References 32