I will begin by defining a canonical family of perturbations of the Dirac equation. These perturbations are complex anti-linear, thus ground states only form a real vector space. A special case of this theory is known as the Jackiw–Rossi theory, which models surface excitations on the surface of a topological insulator placed in proximity to an s-wave superconductor. While the physics of the theory is relatively well-understood, the mathematical side has not been studied, even on surfaces, not to mention its generalizations to higher dimensional and on nontrivial manifolds. I call these equations the generalized Jackiw–Rossi equations.
After the definitions and connections to physics, I will present my current work on the generalized Jackiw–Rossi equations. My main result is a concentration phenomenon which proves the physical expectation that such Majorana fermions concentrate around the vortices of the superconducting order parameter. Moreover, I provide approximate solutions that are exponentially sharp in the large coupling limit.
If time permits, then I will show how these Majorana fermions define a bundle of projective spaces over the simple part of vortex moduli spaces. The holonomies of such bundles give rise to projective representations of (surface) braid groups, and thus, speculatively, can be of interest to quantum information theorists.