Index theorems and domain walls

Dmitri Vassilevich

doi:10.1007/jhep07(2018)108

Abstract

The Atiyah-Patodi-Singer (APS) index theorem relates the index of a Dirac operator to an integral of the Pontryagin density in the bulk (which is equal to global chiral anomaly) and an η invariant on the boundary (which defines the parity anomaly). We show that the APS index theorem holds for configurations with domain walls that are defined as surfaces where background gauge fields have discontinuities.

Highlights

(quite in the spirit of domain walls in ferromagnets)
We show that the APS index theorem holds for configurations with domain walls that are defined as surfaces where background gauge fields have discontinuities
We have demonstrated that a suitably modified APS index theorem (4.12) holds for domain wall geometries in four dimensions that are characterized by gauge fields having a discontinuity at a submanifold Σ of codimension one

Summary

The setup

Let M be a smooth compact four-dimensional manifold, and let Σ be a smooth codimension one submanifold that separates M in two parts, M+ amd M−. In the Atiayh-Singer approach, the index is associated with some elliptic complex over M, which is the spin complex in our case It consists of a spin-bundle S with two subbundles SR and SL of chiral spinors, S = SR ⊕ SL, and of a Dirac operator D/. Since D/ has to anticommute with the chirality matrix γ5, we are left with interaction with an external metric, a gauge field, and an axial vector field. We suppose that the metric gμν is flat, and there is no axial vector field. Since the operator D/ has to be hermitian, D/ ψ should be continuous across Σ This gives us a matching condition for the normal derivatives.

Computation of the index

The APS index theorem

Examples

Discussion and conclusions