Atiyah--Singer Index Theorem for Non-Hermitian Dirac Operators

Abstract

If an operator $H$ anticommutes with a chirality operator $\Gamma_*$ such that $\Gamma_*^2=1$, the null space of $H$ can be decomposed in a direct sum of two spaces having positive and negative chiralities, respectively. When both spaces are finite dimensional, one can define an index, $\mathrm{Ind}(\Gamma_*,H)$, as the difference of dimensions of these two spaces. The key issue is whether $\mathrm{Ind}(\Gamma_*,H)$ is topologically protected, i.e., whether it remains constant under smooth variations of the parameters and background fields entering $H$. For Hermitian Dirac operators, topological protection of the index is guaranteed by the Atiyah--Singer theorem. In this paper, by using the heat kernel methods, we show that $\mathrm{Ind}(\Gamma_*,H)$ is topologically protected also for non-hermitian operators $H$ as long as they are diagonalizable and satisfy some ellipticity conditions.