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

If an operator $H$ anticommutes with a chirality operator ${\mathrm{\Gamma }}_{\ast }$ such that ${\mathrm{\Gamma }}_{\ast }^2=1$, 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, $\mathrm{I}\mathrm{n}\mathrm{d}({\mathrm{\Gamma }}_{\ast },H)$, as the difference of dimensions of these two spaces. The key issue is whether $\mathrm{I}\mathrm{n}\mathrm{d}({\mathrm{\Gamma }}_{\ast },H)$ 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 $\mathrm{I}\mathrm{n}\mathrm{d}({\mathrm{\Gamma }}_{\ast },H)$ is topologically protected also for non-hermitian operators $H$ as long as they are diagonalizable and satisfy some ellipticity conditions.