Computation of the index on orbifold from the Atiyah-Segal-Singer fixed point theorem

TL;DR

This paper applies the Atiyah-Segal-Singer fixed point theorem to compute the index of fermionic zero modes on orbifolds, providing a method that bypasses direct zero mode calculations and extends to complex gauge configurations.

Contribution

It introduces a novel application of the fixed point theorem to determine indices on orbifolds without explicit zero mode computations, including complex cases like Coxeter orbifolds.

Findings

Indices computed for various $T^{2}/ \mathbb{Z}_N$ orbifolds.

Indices computed for various $T^{4}/ \mathbb{Z}_N$ orbifolds.

Extension to Coxeter orbifolds related to the $D_4$ lattice.

Abstract

We investigate the independent chiral zero modes on the orbifolds from the Atiyah-Segal-Singer fixed point theorem. The required information for this calculation includes the fixed points of the orbifold and the manner in which the spatial symmetries act on these points, unlike previous studies that necessitated the calculation of zero modes. Since the fixed point theorem can be applied to any fermionic theory on any orbifold, it allows us to determine the index even on orbifolds where the calculation of zero modes is challenging or in the presence of non-trivial gauge configurations. We compute the indices on the $T^{2}/ \mathbb{Z}_N\,(N=2,3,4,6)$ and $T^{4}/ \mathbb{Z}_N\,(N=2,3,5)$ as examples. Furthermore, we also attempt to compute the indices on a Coxeter orbifold related to the $D_4$ lattice.