Fermion zero modes of Denef Bates black holes

TL;DR

This paper investigates fermion zero modes in Denef Bates black holes within $ ext{N}=2$ Supergravity, demonstrating that only goldstino modes exist and confirming these black holes contribute to the index.

Contribution

It explicitly constructs goldstino zero modes and proves the non-existence of additional normalizable fermion zero modes, clarifying their role in black hole index calculations.

Findings

Goldstino zero modes are explicitly constructed.

No additional normalizable fermion zero modes exist.

Denef Bates black holes contribute to the index.

Abstract

We study the fermion zero modes of Denef Bates black holes in $\mathcal{N}=2$ Supergravity. The goldstino zero modes are constructed explicitly and their boundary conditions, charge, and normalizability are studied. We then study if there can exist fermion zero modes that are normalizable and not gauge equivalent to the goldstino modes. We find that such modes are not possible if we want to preserve the continuity of these solutions. The non-existence of these modes shows that Denef Bates black holes must contribute to the index.