On the Gravitini Zero Modes riding on top of Multiple Black Holes

TL;DR

This paper investigates gravitini zero modes in four-dimensional $ ext{N}=2$ supergravity on multi-black hole backgrounds, revealing solutions related to broken supersymmetries and additional fermionic modes, with implications for black hole microstates.

Contribution

It provides a detailed analysis of gravitini zero modes in multi-black hole backgrounds, identifying both broken supersymmetry modes and new fermionic solutions in this setting.

Findings

Identified gravitini zero modes due to broken supersymmetries.

Discovered additional fermion zero modes beyond those from broken supersymmetries.

Reduced complex equations to familiar electromagnetic-like PDEs on $ ext{R}^3$.

Abstract

We study gravitini zero modes in four dimensional $\mathcal{N} = 2$ pure supergravity theory. The gravitini zero modes we study are solutions to the Rarita-Schwinger equations of motion in the background of the purely bosonic Majumdar-Papapetrou background. We start with a very generic ansatz for the gravitini that involves $32$ ansatz functions and reduce the Rarita-Schwinger equations to a set of linear coupled partial differential equations on $\mathbf{R}^3$: the kind familiar from electromagnetism with divergences, curls etc. We first show how the gravitini zero modes due to broken supersymmetries arise in this set-up and how they solve the equations on $\mathbf{R}^3$. Then we go on and obtain other solutions to these equations: the `extra' fermion zero modes. If we count the fermion zero mode due to broken supersymmetries as one spinor worth of solutions, we have obtained three other spinors worth of solutions.