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 setup 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, parametrized by four integration constants (one of the integration constants corresponds to the gravitini zero modes due to broken supersymmetries). A careful analysis of pure gauge solutions reveals that we have four independent extra zero modes.
Highlights
We study fermion zero modes riding on top of purely bosonic supersymmetric solutions in four dimensional supergravity
The fermion zero modes corresponding to broken supersymmetries are obtained by performing a supersymmetry transformation on the bosonic solution using a supersymmetry parameter, that at asymptotic infinity is orthogonal to the Killing spinors of the bosonic solution; these supersymmetry parameters are referred to as anti-Killing spinors
One can generate fermion zero modes, i.e., solutions to the fermion equations of motion in the purely bosonic supersymmetric background, by performing a supersymmetry transformation whose supersymmetry parameters are the anti-Killing spinors; note that the anti-Killing spinors do not vanish at infinity and they provide for a nontrivial supergauge transformation
Summary
We study fermion zero modes riding on top of purely bosonic supersymmetric solutions in four dimensional supergravity. We first find the fermion zero mode due to broken supersymmetries as one of the solutions to the fermion equations of motion; we obtain new, hitherto unknown, solutions These “extra fermion zero modes” are the main result of this paper. While it is important to recognize that these solutions can be obtained without solving the fermion equations of motion and just by performing a supersymmetry transformation on the bosonic solution, it is even more important to recognize that the other solutions [pertaining to the remaining three integration constants, D1, D2, D3 in (56)] are genuinely new results of our study and could not have been obtained in any other way except by going through the long analysis that will be laid out in this paper. An appendix gathers all the equations of motion in each decoupled sector
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