Abstract
We systematically analyze Riemannian manifolds \( \mathcal{M} \) that admit rigid supersymmetry, focusing on four-dimensional \( \mathcal{N} = {1} \) theories with a U(1) R symmetry. We find that \( \mathcal{M} \) admits a single supercharge, if and only if it is a Hermitian manifold. The supercharge transforms as a scalar on \( \mathcal{M} \). We then consider the restrictions imposed by the presence of additional supercharges. Two supercharges of opposite R-charge exist on certain fibrations of a two-torus over a Riemann surface. Upon dimensional reduction, these give rise to an interesting class of supersymmetric geometries in three dimensions. We further show that compact manifolds admitting two supercharges of equal R-charge must be hyperhermitian. Finally, four supercharges imply that \( \mathcal{M} \) is locally isometric to \( {\mathcal{M}_3} \times \mathbb{R} \), where \( {\mathcal{M}_3} \) is a maximally symmetric space.
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ArXiv ePrint: 1205.1115
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Dumitrescu, T.T., Festuccia, G. & Seiberg, N. Exploring curved superspace. J. High Energ. Phys. 2012, 141 (2012). https://doi.org/10.1007/JHEP08(2012)141
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DOI: https://doi.org/10.1007/JHEP08(2012)141