Exploring curved superspace (II)

Abstract

We extend our previous analysis of Riemannian four-manifolds $\mathcal{M}$ admitting rigid supersymmetry to $\mathcal{N}$ = 1 theories that do not possess a U(1) _R symmetry. With one exception, we find that $\mathcal{M}$ must be a Hermitian manifold. However, the presence of supersymmetry imposes additional restrictions. For instance, a supercharge that squares to zero exists, if the canonical bundle of the Hermitian manifold $\mathcal{M}$ admits a nowhere vanishing, holomorphic section. This requirement can be slightly relaxed if $\mathcal{M}$ is a torus bundle over a Riemann surface, in which case we obtain a supercharge that squares to a complex Killing vector. We also analyze the conditions for the presence of more than one supercharge. The exceptional case occurs when $\mathcal{M}$ is a warped product S ³ × $\mathbb{R}$, where the radius of the round S ³ is allowed to vary along $\mathbb{R}$. Such manifolds admit two supercharges that generate the superalgebra OSp(1|2). If the S ³ smoothly shrinks to zero at two points, we obtain a squashed four-sphere, which is not a Hermitian manifold.