Torus bundle

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.