Exploring curved superspace (II)

We extend our previous analysis of Riemannian four-manifolds M admitting rigid supersymmetry to N = 1 theories that do not possess a U(1)(R) symmetry. With one exception, we find that 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 M admits a nowhere vanishing, holomorphic section. This requirement can be slightly relaxed if 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 M is a warped product S-3 x R, where the radius of the round S-3 is allowed to vary along. Such manifolds admit two supercharges that generate the superalgebra OSp(1 vertical bar 2). If the S-3 smoothly shrinks to zero at two points, we obtain a squashed four-sphere, which is not a Hermitian manifold.