Comments on N=(2,2) Supersymmetry on Two-Manifolds

Understand

We study curved-space rigid supersymmetry for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric fields theories with a vector-like $R$-symmetry by coupling such theories to background supergravity.

• The associated Killing spinors can be viewed as holomorphic sections of particular complex line bundles over Euclidean space-time, which severely restrict the allowed supersymmetric couplings on compact orientable Riemann surfaces without boundaries.
• For genus $g>1$, the only consistent non-singular couplings are the ones dictated by the topological $A$-twist.
• On spaces with $S^2$ topology, there exist additional supersymmetric backgrounds with $m=0$ or $\pm 1$ unit of flux for the $R$-symmetry gauge field.