On N=1 partition functions without R-symmetry

Understand

We examine the dependence of four-dimensional Euclidean $\mathcal{N}=1$ partition functions on coupling constants.

• In particular, we focus on backgrounds without R-symmetry, which arise in the rigid limit of old minimal supergravity.
• Backgrounds preserving a single supercharge may be classified as having either trivial or SU(2) structure, with the former including $S^{4}$.
• We show that, in the absence of additional symmetries, the partition function depends non-trivially on all couplings in the trivial structure case, and (anti)-holomorphically on couplings in the SU(2) structure case.