On N=1 partition functions without R-symmetry

TL;DR

This paper investigates how four-dimensional N=1 supersymmetric partition functions depend on couplings in backgrounds lacking R-symmetry, revealing conditions for their physicality and limitations of localization.

Contribution

It classifies supersymmetric backgrounds without R-symmetry and analyzes the dependence of partition functions on couplings, highlighting when they are physical or ambiguous.

Findings

Partition functions depend non-trivially or holomorphically on couplings depending on background structure.

Ambiguities in partition functions are limited to finite powers of couplings, making them controllable.

Localization is not possible for backgrounds without R-symmetry, affecting computational approaches.

Abstract

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. In both cases, this allows for ambiguities in the form of finite counterterms, which in principle render the partition function unphysical. However, we argue that on dimensional grounds, ambiguities are restricted to finite powers in relevant couplings, and can therefore be kept under control. On the other hand, for backgrounds preserving supercharges of opposite chiralities, the partition function is completely independent of all couplings. In this case, the background admits an R-symmetry, and the partition function is physical, in agreement with the results obtained in the rigid limit of new minimal supergravity. Based on a systematic analysis of supersymmetric invariants, we also demonstrate that $\mathcal{N}=1$ localization is not possible for backgrounds without R-symmetry.