Supersymmetric counterterms from new minimal supergravity

TL;DR

The paper classifies finite supersymmetric counterterms for four-dimensional $\mathcal{N}=1$ theories on curved spaces by using the rigid limit of Euclidean new minimal supergravity. It builds a complete set of curvature- and background-field–based invariants via D-, F-, and $\widetilde{F}$-terms, and identifies a five-term basis of marginal counterterms tied to Euler, Pontryagin, Weyl-squared, and related curvature structures. A central result is that backgrounds preserving two supercharges of opposite R-charge annihilate all finite marginal counterterms, rendering the supersymmetric partition function unambiguous, while backgrounds with a single supercharge exhibit controlled ambiguities linked to topological invariants and holomorphic dependence on marginal couplings. The analysis clarifies how anomalies behave under these constraints and highlights the Einstein–Hilbert term as a dimensionful counterterm with subtleties tied to background fields, suggesting a constrained structure for quadratic divergences and guiding future extensions to higher dimensions. Overall, the work provides a precise framework for understanding scheme dependence and anomaly structure in rigid SUSY theories on curved manifolds.

Abstract

We present a systematic classification of counterterms of four-dimensional supersymmetric field theories on curved space, obtained as the rigid limit of new minimal supergravity. These are supergravity invariants constructed using the field theory background fields. We demonstrate that if the background preserves two supercharges of opposite chirality, then all dimensionless counterterms vanish, implying that in this case the supersymmetric partition function is free of ambiguities. When only one Euclidean supercharge is preserved, we describe the ambiguities that appear in the partition function, in particular in the dependence on marginal couplings.