Supersymmetric counterterms from new minimal supergravity

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.