’t Hooft anomalies and the holomorphy of supersymmetric partition functions

We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, GF, for 2d N = (0, 2) and 4d N = 1 supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of GF ’t Hooft anomalies, the supersymmetric Ward identities imply that the partition function has a non-holomorphic dependence on the flavor parameters. We show this explicitly for the 2d torus partition function, ZT2, and for a large class of 4d partition functions on half-BPS four-manifolds, ZM4— in particular, for M4 = S3 × S1 and M4 = Σg × T2. We propose a new expression for ZMd−1×1, which differs from earlier holomorphic results by the introduction of a non-holomorphic “Casimir” pre-factor. The latter is fixed by studying the “high temperature” limit of the partition function. Our proposal agrees with the supersymmetric Ward identities, and with explicit calculations of the absolute value of the partition function using a gauge-invariant zeta-function regularization.