Supersymmetric partition functions, 't Hooft anomalies and Holomorphy

Supersymmetric partition functions on compact spaces provide an important tool to study the non-perturbative aspects of quantum field theories. In this talk, I discuss universal properties of the partition functions for 4d N=1 gauge theories. After reviewing the dependence of the partition function on various discrete and continuous parameters, I introduce a novel way to construct the partition function on a large class of four-manifolds which preserves two supersymmetries. In the second part of the talk, I discuss the existence of a mixed anomaly between supersymmetry and background gauge symmetry, which implies an anomaly in the holomorphy of the partition function with respect to the background gauge parameters. I propose a way to compute the full non-holomorphic dependence of the partition functions, which provides an important correction to earlier holomorphic results.

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