# 't Hooft anomalies and the holomorphy of supersymmetric partition functions

## TL;DR

This paper investigates how supersymmetric partition functions depend on flavor parameters in theories with 't Hooft anomalies, revealing a non-holomorphic dependence and proposing a new expression involving a non-holomorphic Casimir factor.

## Contribution

It demonstrates the non-holomorphic dependence of partition functions due to anomalies and introduces a new formula with a non-holomorphic Casimir factor consistent with Ward identities.

## Key findings

- Partition functions depend non-holomorphically on flavor parameters in the presence of anomalies.
- A new expression for partition functions includes a non-holomorphic Casimir pre-factor.
- The proposed formula matches Ward identities and gauge-invariant regularization results.

## Abstract

We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, $G_F$, for 2d $\mathcal{N} = (0,2)$ and 4d $\mathcal{N}=1$ supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of $G_F$ '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, $Z_{T^2}$, and for a large class of 4d partition functions on half-BPS four-manifolds, $Z_{\mathcal{M}_4}$---in particular, for $\mathcal{M}_4=S^3 \times S^1$ and $\mathcal{M}_4=\Sigma_g \times T^2$. We propose a new expression for $Z_{\mathcal{M}_{d-1} \times S^1}$, 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.