't Hooft anomalies and the holomorphy of supersymmetric partition functions

TL;DR

The paper clarifies how supersymmetric partition functions in 2d and 4d depend on continuous flavor and geometric parameters in the presence of ’t Hooft anomalies. It shows that, in a diffeomorphism-invariant scheme, the SUSY Ward identities acquire anomaly-induced corrections that render the partition functions non-holomorphic in the flavor parameters, while preserving gauge invariance of their absolute values. The authors distinguish between a holomorphic, scheme-dependent partition function and a gauge-invariant one, related by non-holomorphic Casimir-type prefactors, and they derive explicit forms for the non-holomorphic terms across 2d and 4d settings, including their behavior under large gauge and modular transformations and their high-temperature (small-$\beta$) limits. They further propose a general gauge-invariant construction for $Z_{\mathcal M_{d-1}\times S^1}$, given by $Z_{\mathcal M_{d-1}\times S^1}(\nu,\bar{\nu},\tau) = e^{-\beta W_{d-1}^{(1)}(\nu,\bar{\nu},\tau)} \mathcal I_{\mathcal M_{d-1}}(\nu,\tau)$, which unifies anomaly constraints with supersymmetry and provides a practical framework for computing partition functions on half-BPS manifolds. The work highlights a universal mechanism in which supersymmetry anomalies, fixed by ’t Hooft anomalies, govern non-holomorphic corrections that ensure gauge invariance and consistency of the quantum theory, with concrete implications for dimensionally reduced theories and future extensions to higher dimensions.

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 $\mathcal{M}_4=Σ_g \times T^2$. 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.