't Hooft anomalies and the holomorphy of supersymmetric partition functions

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.

62 pages plus appendix. v2: added references and comments

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Cited by in corpus (21)

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• The large-$N$ limit of the 4d $\mathcal{N}=1$ superconformal index
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• Localization of 4d $\mathcal{N}=1$ theories on ${\mathbb{D}}^2\times {\mathbb{T}}^2$
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• Holographic Weyl Anomalies for 4d Defects in 6d SCFTs
• Comments on Anomalies in Supersymmetric Theories
• Supersymmetry anomaly in the superconformal Wess-Zumino model
• Consistency of supersymmetric 't Hooft anomalies
• BPS States Meet Generalized Cohomology
• Localization on $Ad{S}_3\times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity
• Remarks on Berry Connection in QFT, Anomalies, and Applications
• From Exact Results to Gauge Dynamics on ${\mathbb{R}}^3\times S^1$
• Topological AdS/CFT and the $\Omega$ deformation
• CP-violating supersymmetry anomaly
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Anomalies and Supersymmetry
• Boundary Conditions in Topological AdS${}_4$/CFT${}_3$