Abstract
We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, GF, 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 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, {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 = S3 × S1 and mathcal{M} 4 = Σg × T2. 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.
Highlights
In this paper, we explore intriguing properties of supersymmetric partition functions in even space-time dimensions.1 In two dimensions, we consider the T 2 partition function of2d N = (0, 2) supersymmetric theories, or elliptic genus [2]
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
For simplicity, we focus on the flavor sector and, on the consequences of the flavor ’t Hooft anomalies in the presence of supersymmetry
Summary
We explore intriguing properties of supersymmetric partition functions in even space-time dimensions. In two dimensions, we consider the T 2 partition function of. The relation (1.7) is modified, and the partition function acquires a non-holomorphic dependence on the flavor parameters, that is precisely as required to make its absolute value gauge-invariant These quantum corrections to supersymmetric Ward identities, referred to as “supersymmetry anomalies,” were studied long ago in [14, 15], and more recently in [16,17,18,19,20,21,22]. As an interesting warm-up to the four-dimensional case, let us study the partition function of a 2d N = (0, 2) supersymmetric theory on the torus, in the presence of flat connections for background gauge fields coupling to the flavor symmetry — this is known as the flavored N = (0, 2) elliptic genus
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