’t Hooft anomalies and the holomorphy of supersymmetric partition functions
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Abstract
A bstract We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, G F , for 2d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = (0, 2) and 4d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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, $$ {Z}_{T^2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> </mml:math> , and for a large class of 4d partition functions on half-BPS four-manifolds, $$ {Z}_{{\mathcal{M}}^4} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:msub> </mml:math> — in particular, for $$ \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> 4 = S 3 × S 1 and $$ \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> 4 = Σ g × T 2 . We propose a new expression for $$ {Z}_{{\mathcal{M}}_{d-1}\times {S}^1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:msub> </mml:math> , 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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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