$\mathcal{N}{=}1$ supersymmetric indices and the four-dimensional A-model

Cyril Closset; Heeyeon Kim; Brian Willett

doi:10.1007/JHEP08(2017)090

paper

$N = 1$ supersymmetric indices and the four-dimensional A-model

arXiv:1707.05774 · doi:10.1007/JHEP08(2017)090

Abstract

We compute the supersymmetric partition function of $N = 1$ supersymmetric gauge theories with an $R$ -symmetry on $M_{4} ≅ M_{g, p} \times S^{1}$ , a principal elliptic fiber bundle of degree $p$ over a genus- $g$ Riemann surface, $Σ_{g}$ . Equivalently, we compute the generalized supersymmetric index $I_{M_{g, p}}$ , with the supersymmetric three-manifold $M_{g, p}$ as the spatial slice. The ordinary $N = 1$ supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological $A$ -model for the abelianized gauge fields on the base $Σ_{g}$ . This $A$ -model---or $A$ -twisted two-dimensional $N = (2, 2)$ gauge theory---encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on $T^{2}$ inside $Σ_{g} \times T^{2}$ . Being defined by compactification on the torus, the $A$ -model also enjoys natural modular properties, governed by the four-dimensional 't Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.

arXiv comment91 pages including appendices; v2: corrected typos and added references, JHEP version

$N = 1$ supersymmetric indices and the four-dimensional A-model

Abstract

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N=1 supersymmetric indices and the four-dimensional A-model

Abstract

References in corpus (65)

Cited by in corpus (180)

$N = 1$ supersymmetric indices and the four-dimensional A-model