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- Title
$$ \mathcal{N} $$ = 1 supersymmetric indices and the four-dimensional A-model.
- Authors
Closset, Cyril; Kim, Heeyeon; Willett, Brian
- Abstract
We compute the supersymmetric partition function of $$ \mathcal{N} $$ = 1 supersymmetric gauge theories with an R-symmetry on $$ {\mathrm{\mathcal{M}}}_4\cong {\mathrm{\mathcal{M}}}_{g,p}\times {S}^1 $$ , a principal elliptic fiber bundle of degree p over a genus- g Riemann surface, Σ . Equivalently, we compute the generalized supersymmetric index $$ {I_{\mathrm{\mathcal{M}}}}_{{}_{g,p}} $$ , with the supersymmetric three-manifold $$ {\mathrm{\mathcal{M}}}_{g,p} $$ as the spatial slice. The ordinary $$ \mathcal{N} $$ = 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. This A-model - or A-twisted two-dimensional $$ \mathcal{N} $$ = (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 inside Σ × T . 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.
- Subjects
SUPERSYMMETRY; GAUGE field theory; RIEMANN surfaces; W bosons; FIELD theory (Physics)
- Publication
Journal of High Energy Physics, 2017, Vol 2017, Issue 8, p1
- ISSN
1126-6708
- Publication type
Article
- DOI
10.1007/JHEP08(2017)090