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

By Cyril Closset, Hee-Cheol Kim, Brian Willett. 2017

370 citations recorded by the source corpus.

A bstractWe compute the supersymmetric partition function of N$$ \mathcal{N} $$ = 1 supersymmetric gauge theories with an R-symmetry on ℳ4≅ℳg,p×S1$$ {\mathrm{\mathcal{M}}}_4\cong {\mathrm{\mathcal{M}}}_{g,p}\times {S}^1 $$, a principal elliptic fiber bundle of degree p over a genus-g Riemann surface, Σg . Equivalently, we compute the generalized supersymmetric index Iℳg,p$$ {I_{\mathrm{\mathcal{M}}}}_{{}_{g,p}} $$, with the supersymmetric three-manifold ℳg,p$$ {\mathrm{\mathcal{M}}}_{g,p} $$ as the spatial slice. The ordinary N$$ \mathcal{N} $$ = 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 Σg. This A-model — or A-twisted two-dimensional N$$ \mathcal{N} $$ = (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 T2 inside Σg × T2. 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.

Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/83ce412d1b442adaba2dbbc0670ca87a9e56a5ea