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

Cyril Closset, Heeyeon Kim, Brian Willett

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


We compute the supersymmetric partition function of N = 1 supersymmetric gauge theories with an R-symmetry on ℳ 4≅ ℳ g , p× S1, a principal elliptic fiber bundle of degree p over a genus-g Riemann surface, Σg. Equivalently, we compute the generalized supersymmetric index Iℳg,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 Σg. 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 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.

Original languageEnglish (US)
Article number90
JournalJournal of High Energy Physics
Issue number8
StatePublished - Aug 1 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


  • Anomalies in Field and String Theories
  • Duality in Gauge Field Theories
  • Supersymmetric Gauge Theory
  • Supersymmetry and Duality


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