Letters in Mathematical Physics

, Volume 98, Issue 3, pp 225–287 | Cite as

Vortex Counting and Lagrangian 3-Manifolds



To every 3-manifold M one can associate a two-dimensional \({\mathcal{N}=(2, 2)}\) supersymmetric field theory by compactifying five-dimensional \({\mathcal{N}=2}\) super-Yang–Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional \({\mathcal{N}=2}\) gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional \({\mathcal{N}=(2, 2)}\) supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be mapped to the computation of degenerate conformal blocks in two-dimensional CFT’s. Degenerate limits of vertex operators in CFT receive a simple interpretation via geometric transitions in BPS counting.

Mathematics Subject Classification (2000)

81T60 81T30 81T40 14N35 


gauge theory vortex equations BPS invariants D-branes conformal field theory 


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Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.University of CaliforniaSanta BarbaraUSA

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