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Vortex Counting and Lagrangian 3-manifolds

To every 3-manifold M one can associate a two-dimensional N=(2,2) supersymmetric field theory by compactifying five-dimensional N=2 super-Yang-Mills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional 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 N=(2,2) supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can be also 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.