Vortex Counting and Lagrangian 3-manifolds

TL;DR

This paper explores the connection between vortex counting in 2D supersymmetric theories, refined BPS invariants of dual geometries, and conformal blocks in 2D CFTs, revealing geometric transitions in BPS counting.

Contribution

It establishes a novel link between vortex counting, BPS invariants, and conformal blocks, advancing the understanding of geometric and physical dualities.

Findings

Vortex counting relates to refined BPS invariants in dual geometries.

Degenerate conformal blocks correspond to geometric transitions in BPS counting.

The framework connects supersymmetric field theories, knot homologies, and CFTs.

Abstract

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.