Vortex Counting and Lagrangian 3-manifolds

TL;DR

<3-5 sentence high-level summary> The paper investigates a deep, holography-like connection between 3-manifold quantum invariants and two-dimensional vortex counting arising from half-BPS surface operators in four-dimensional N=2 gauge theories. It develops a framework where 5d N=2 super-Yang–Mills on R^2×M yields a 2d N=(2,2) theory whose vortex partition function mirrors refined BPS invariants and relates to open/closed topological string data via geometric transitions. By geometric engineering with Lagrangian branes, it provides explicit checks in U(1) and SU(2) theories, relates surface operators to knots/links, and connects to conformal field theory through degenerate insertions in AGT-type correspondences. The work thus ties 3d manifold invariants, 2d vortex counting, knot homologies, and CFT blocks into a unified 3D–2D–CFT geometric picture with potential broad implications for topological strings and gauge theory dualities.

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.

Figures (29)

• Figure 1: The duality web.
• Figure 2: The brane construction of ${\cal N}=2$ super Yang-Mills theory with a half-BPS surface operator in type IIA string theory $(a)$ and its M-theory lift $(b)$.
• Figure 3: $U(1)$ toric geometry with a single Lagrangian brane.
• Figure 4: a) Possible phases (placements) of a Lagrangian brane. b) The phase for instanton counting.
• Figure 5: Regulating the surface-operator theory in the brane construction. (This setup is dual to the geometry in Figure \ref{['fig:branechoices2']}(b).)