Fivebranes and Knots

TL;DR

Fivebranes and Knots builds a bridge from 3D Chern-Simons knot invariants to a categorified invariant—Khovanov homology—via a sequence of gauge-theoretic dualities. Starting from a D3-NS5 boundary system with a theta-angle, the paper uses S- and T-dualities to recast the Jones polynomial in terms of a boundary-theory observable, then expresses the theory in terms of higher-dimensional (N=4) and ultimately six-dimensional (0,2) frameworks with surface- and monodromy-defect insertions. Localization, boundary conditions, and the interplay between Wilson lines, ’t Hooft operators, and surface operators yield a natural bigrading and a cohomological structure whose Euler characteristic recovers the Jones polynomial, while its full cohomology yields Khovanov homology. The work also extends to non-simply-laced groups and discusses ultraviolet completions, framing anomalies, and two complementary routes (bottom-up 5D/6D and top-down 6D) to categorification, with potential applications to broader knot invariants and geometric Langlands-type dualities.

Abstract

We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS states). The starting point is a system of D3-branes ending on an NS5-brane with a nonzero theta-angle. On the one hand, this system can be related to a Chern-Simons gauge theory on the boundary of the D3-brane worldvolume; on the other hand, it can be studied by standard techniques of $S$-duality and $T$-duality. Combining the two approaches leads to a new and manifestly invariant description of the Jones polynomial of knots, and its generalizations, and to a manifestly invariant description of Khovanov homology, in terms of certain elliptic partial differential equations in four and five dimensions.