Fivebranes and Knots

TL;DR

This paper introduces a gauge theory approach to Khovanov homology of knots, connecting D-brane systems with Chern-Simons theory and dualities, resulting in a new invariant description of knot polynomials and homology.

Contribution

It develops a novel gauge-theoretic framework for Khovanov homology using D-branes, dualities, and elliptic PDEs, providing a manifestly invariant formulation.

Findings

New invariant description of Jones polynomial and Khovanov homology.

Connection between D-brane systems and knot invariants.

Use of elliptic PDEs in four and five dimensions for knot theory.

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.