We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approaches 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.
Cite this article
Edward Witten, Fivebranes and knots. Quantum Topol. 3 (2012), no. 1, pp. 1–137DOI 10.4171/QT/26