# Fivebranes and Knots

@article{Witten2011FivebranesAK, title={Fivebranes and Knots}, author={Edward Witten}, journal={arXiv: High Energy Physics - Theory}, year={2011} }

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…

## 260 Citations

Octonions, Monopoles, and Knots

- Mathematics, Physics
- 2015

Witten’s approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys–Witten equations. We argue for a one-to-one…

Khovanov homology and gauge theory

- Mathematics, Physics
- 2011

In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions.…

Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology

- Physics, Mathematics
- 2016

We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional…

Fivebranes and 3-manifold homology

- Physics, Mathematics
- 2016

A bstractMotivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of…

Knot invariants and M-theory: Hitchin equations, Chern-Simons actions, and surface operators

- Physics, Mathematics
- 2017

Recently Witten introduced a type IIB brane construction with certain boundary conditions to study knot invariants and Khovanov homology. The essential ingredients used in his work are the…

Knot Invariants from Four-Dimensional Gauge Theory

- Mathematics, Physics
- 2012

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory…

Two lectures on gauge theory and Khovanov
homology

- Mathematics, PhysicsProceedings of Symposia in Pure Mathematics
- 2018

In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the…

Gauge Theory and Integrability, II

- Physics, Mathematics
- 2018

Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang-Baxter equation, we determine the corresponding quantum group deformations to all…

Branes and categorifying integrable lattice models

- Physics, Mathematics
- 2018

We elucidate how integrable lattice models described by Costello's 4d Chern-Simons theory can be realized via a stack of D4-branes ending on an NS5-brane in type IIA string theory, with D0-branes on…

3d-3d correspondence revisited

- Physics, Mathematics
- 2016

A bstractIn fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2$$ \mathcal{N}=2 $$ theory. The Lagrangians…

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