# Fivebranes and Knots

@article{Witten2011FivebranesAK,
title={Fivebranes and Knots},
author={Edward Witten},
journal={arXiv: High Energy Physics - Theory},
year={2011}
}
• E. Witten
• Published 17 January 2011
• Mathematics, Physics
• arXiv: High Energy Physics - Theory
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… Expand
259 Citations
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• Mathematics, Physics
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• 2018
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#### References

SHOWING 1-10 OF 132 REFERENCES
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 theoryExpand
Surface Operators and Knot Homologies
Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braidExpand
Three-Dimensional Quantum Gravity
We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds.Expand
D-branes, monopoles and Nahm equations
Abstract We study the correspondence between 11b solitonic 1-branes and monopoles in the context of the 3-brane realization of D = 4 N = 4 super Yang-Mills theory. We show that a bound state ofExpand
Localization for Wilson Loops in Chern-Simons Theory
We reconsider Chern-Simons gauge theory on a Seifert manifold M , which is the total space of a nontrivial circle bundle over a Riemann surface Σ, possibly with orbifold points. As shown in previousExpand
Knot Invariants and Topological Strings
• Physics
• 1999
We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of aExpand
M-theory and a topological string duality
• Physics
• 2006
We show how the topological string partition function, which is known to capture the degeneracies of a gas of BPS spinning M2-branes in M-theory compactified to 5 dimensions, is related to aExpand
Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
We study three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds.Expand
Gauge Theory, Ramification, And The Geometric Langlands Program
• Mathematics, Physics
• 2006
In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson orExpand
Link Homologies and the Refined Topological Vertex
• Physics, Mathematics
• 2007
We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation providesExpand