# Surface Operators and Knot Homologies

@inproceedings{Gukov2007SurfaceOA, title={Surface Operators and Knot Homologies}, author={Sergei Gukov}, year={2007} }

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 braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N = 2 and N = 4 twisted gauge theories which lead to a categorification of the…

## Figures and Tables from this paper

## 10 Citations

Homological algebra of knots and BPS states

- Mathematics
- 2013

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves…

Topological Strings, Double Affine Hecke Algebras, and Exceptional Knot Homology

- Mathematics
- 2015

In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras ("DAHA") and…

Selfduality and Chern-Simons Theory

- Mathematics
- 2008

We propose a relation between the operator of S-duality (of N=4 super Yang-Mills theory in 3+1D) and a topological theory in one dimension lower. We construct the topological theory by compactifying…

Instanton counting with a surface operator and the chain-saw quiver

- Mathematics
- 2011

We describe the moduli space of SU(N) instantons in the presence of a general surface operator of type N = n1 + ⋯ + nM in terms of the representations of the so-called chain-saw quiver, which allows…

Q A ] 1 1 A pr 2 01 3 Quadruply-graded colored homology of knots

- Mathematics
- 2013

We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative…

Hitchin systems in N = 2 field theory

- Mathematics
- 2014

This is the second article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J. Teschner. The space of vacua of class $\cal S$ theories on $R^3 \times S^1$ can be…

Brane tilings and their applications

- Physics
- 2008

We review recent developments in the theory of brane tilings and four‐dimensional 𝒩 = 1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of…

A sheaf-theoretic model for SL(2,C) Floer homology

- Mathematics
- 2017

Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth…

AP Theory II:Intrinsic 4D Quantum YM Theory with Mass Gap

- Mathematics
- 2007

We describe a sub-theory of Artin Presentation Theory (AP Theory), which has many genuine,discrete,group-theoretic,non-infinitesimal, qualitative analogues (including with the mass gap) of the main…

Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde Formula

- Mathematics
- 2019

Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the…

## References

SHOWING 1-10 OF 78 REFERENCES

Gauge Theoretic Invariants of, Dehn Surgeries on Knots

- Mathematics
- 1999

New methods for computing a variety of gauge theoretic invariants for homology 3{spheres are developed. These invariants include the Chern{Simons invariants, the spectral flow of the odd signature…

KNOT AND LINK INVARIANTS AND MODULI SPACE OF PARABOLIC BUNDLES

- Mathematics
- 2001

In this paper, we show that the representation variety of the fundamental group of a 2n-punctured S2 with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from…

Link Homologies and the Refined Topological Vertex

- 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 provides…

Gauge Theory, Ramification, And The Geometric Langlands Program

- Mathematics
- 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 or…

Knots, links and branes at large N

- Mathematics
- 2000

We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting…

Flat connections, the Alexander invariant, and Casson’s invariant

- Mathematics
- 1997

This paper explores the relationship between the flat moduli space of a homology knot complement and other topological invariants of the knot. We define an invariant by counting the flat orbits with…

A link invariant from the symplectic geometry of nilpotent slices

- Mathematics
- 2004

We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid,…

An Area-Preserving Action of the Modular Group on Cubic Surfaces and the Painlevé VI Equation

- Mathematics
- 2003

AbstractWe construct an area-preserving action of the modular group on a general 4-parameter family of affine cubic surfaces. We present a geometrical background behind this construction, that is, a…