# Junctions of surface operators and categorification of quantum groups

@article{Chun2015JunctionsOS,
title={Junctions of surface operators and categorification of quantum groups},
author={Sungbong Chun and Sergei Gukov and Daniel Roggenkamp},
journal={arXiv: High Energy Physics - Theory},
year={2015}
}
• Published 22 July 2015
• Physics, Mathematics
• arXiv: High Energy Physics - Theory
We show how networks of Wilson lines realize quantum groups U_q(sl_m), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface operators we reproduce known mathematical constructions of categorical representations and categorified quantum groups.
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## References

SHOWING 1-10 OF 146 REFERENCES
The quantum sl(n) graph invariant and a moduli space
• Mathematics
• 2012
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We
An introduction to diagrammatic algebra and categorified quantum sl(2)
This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of
A diagrammatic approach to categorification of quantum groups II
• Mathematics
• 2011
To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the Kac-Moody Lie algebra associated
A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products
• Mathematics
• 2005
The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain
Introduction to Quantum Groups
• Mathematics
• 1998
We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra
Quantum Schubert Calculus
String theorists (notably Witten [W]) recently introduced the notion of a “quantum” deformation of the cohomology ring of a smooth projective variety X. This quantum deformation, or quantum
Gauge theory 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 braid
Eigenvalues of products of unitary matrices and quantum Schubert calculus
• Mathematics
• 1997
We describe the inequalities on the possible eigenvalues of products of unitary matrices in terms of quantum Schubert calculus. Related problems are the existence of flat connections on the punctured