Junctions of surface operators and categorification of quantum groups

TL;DR

The paper develops a physical program to categorify quantum groups by elevating networks of Wilson lines in 3d SU(N) Chern-Simons theory to foams of surface operators in 4d, realizing a 2-category ${\dot{\mathcal{U}}}_q(\mathfrak{sl}_m)$. It unifies two complementary 2d descriptions—Landau-Ginzburg interfaces and Grassmannian sigma-models—into a 4d foam framework where line and surface defects form a 2-category whose 2-morphisms encode categorified relations. Through skew Howe duality, MOY graph polynomials, and matrix-factorization constructions, the authors derive the categorified actions of the quantum group generators $E_i,F_i$ and their relations, providing a concrete bridge between physical brane setups and diagrammatic categorification. The work illuminates rich connections with Schubert calculus, Grassmannians, and Horn-type OPE constraints, and points to further avenues for generalizations, holographic realizations, and broader algebraic structures. Overall, it offers a cohesive physical realization of categorified quantum groups and their representations via surface-operator junctions and LG interfaces, with potential applications to knot homologies and geometric representation theory.

Abstract

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.

Figures (12)

• Figure 1: Categorified representation theory and categorified skew Howe duality.
• Figure 2: Projecting $\Sigma$ onto $(x^0,x^1)$ plane gives a product of Landau-Ginzburg theories, in which junctions (singular edges of $\Sigma$) are represented as interfaces.
• Figure 3: Relations between vertices in the conventions of Witten:1989wf. Our choice for the vertices between totally antisymmetric representations $\Lambda^{k_1}\square$, $\Lambda^{k_2}\square$ and $\Lambda^{k_1+k_2}\square$ involves an extra sign factor $(-1)^{k_1k_2}$.
• Figure 4: Relations among networks of Wilson lines.
• Figure 5: (a) Two linear relations among three Wilson lines in ${\mathcal{H}}_{S^{2},\{1,m,\overline{1},\overline{m}\}}$, (b) two linear relations among three Wilson lines containing those of (a) in the red dashed box, and (c) proportionality relations between two vectors in ${\mathcal{H}}_{S^{2},\{G,j,\overline{j} \}}$.