Higher laminations, webs and N=2 line operators

TL;DR

This work extends the lamination approach for half-BPS line operators from the rank-1 case to higher rank $A_{N-1}$ theories by encoding UV line operators as higher laminations—irreducible bipartite webs built from three-junctions—and IR line operators as cluster $X$-coordinates on the corresponding $SL(N,\mathbb{C})$ moduli. A canonical map ties these pictures, yielding positive Laurent expressions for UV operator expectation values on the Coulomb branch and enabling a finite, positive OPE expansion governed by higher-rank Skein relations; this also leads to a description of line operators via webs and a higher Teichmüller–type coordinate system. The paper provides explicit constructions for $A_2$ on a torus with one full puncture, identifies missing internal line operators via OPEs, and verifies Poisson brackets and integrability, with generalizations to arbitrary bordered surfaces and $T_N$ and Argyres–Douglas theories. The framework unifies geometric (lamination/web) and algebraic (cluster) descriptions of line operators, offers a path to quantization and connections to Hitchin integrable systems, and has potential implications for geometric Langlands, 2d Toda, and related mathematical structures. Overall, the work establishes a coherent higher-rank extension of laminations and Skein calculus as a foundational tool for studying line operators in class $\mathcal{S}$ theories.

Abstract

A detailed study of half-BPS line operators of higher rank 4d N=2 theory engineered from six dimensional A_{N-1} (2,0) theory on a bordered Riemann surface with full marked points is performed. Geometrically, each 4d UV line operator is represented by an irreducible bipartite web formed by three junctions on Riemann surface, and such web structure is called higher lamination. Algebraically, the space of UV line operators is identified with the integral tropical a coordinates of the corresponding PGL(N,C) local system, and the space of IR line operator is identified with the cluster X coordinates of SL(N.C) local system. The expectation value of UV line operator at Coulomb branch parameterized by X coordinates is calculated, and the result is a positive Laurent polynomial in X. Using the expectation values, we calculate the operator product expansion (OPE) between the line operators, which is then represented geometrically by higher rank Skein relations. We also calculate the Poisson brackets of these line operators, and Frenchel-Nielson type coordinates are constructed for Higher Teichmuller space, etc.

Figures (37)

• Figure 1: Top left: the BPS quiver for $A_2$ theory on once punctured torus, here the quiver nodes on the edges labeled by same red lines are identified. There is a cluster $X$ variable on each quiver node. Top right: the tropical $a$ coordinate of a line operator $L$. Bottom: the OPE between Wilson and 't Hooft line, and a bipartite web must appear in the OPE.
• Figure 2: A: Skein relations for $A_2$ theory. B. Bipartite webs on once punctured torus of $A_2$ theory, and edges of three junctions with same label are connected.
• Figure 3: A: The two basic matter systems for four dimensional theory built from six dimensional $A_1$ theory: the tri-fundamental represented by a sphere with three regular singularity and the D type Argyres-Douglas theory represented by a sphere with one irregular and one regular singularity. B: In one weakly coupled gauge group duality frame, the theory is formed by gluing above two pieces together: physically this is achieved by gauging the diagonal flavor symmetries. C: We replace each irregular singularity with a boundary with marked points.
• Figure 4: The Dehn-Thurston coordinates of a set of closed curves are defined using a pants decomposition: the $p_i$ coordinates are defined by simply counting the weighted intersecting number, while the $q_i$ coordinates are defined by counting the oriented winding number.
• Figure 5: A: Laminations on an annulus with one marked points on each boundary, which represent the pure $SU(2)$ theory. B: Constructing coordinates for the lamination using the triangulation.