Higher laminations, webs and N=2 line operators

TL;DR

This paper explores the geometric and algebraic structures of line operators in 4d N=2 theories derived from 6d (2,0) theory, introducing higher laminations and webs to understand their properties and interactions.

Contribution

It introduces the concept of higher laminations and webs for line operators, linking geometric web structures with algebraic coordinate systems in higher Teichmüller spaces.

Findings

Representation of line operators as bipartite webs (higher laminations)

Explicit calculation of expectation values as positive Laurent polynomials

Derivation of Skein relations and Poisson brackets for line operators

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.