Higher laminations, webs and N=2 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.