High Energy Theory Seminar

Line operators in class S theories correspond to network operators on the associated Riemann surface. I will describe the algebra of functions on the moduli space of flat SL(N)-connections in terms of a set of networks that are naturally associated with a pants decomposition. These generators satisfy some polynomial relations, implied by skein relations. I will present an explicit representation of this algebra in terms of Fock-Goncharov holonomies. Their quantization involves the calculation of framed BPS indices, which count the BPS states supported by a line operator, keeping track of their spins. Another representation of the algebra is provided by Verlinde operators in SU(N) Toda field theory.