Schur indices with class S line operators from networks and further skein relations

Abstract: We compute the Schur indices in the presence of some line operators based on our con- jectural formula introduced in [1]. In particular, we focus on the rank 1 superconformal field theories with the enhanced global symmetry and the free hypermultiplets with the elementary pants networks defined on the three punctured sphere in the class S context. From the observations on the concrete computations, we propose new kinds of the class S skein relations in the sense that they include the generic puncture non-trivially. We also give a general formula to unify all the relations we have exhibited.