Loop and surface operators in gauge theory and Liouville modular geometry

Recently, a duality between Liouville theory and four dimensional $$ \mathcal{N} = 2 $$ gauge theory has been uncovered by some of the authors. We consider the role of extended objects in gauge theory, surface operators and line operators, under this correspondence. We map such objects to specific operators in Liouville theory. We employ this connection to compute the expectation value of general supersymmetric ’t Hooft-Wilson line operators in a variety of $$ \mathcal{N} = 2 $$ gauge theories.