The virtue of defects in 4D gauge theories and 2D CFTs

We advance a correspondence between the topological defect operators in Liouville and Toda conformal field theories — which we construct — and loop operators and domain wall operators in four dimensional $$ \mathcal{N} = 2 $$ supersymmetric gauge theories on S4. Our computation of the correlation functions in Liouville/Toda theory in the presence of topological defect operators, which are supported on curves on the Riemann surface, yields the exact answer for the partition function of four dimensional gauge theories in the presence of various walls and loop operators; results which we can quantitatively substantiate with an independent gauge theory analysis. As an interesting outcome of this work for two dimensional conformal field theories, we prove that topological defect operators and the Verlinde loop operators are different descriptions of the same operators.