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We advance a correspondence between the topological defect operators inLiouville and Toda conformal field theories - which we construct - and loopoperators and domain wall operators in four dimensional N=2 supersymmetricgauge theories on S^4. Our computation of the correlation functions inLiouville/Toda theory in the presence of topological defect operators, whichare supported on curves on the Riemann surface, yields the exact answer for thepartition function of four dimensional gauge theories in the presence ofvarious walls and loop operators; results which we can quantitativelysubstantiate with an independent gauge theory analysis. As an interestingoutcome of this work for two dimensional conformal field theories, we provethat topological defect operators and the Verlinde loop operators are differentdescriptions of the same operators.

The Virtue of Defects in 4D Gauge Theories and 2D CFTs