  We propose a correspondence between loop operators in a family of four
dimensional N=2 gauge theories on S^4 -- including Wilson, 't Hooft and dyonic
operators -- and Liouville theory loop operators on a Riemann surface. This
extends the beautiful relation between the partition function of these N=2
gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa.
We show that the computation of these Liouville correlators with the insertion
of a Liouville loop operator reproduces Pestun's formula capturing the
expectation value of a Wilson loop operator in the corresponding gauge theory.
We prove that our definition of Liouville loop operators is invariant under
modular transformations, which given our correspondence, implies the
conjectured action of S-duality on the gauge theory loop operators. Our
computations in Liouville theory make an explicit prediction for the exact
expectation value of 't Hooft and dyonic loop operators in these N=2 gauge
theories. The Liouville loop operators are also found to admit a simple
geometric interpretation within quantum Teichmuller theory as the quantum
operators representing the length of geodesics. We study the algebra of
Liouville loop operators and show that it gives evidence for our proposal as
well as providing definite predictions for the operator product expansion of
loop operators in gauge theory.
