The integrable structure of nonrational conformal field theory

Using the example of Liouville theory, we show how the separation into left- and rightmoving degrees of freedom of a nonrational conformal field theory can be made explicit in terms of its integrable structure. The key observation is that there exist separate Baxter Q-operators for left- and right-moving degrees of freedom. Combining a study of the analytic properties of the Q-operators with Sklyanin’s Separation of Variables Method leads to a complete characterization of the spectrum. Taking the continuum limit allows us in particular to rederive the Liouville reflection amplitude using only the integrable structure.