String Equation for String Theory on a Circle

We derive a constraint (string equation) which together with the Toda Lattice hierarchy determines completely the integrable structure of the compactified 2D string theory. The form of the constraint depends on a continuous parameter, the compactification radius R. We show how to use the string equation to calculate the free energy and the correlation functions in the dispersionless limit. We sketch the phase diagram and the flow structure and point out that there are two UV critical points, one of which (the sine-Liouville string theory) describes infinitely strong vortex or tachyon perturbation.