Notes on Liouville theory at c ≤ 1

The continuation of the Liouville conformal field theory to c <= 1 is considered. The viability of an interpretation involving a timelike boson which is the conformal factor for two-dimensional asymptotically de Sitter geometries is examined. The conformal bootstrap leads to a three-point function with a unique analytic factor which is the same as that which appears along with the fusion coefficients in the minimal models. A corresponding nonanalytic factor produces a well-defined metric on fields only when the central charge is restricted to those of the topological minimal models, and when the conformal dimensions satisfy h>(c-1)/24. However, the theories considered here have a continuous spectrum which excludes the degenerate representations appearing in the minimal models. The c = 1 theory has been investigated previously using similar techniques, and is identical to a nonrational conformal field theory ( CFT) which arises as a limit of unitary minimal models. When coupled to unitary matter fields, the nonunitary theories with c <= - 2 produce string amplitudes which are similar to those of the minimal string.