Three-Point Functions in c ≤ 1 Liouville Theory and Conformal Loop Ensembles

The possibility of extending the Liouville conformal field theory from values of the central charge c >= 25 to c <= 1 has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension-involving a real spectrum of critical exponents as well as an analytic continuation of the Dorn-Otto-Zamolodchikov-Zamolodchikov formula for three-point couplings-does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators V-(alpha) over cap in c <= 1 Liouville theory. We interpret geometrically the limit (alpha) over cap -> 0 of V-(alpha) over cap and explain why it is not the identity operator (despite having conformal weight Delta = 0).