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_{α[over ^]} in c≤1 Liouville theory. We interpret geometrically the limit α[over ^]→0 of V_{α[over ^]} and explain why it is not the identity operator (despite having conformal weight Δ=0).