The possibility of extending the Liouville conformal field theory from values of the central charge $c\ge 25$ to $c\le 1$ 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 $V_{\widehat{\alpha }}$ in $c\le 1$ Liouville theory. We interpret geometrically the limit $\widehat{\alpha }\to 0$ of $V_{\widehat{\alpha }}$ and explain why it is not the identity operator (despite having conformal weight $\mathrm{\Delta }=0$).

• Received 14 November 2015

DOI:https://doi.org/10.1103/PhysRevLett.116.130601