Three-point functions in c <= 1 Liouville theory and conformal loop ensembles

Yacine Ikhlef (1) , Jesper Lykke Jacobsen (2) , Hubert Saleur (3, 4)

1 LPTHE - Laboratoire de Physique Théorique et Hautes Energies
2 LPTENS - Laboratoire de Physique Théorique de l'ENS [École Normale Supérieure]
3 USC - University of Southern California
4 IPHT - Institut de Physique Théorique - UMR CNRS 3681

The possibility of extending the Liouville Conformal Field Theory from values of the central charge $c \geq 25$ to $c \leq 1$ 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 DOZZ 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 $V_{\hat{\alpha}}$ in $c \leq 1$ Liouville. We interpret geometrically the limit $\hat{\alpha} \to 0$ of $V_{\hat{\alpha}}$ and explain why it is not the identity operator (despite having conformal weight $\Delta=0$).

Domaines: Mécanique statistique [cond-mat.stat-mech]

Yacine Ikhlef, Jesper Lykke Jacobsen, Hubert Saleur. Three-point functions in c <= 1 Liouville theory and conformal loop ensembles. Physical Review Letters, 2015, 116 (13), pp.130601. ⟨10.1103/PhysRevLett.116.130601⟩. ⟨cea-01468523⟩