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

Abstract : 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 $\Delta=0$).
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Journal articles

Cited literature [15 references]

https://hal-cea.archives-ouvertes.fr/cea-01468523
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### Citation

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

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