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 $c\ge 25$ 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 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 $c\le 1$ 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 $\Delta =0$).

11 pages, 6 figures. Version 2: minor improvements

References in corpus (3)

• Liouville theory with a central charge less than one
• Conformal Field Theory and Statistical Mechanics
• Correlation functions in loop models

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