Geometry and Integrability in Mathematical Physics GIMP'06

May 15 - 19, 2006, Moscow, Russia

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Alexei Zamolodchikov

Univ. Montpellier, France

Perturbed CFT coupled to 2D gravity and dynamical triangulations

As an introduction, we consider the ``critical'' 2D gravity, i.e., a ``matter'' conformal field theory coupled to quantized two-dimensional metric space. The Liouville field theory is constructed as the effective CFT which describes the dynamics of the metric in the conformal gauge. This gives rise to the KPZ critical exponents, which can be compared with those of statistical systems on dynamical triangulations (when the latter are solvable by the matrix model techniques). Perturbed Liouville reveils more detailed information, in particular certain universal scaling functions. An example of the ``random lattice Ising model'', in particular the scaling near its phase transition point, is discussed in more details both from matrix model and perturbed Liouville grantiy points of view.