Non-rational 2D quantum gravity II. Target space CFT

We explore the formulation of non-rational 2D quantum gravity in terms of a chiral CFT on a Riemann surface associated with the target space. The CFT in question is constructed as the collective theory for a matrix chain, which is dual to a statistical height model on dynamical triangulations. The heights are associated with the sheets of the Riemann surface, which represents an infinite branched cover of the spectral plane. We consider two examples of height models: the SOS model and the semi-restricted SOS (SRSOS) model, in which the heights are restricted from below. Both models are described in the continuum limit by theories of 2D quantum gravity with conformal matter, perturbed by a thermal operator (1,3). We give a compact operator expression for the n-loop amplitudes as a collection of target space Feynman rules. The n-point functions of local fields are obtained by shrinking the loops. In particular, we show that the 4-point function of order operators in the SRSOS model coincides with the 4-point function of the ''diagonal'' world sheet CFT studied in [I. Kostov, V. Petkova, Non-rational 2D quantum gravity I. World sheet CFT, hep-th/0512346]