Moduli integrals in Liouville gravity*) YUKITAKA ISHIMOTO Okayama Institute for Quantum Physics, 1-9-1 Kyouyama, Okayama 700-0015, Japan Received 15 August 2006 We consider moduli integrals appearing in four-point correlation functions of the (p, q) minimal models coupled to Liouville gravity on a sphere, which is sometimes called 2D minimal gravity or minimal string theory on a sphere. Liouville gravity on a sphere is the quantized metric of the spherical topology in the conformal gauge. Reviewing the previous results on such four-point functions (Y. Ishimoto and Sh. Yamaguchi: Phys. Lett. B 607 (2005) 172), we show logarithmic correlation functions of 'tachyons' in the Liouville sector, and its moduli integrals of the full correlation functions, in particular in the Majorana fermion model coupled to 2D gravity. Further discussions and related results are given in the final section and in Y. Ishimoto and A1. Zamolodchikov: Theor. Math. Phys. 147 (2006) 755. PACS: 11.25.Hf Key words: conformal field theory, Liouville field theory, quantum gravity, noncritical string theory, random lattice 1 Introduction The (p, q) minimal models coupled to the quantized metric of 2D surface have drawn much attention in the literature (see for example [1-19]). Partially, it is be- cause the models, or 2D minimal gravity, are nothing but noncritical string theories and serve as string laboratories, whose target space is two-dimensional in a naive sense. On the other hand, much attention has also been paid to the models from the statistical physics point of view, because they deal with conformal matters on random surface. For example, Ising spins on random lattices at its criticality. It surely contains some information on quantum gravity and is expected to provide more interesting features of quantum gravity. There has been a number of works devoted to this problem, and they can be roughly classified into two species according to their approaches to the problem. One is the discrete approach, by which the conformal matter on random lattices is described by the dynamics of random matrices: matrix models (many references in- cluding [3,4]). The other is the continuum approach, where the theory is formulated in the field theory language (for example, see [1, 2, 5-19]). The former can handle the free energy directly, summing over all topologies of the two-dimensional surface in question. However, it cannot explain local behaviours of the theory and local algebraic structures therein. On the other hand, the latter can do such behaviours by definition and, furthermore, perturbations from criticality [5, 6], whereas it is not suitable for treating all possible topologies at once. Both approaches have their *) Presented at the 15 TM International Colloquium on "Integrable Systems and Quantum Sym- metries", Prague, 15 17 June 2006. Czechoslovak Journal of Physics, Vol. 56 (2000), No. I0/ii 1203