Boundary correlators in 2D quantum gravity: Liouville versus discrete approach

by Ivan KOSTOV (CEN Saclay)

Tuesday 25 Feb 2003, 14:00 → 15:00 Europe/Zurich

TH Auditorium (CERN)

The 2D quantum gravity, known also as non-critical string theory, has two complementary formulations: the continuous one based on Liouville theory and the discrete formulation through planar graphs and matrix models. There is no formal proof of the equivalence of the two approaches, but it is widely believed that they describe the same theory. In the theory of closed strings (world sheet without boundary), the only quantities calculable in both approaches are the scaling exponents (given by the KPZ rule), which indeed coincide. Much more interesting is the case of world sheets with boundaries, where one can compare not only critical exponents, but also functions of several dimensionless quantities. I will show here that the two-point boundary correlation function in Liouville theory, obtained by V. Fateev, A. Zamolodchikov and Al. Zamolodchikov can be also obtained from the discrete formulation of 2D quantum gravity as loop gas on a random graph. The mysterious functional equation found by these authors can be given a nice geometrical meaning in the discrete approach.