Fusion rules and macroscopic loops from discretized approach to two-dimensional gravity

Abstract

We investigate the multi-loop correlators and the multi-point functions for all of the scaling operators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for these scaling operators exist, and these are summarized in a compact form as fusion rules for loops. We clarify the role of the boundary operators and discuss its connection to how loops touch each other. We derive a general formula for the n-resolvent correlators, and point out the structure similar to the crossing symmetry of underlying conformal field theory. We discuss the connection of the boundary conditions of the loop correlators to the touching of loops for the case of the four-loop correlators.