Macroscopic $n$-Loop Amplitude for Minimal Models Coupled to Two-Dimensional Gravity: Fusion Rules and Interactions

Abstract

We investigate the structure of the macroscopic $n$-loop amplitude obtained from the two-matrix model at the unitary minimal critical point $(m+1,m)$. We derive a general formula for the $n$-resolvent correlator at the continuum planar limit whose inverse Laplace transform provides the amplitude in terms of the boundary lengths ${\ell }_i$ and the renormalized cosmological constant $t$. The amplitude is found to contain a term consisting of ${\left(\frac{\partial }{\partial t}\right)}^{n-3}$ multiplied by the product of modified Bessel functions summed over their degrees which conform to the fusion rules and the crossing symmetry. This is found to be supplemented by an increasing number of other terms with $n$ which represent residual interactions of loops. We reveal the nature of these interactions by explicitly determining them as the convolution of modified Bessel functions and their derivatives for the case $n=4$ and the case $n=5$. We derive a set of recursion relations which relate the terms in the $n$-resolvents to those in the $(n-1)$-resolvents.

30 pages, Latex, figures: figures have been introduced to represent our results on the resolvents. A better formula for the resolvents has been put and the section on residual interactions has been expanded to a large extent

References in corpus (6)

• Rational Theories of 2D Gravity from the Two-Matrix Model
• Combinatorial Solution of the Two-Matrix Model
• 2D Quantum Gravity in the Proper-Time Gauge
• Two-matrix model and c=1 string theory
• Macroscopoic Three-Loop Amplitudes and the Fusion Rules from the Two-Matrix Model
• Continuum Annulus Amplitude from the Two-Matrix Model

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• Fractional-Superstring Amplitudes, Multi-Cut Matrix Models and Non-Critical M Theory
• Boundary operators and touching of loops in 2d gravity
• Loop Equations for + and - Loops in c = 1/2 Non-Critical String Theory