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 ℓ i and the renormalized cosmological constant t. The amplitude is found to contain a term consisting of ∂ ∂t 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.

Fractional-superstring amplitudes, multi-cut matrix models and non-critical M theory

“…Therefore, the fractional superstring theory includes ⌊ k 2 ⌋ different perturbatively isolated sectors in its vacuum or in the weak-coupling Landscape of this string theory. 47 • From our results about the macroscopic loop amplitudes (5.1), one can conclude that every perturbatively isolated sector (or perturbative string theory) included in the k-cut fractional-superstring (p, q) critical points is also realized as the perturbatively isolated sector in k ′ (∈ k Z)-cut fractional-superstring (p, q) critical points. In particular, the fractional (p, q) critical points of the infinite-cut two-matrix models includes all the perturbatively isolated sectors of (p, q) minimal fractional superstring theory.…”

“…This extra discrete spacetime becomes the continuum third dimension in the k → ∞ limit. In this sense, the appearance of the third dimension has no contradiction with the 47 As an actual check of this isolation, one can see the calculation of annulus amplitudes in type 0 superstring theory [61], which shows that the two-point functions of local operators vanish in the one-cut phase of two-cut critical points. Note that annulus amplitudes between macroscopic loop amplitudes can have non-zero contributions which however depend on the regularization or definition of the operators.…”

“…The actual quantities investigated here are macroscopic loop amplitudes [36][37][38][39][40][41][42][43][44][45][46][47], 3 Q(x) ∼ 1 N tr 1 x − X = dXdY e −N tr w(X,Y ) 1 N tr 1 x − X , (1.4) and they are known to play important roles in extracting various information about the matrix models: eigenvalue distributions [36], generating functions of local (on-shell) operators on worldsheets, and also effective potentials of a single (pair of) matrix-model eigenvalues (x, y) [48,49]. Furthermore, since these amplitudes directly correspond to boundary states of FZZT branes [6], it is through the study of these amplitudes that we can perform a direct comparison with the amplitudes in the Liouville theory [50].…”

“…The actual quantities investigated here are macroscopic loop amplitudes [36][37][38][39][40][41][42][43][44][45][46][47], 3…”

Boundary operators and touching of loops in 2D gravity

“…We generalize this idea to the loop correlators [4,12,13,14] in the cases of the general unitary minimal models, and derive the fusion rules for all of the scaling operators. First we derive the explicit form of the expansion of loops in terms of the scaling operators, and then deduce the three-point correlators from the loop correlators which were calculated in [13,14] from the two-matrix model. We show that the three-point correlators of all of the scaling operators satisfy certain simple fusion rules and these fusion rules are summarized in a compact form as the fusion rules for three-loop correlators [13].…”