A matrix model for strings beyond the c=1 barrier: the spin-s Heisenberg model on random surfaces

TL;DR

This paper develops a matrix model for the spin-s Heisenberg model coupled to 2D quantum gravity, incorporating random geometries and spin configurations, and introduces a technique to simplify the matrix integration process.

Contribution

It introduces a novel matrix model that captures the spin-s Heisenberg model on random surfaces, extending beyond the c=1 barrier in string theory.

Findings

Formulation of a matrix model reproducing the annealed average over random geometries.

Development of a technique to reduce matrix integration to eigenvalue integration.

Integration of spin configurations with 2D quantum gravity in a unified framework.

Abstract

We consider a spin-s Heisenberg model coupled to two-dimensional quantum gravity. We quantize the model using the Feynman path integral, summing over all possible two-dimensional geometries and spin configurations. We regularize this path integral by starting with the R-matrices defining the spin-s Heisenberg model on a regular 2d Manhattan lattice. 2d quantum gravity is included by defining the R-matrices on random Manhattan lattices and summing over these, in the same way as one sums over 2d geometries using random triangulations in non-critical string theory. We formulate a random matrix model where the partition function reproduces the annealed average of the spin-s Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.