From Matrix Models to Gaussian Molecules and the Einstein-Hilbert Action

TL;DR

This paper introduces a matrix model as a non-perturbative framework for discretized string theory, deriving the Einstein-Hilbert action and related gravitational constants from graph invariants.

Contribution

It establishes a novel connection between matrix models, Gaussian molecules, and the Einstein-Hilbert action without requiring on-shell metric conditions.

Findings

Free energy expressed via invariant graph polynomials

Relation of vacuum diagrams to Gaussian molecules

Derivation of Einstein-Hilbert action from matrix model

Abstract

A matrix model on a D-dimensional Euclidean space is introduced as a generalization of random matrix models and as a non-perturbative definition of discretized closed string theory. The free energy of the matrix model is formally derived to all orders in string perturbation expansion and shown to be given in terms of invariant graph polynomials, whose coefficients enumerate ribbon graphs and are a refinement of the generalized Catalan numbers. The vacuum diagrams contributing to the free energy are found to be related to Gaussian molecules, known from the study of polymer structures. Coupling the matrix field to a curved background with Riemannian metric yields a non-perturbative definition of discretized string theory on this background. No on-shell condition for the metric is required to arrive at the free energy. Rather, it is shown that the free energy of the matrix model is the Einstein-Hilbert action with cosmological constant term. The gravitational and the cosmological constants are both formally determined to all orders in the string perturbation expansion. In fact, they are explicitly given by the expectation value of a particular graph invariant. Introducing a vector field, minimally coupled to a background gauge field, provides a discretized open-closed string theory, leading to the Yang-Mills action as well as intrinsic and extrinsic curvature terms.