Random volumes from matrices

TL;DR

This paper introduces matrix-based models for generating three-dimensional random volumes, which can be restricted to manifolds with tetrahedral decompositions through a color structure and large N limit, revealing a strong-weak duality.

Contribution

The paper presents a new class of matrix models for 3D random volumes that can produce manifold configurations and explores their duality properties.

Findings

Models generate 3D volumes with tetrahedral decompositions.

Introduction of color structure reduces diagrams to manifolds.

Discovery of a strong-weak duality in matrix ring models.

Abstract

We propose a class of models which generate three-dimensional random volumes, where each configuration consists of triangles glued together along multiple hinges. The models have matrices as the dynamical variables and are characterized by semisimple associative algebras A. Although most of the diagrams represent configurations which are not manifolds, we show that the set of possible diagrams can be drastically reduced such that only (and all of the) three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate large N limit. We examine the analytic properties when A is a matrix ring or a group ring, and show that the models with matrix ring have a novel strong-weak duality which interchanges the roles of triangles and hinges. We also give a brief comment on the relationship of our models with the colored tensor models.