Matrix Models as Non-commutative Field Theories on R^3

TL;DR

This paper explores the connection between matrix models, non-commutative field theories, and quantum gravity, providing new insights into their relationships and deriving effective theories with potential for further analysis.

Contribution

It establishes explicit links between 2d group field theories, matrix models, and non-commutative geometry, including derivations of effective theories from 3d quantum gravity models.

Findings

Relation between matrix models and non-commutative field theories.

Derivation of effective 2d group field theories from 3d quantum gravity.

Expression of these theories as multi-matrix models with non-trivial couplings.

Abstract

In the context of spin foam models for quantum gravity, group field theories are a useful tool allowing on the one hand a non-perturbative formulation of the partition function and on the other hand admitting an interpretation as generalized matrix models. Focusing on 2d group field theories, we review their explicit relation to matrix models and show their link to a class of non-commutative field theories invariant under a quantum deformed 3d Poincare symmetry. This provides a simple relation between matrix models and non-commutative geometry. Moreover, we review the derivation of effective 2d group field theories with non-trivial propagators from Boulatov's group field theory for 3d quantum gravity. Besides the fact that this gives a simple and direct derivation of non-commutative field theories for the matter dynamics coupled to (3d) quantum gravity, these effective field theories can be expressed as multi-matrix models with non-trivial coupling between matrices of different sizes. It should be interesting to analyze this new class of theories, both from the point of view of matrix models as integrable systems and for the study of non-commutative field theories.