Thoughts on Membranes, Matrices and Non-Commutativity

TL;DR

This paper reviews the transition from supermembrane theory to matrix models through non-commutative deformations, exploring potential generalizations involving three-index non-commutativity and their algebraic structures.

Contribution

It introduces a framework for understanding membrane to matrix theory transition via non-commutative deformation and suggests possible extensions with three-index non-commutativity.

Findings

Non-commutative deformation facilitates membrane to matrix transition.

Indications of a three-index theta generalization of non-commutativity.

Discussion of algebraic structures related to higher-order non-commutativity.

Abstract

We review the passage from the supermembrane to matrix theory via a consistent truncation following a non-commutative deformation in light-cone gauge. Some indications are given that there should exist a generalisation of non-commutativity involving a three-index theta on the membrane, and we discuss some possible ways of investigating the corresponding algebraic structure.