On Non Commutative G2 structure

TL;DR

This paper explores the non-commutative structures of seven-dimensional $G_2$ manifolds using algebraic orbifold methods, identifying solutions, matrix representations, and potential applications to M-theory models.

Contribution

It introduces a novel algebraic approach to non-commutative $G_2$ structures, solving constraint equations and providing matrix representations with implications for M-theory.

Findings

Eight possible solutions for non-commutative $G_2$ algebras identified

Matrix representations constructed using combinatorial methods

Application discussed in the context of M-theory matrix models

Abstract

Using an algebraic orbifold method, we present non-commutative aspects of $G_2$ structure of seven dimensional real manifolds. We first develop and solve the non commutativity parameter constraint equations defining $G_2$ manifold algebras. We show that there are eight possible solutions for this extended structure, one of which corresponds to the commutative case. Then we obtain a matrix representation solving such algebras using combinatorial arguments. An application to matrix model of M-theory is discussed.