Noncommutative Geometry and Matrix Theory: Compactification on Tori

TL;DR

This paper explores the use of noncommutative geometry in the toroidal compactification of Matrix theory, connecting it to supergravity backgrounds with constant three-form fields and providing mathematical background for physicists.

Contribution

It extends Matrix theory compactification to noncommutative tori, classifies these backgrounds, and relates them to supergravity solutions, offering a bridge between mathematics and physics.

Findings

Generalization of compactification to noncommutative tori

Classification of noncommutative torus backgrounds

Connection to supergravity with constant three-form fields

Abstract

We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory.