Matrix theory compactifications on twisted tori

TL;DR

This paper explores Matrix theory compactifications on twisted and non-commutative tori, analyzing their mathematical structures, solutions, and connections to supergravity flux compactifications, revealing new consistent backgrounds.

Contribution

It introduces new solutions for Matrix theory compactifications on twisted tori, including non-commutative and non-associative structures, and discusses their relation to supergravity flux backgrounds.

Findings

Found consistent commutative and non-commutative solutions for 3- and 6-dimensional twisted tori.

Identified non-constant non-commutativity and non-associative phase space structures.

Explored relations among solutions via Seiberg-Witten maps and T-duality.

Abstract

We study compactifications of Matrix theory on twisted tori and non-commutative versions of them. As a first step, we review the construction of multidimensional twisted tori realized as nilmanifolds based on certain nilpotent Lie algebras. Subsequently, matrix compactifications on tori are revisited and the previously known results are supplemented with a background of a non-commutative torus with non-constant non-commutativity and an underlying non-associative structure on its phase space. Next we turn our attention to 3- and 6-dimensional twisted tori and we describe consistent backgrounds of Matrix theory on them by stating and solving the conditions which describe the corresponding compactification. Both commutative and non-commutative solutions are found in all cases. Finally, we comment on the correspondence among the obtained solutions and flux compactifications of 11-dimensional supergravity, as well as on relations among themselves, such as Seiberg-Witten maps and T-duality.