Compactification of M(atrix) theory on noncommutative toroidal orbifolds

TL;DR

This paper explores the structure of noncommutative toroidal orbifolds in M(atrix) theory, focusing on projective modules and Morita equivalence, extending the understanding of noncommutative geometry in string theory compactifications.

Contribution

It introduces the algebra B_{θ} as a crossed product of a noncommutative torus with Z_{2} and studies projective modules and Morita equivalence for these orbifolds.

Findings

Analysis of projective modules over B_{θ}

Morita equivalence in the two-dimensional case

Extension of noncommutative torus theory to orbifolds

Abstract

It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal Z_{2} orbifolds leads to the algebra B_{\theta} that can be defined as a crossed product of noncommutative torus and the group Z_{2}. Our paper is devoted to the study of projective modules over B_{\theta} (Z_{2}-equivariant projective modules over a noncommutative torus). We analyze the Morita equivalence (duality) for B_{\theta} algebras working out the two-dimensional case in detail.