Morita equivalence and duality

TL;DR

The paper establishes that compactifications of M(atrix) theory on noncommutative tori are physically equivalent whenever the tori are Morita equivalent, formalized by $\widehat{\theta}=g\theta$ with $g\in SO(n,n|\mathbb{Z})$. Using equivalence bimodules and constant-curvature connections, it shows how modules, their Chern characters, and the associated action functionals transform under Morita equivalence, preserving BPS structure and yielding a higher-dimensional generalization of known dualities. A detailed, explicit construction of the correspondence between modules and their curvatures is provided, along with monodromy interpretations in a fermionic Fock-space setting. Examples in low dimensions illustrate the mechanism, including the $\theta \leftrightarrow \theta^{-1}$ duality, and the results extend to BFSS theory via Wick rotation, indicating a robust, geometry-driven duality framework for matrix models on noncommutative tori.

Abstract

It was shown by Connes, Douglas, Schwarz[1] that one can compactify M(atrix) theory on noncommutative torus. We prove that compactifications on Morita equivalent tori are physically equivalent. This statement can be considered as a generalization of non-classical duality conjectured in [1] for two-dimensional tori.