Noncommutative tori and the Riemann–Hilbert correspondence

Abstract

We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on $C^{\ast }$. We functorially relate this category to the category of holomorphic vector bundles on noncommutative tori as introduced by Polishchuk and Schwarz and study the induced map between the corresponding K-theories. In addition, there is a forgetful functor to the category of noncommutative elliptic curves of Soibelman and Vologodsky, as well as the forgetful functor to the category of vector bundles on $C^{\ast }$ with regular singular connections.

The category that we consider has the nice property of being a Tannakian category, hence it is equivalent to the category of representations of an affine group scheme. Via an equivariant version of the Riemann–Hilbert correspondence we determine this group scheme to be (the algebraic hull of) $Z^2$ . We also obtain a full subcategory of the holomorphic vector bundles on the noncommutative torus which is equivalent to the category of representations of $Z$. This group is the proposed topological fundamental group of the noncommutative torus (understood as a degenerate elliptic curve) and we study Nori’s notion of étale fundamental group in this context.