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 ${\mathbb{C}}^{\ast }$. 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 a forgetful functor to the category of vector bundles on ${\mathbb{C}}^{\ast }$ 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) ${\mathbb{Z}}^2$. We also obtain a full subcategory of the category of holomorphic bundles of the noncommutative torus, which is equivalent to the category of representations of $\mathbb{Z}$. 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.

22 pages with major revisions. Some preliminary material removed. Section 4 on the étale fundamental group of noncommutative tori is entirely new. References changed accordingly, to appear in JNCG

References in corpus (1)

• Classification of holomorphic vector bundles on noncommutative two-tori

Cited by in corpus (5)

• Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier
• Noncommutative Coverings of Quantum Tori
• Modules over the Noncommutative Torus and Elliptic Curves
• Local analytic classification of $q$-difference equations with $|q|=1$
• On a Teichmueller functor between the categories of complex tori and the Effros-Shen algebras