Categories of holomorphic line bundles on higher dimensional noncommutative complex tori

Abstract

We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex∕holomorphic structures on them introduced by Schwarz [“Theta functions on noncommutative tori,” Lett. Math. Phys. 58, 81–90 (2001)]. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommutative tori. Also, we present the explicit composition formula of morphisms, which, in fact, depends on the noncommutativity.