Categories of Holomorphic Vector Bundles on Noncommutative Two-Tori

Abstract

Abstract. In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.

Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

“…We know from [22], [21] that Hol(T θ,τ ) is equivalent to C θ (E). In this section we will show that the construction of this equivalence can be adjusted to be compatible with the action of a finite group G.…”

“…1 on such a vector bundle. As in [22], [21], let us consider a complex structure on T θ associated with a complex number τ ∈ C \ R. It is given by a derivation…”

“…Recall that in [22], [21] we studied the category Hol(T θ,τ ) of holomorphic bundles on T θ . By definition, these are pairs (P, ∇) consisting of a finitely generated projective right A θ -module P and an operator ∇ : P → P satisfying the Leibnitz identity…”

“…Recall that the combined results of [22] and [21] imply that the category Hol(T θ,τ ) is abelian and one has an equivalence of bounded derived categories…”

“…Note that in [22] we made one possible choice of (c v (z)), however, it is not the only choice. In fact, one can easily see that the equivalences corresponding to different choices of (c v (z)) differ by tensoring with a holomorphic line bundle on E. One of possible solutions of (2.1) is c 0 mτ +n (z) = −2mz − m(mτ + n).…”