Analogues of the Exponential Map Associated with Complex Structures on Noncommutative Two-tori

We define and study analogues of exponentials for functions on noncommutative two-tori that depend on a choice of a complex structure. The major difference with the commutative case is that our noncommutative exponentials can be defined only for sufficiently small functions. We show that this phenomenon is related to the existence of certain discriminant hypersurfaces in an irrational rotation algebra. As an application of our methods we give a very explicit characterization of connected components in the group of invertible elements of this algebra. Introduction In this paper we study some natural constructions for functions on noncommutative two-tori equipped with a complex structure. Recall that for every number θ ∈ R \ Q the algebra Aθ of smooth functions on the noncommutative torus Tθ (also known as irrational rotation algebra) consists of expressions ∑ (m,n)∈Z2 am,nU m 1 U n 2 where coefficients am,n ∈ C rapidly decrease at infinity and the multiplication is performed using the rule U1U2 = exp(2πiθ)U2U1. Given an element τ ∈ C \ R (following [5] we will always assume that Im(τ) < 0) we define a derivation δτ : Aθ → Aθ : ∑ am,nU m 1 U n 2 7→ 2πi · ∑ m,n (mτ + n)am,nU m 1 U n 2 We consider δτ as a complex structure on Tθ and denote the obtained complex noncommutative torus by Tθ,τ . The main object of our study is the equation δτ (x) = xa (0.1) for x ∈ Aθ, where a ∈ Aθ is given. In the commutative case this equation is clearly related with the exponential map on smooth functions. It turns out that there is a local analogue of this map for Aθ. However, in noncommutative case there seem to be serious reasons why the exponential map does not extend to all functions. For example, we show that (0.1) has a nonzero solution iff tr(a) ∈ 2πi(Z + Zτ), where tr( ∑ am,nU m 1 U n 2 ) = a0,0 (see Corollary 2.7), but these solutions are not necessarily invertible in Aθ (as in commutative case). The study of equation (0.1) turns out to be closely related to the study of holomorphic structures on the trivial holomorphic bundle over Tθ,τ . By a holomorphic bundle on Tθ,τ we mean a right projective module E over Aθ equipped with a δτ -connection, i.e., a linear Supported in part by NSF grant.