Differential and complex geometry of two-dimensional noncommutative tori (Q1868228)

In [\textit{A. Schwarz}, Lett. Math. Phys. 58, 81-90 (2001; Zbl 1032.53082)], complex geometry of noncommutative tori and of projective modules over them in connection with noncommutative generalization of theta-functions are studied. In this paper, general results are illustrated using examples of two-dimensional tori. Noncommutative theta functions appear in the coordinate description of the tensor product of right and left projective modules over a noncommutative torus. If the torus \(T_\theta\) is two-dimensional, a right (resp. left) projective module over \(T_\theta\) is the Schwartz space \(E_{n,m}= {\mathcal S}(\mathbb{R}\times \mathbb{Z}_m)\), \((n, m)=1\), (denoted \(E_{n,m}'\) when a left module) with the actions \[ U_1 f(x,\mu)= f \left(x-{n\pm m\theta \over m},\mu-1 \right),\quad U_2f(x,\mu)= e^{2\pi i(x-\mu n/m)} f(x,\mu). \] After explaining properties of \(E_{n,m}\), the tensor product \(h\in E_{n,m} \otimes_{T_\theta}E_{k,l}'\) of \(f\in E_{n,m}\) and \(g \in E_{k,l}'\) is calculated (1). (Precise calculation can be seen in the arXiv version of this paper (math.QA/0203160)). Then it is shown that \(h=\Theta\cdot\xi\), where \(\Theta\) is a classical theta function for suitable \(f\) and \(g\). An explicit form of \(\xi\) is also given. In Section 3, introducing complex structure on \(E_{n,m}\), fixing a \(\overline\partial\)-connection \[ \overline \nabla= \lambda_1 \nabla_1+ \lambda_2\nabla_2,\;\tau={\lambda_1 \over\lambda_2} \notin\mathbb{R}, \quad \nabla_1= 2\pi {m\over n+m \theta}x, \quad \nabla_2=2\pi {d\over dc}, \] the holomorphic vector \(\Theta\in E_{n,m}\) is introduced to be a vector that satisfies \(\overline \nabla \Theta=0\). The space of holomorphic vectors \({\mathcal H}_{n,m}\) is an \(m\)-dimensional space if holomorphic vectors exist (Existence condition is given). Then the tensor product of the basis of the modules of holomorphic vectors is calculated, and it is shown that theta functions appear as coefficients. To be selfcontained, the paper gives necessary definitions. Auth authors say that Manin remarked that the results of this paper could be useful in number theory [\textit{Yu. Manin}, Real multiplication and noncommutative geometry, arXiv:math. AG/0202109]. The authors also say that to analyze the connection of the results of this paper and Manin's version of noncommutative theta-functions [\textit{Yu. Manin}, Lett. Math. Phys. 56, No. 3, 295-320 (2001; Zbl 1081.14501)] is an interesting problem.