Differential and complex geometry of two-dimensional noncommutative tori
type: Item
instance of: scholarly article
title: Differential and complex geometry of two-dimensional noncommutative tori
zbMATH Open document ID: 1019.58004
DOI: 10.1023/A:1021272314232
author: Momar Dieng
published in: Letters in Mathematical Physics
publication date: 2003-04-27T00:00:00Z
review text: 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 selfcontain…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 1901314
zbMATH Keywords: noncommutative theta functions
zbMATH Keywords: noncommutative torus
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1901314