##
**Theta functions on noncommutative tori.**
*(English)*
Zbl 1032.53082

The author considers \(d\)-dimensional (where \(d\) is even) noncommutative tori, i.e. ordinary tori \(T^d= \mathbb R^d / \mathbb Z^d\) together with the associative algebra \(T^d_\theta\) of smooth functions on \(T^d\) equipped with the Weyl-Moyal star product (where \(\hbar = 1\)) corresponding to the anti-symmetric, non-degenerate matrix \(\theta^{\alpha\beta}\). In the article under review he explicitly constructs projective modules over the algebra \(T^d_\theta\) which can be considered as the natural generalizations of vector bundles over the manifold \(T^d\) since the latter are projective modules over the algebra of smooth functions with the commutative, pointwise product. (This generalization is also suggested by the observation that in case \(\theta^{\alpha\beta}\) has integer entries the Weyl-Moyal star product actually coincides with the undeformed product and hence in this case a projective module over \(T^d_\theta\) is a vector bundle over \(T^d\).) In the context of formal deformation quantization [cf. H. Bursztyn, and S. Waldmann, Lett. Math. Phys. 53, 349-365 (2000; Zbl 0982.53073)] such constructions which can be carried out for arbitrary Poisson manifolds are referred to as deformed vector bundles. Furthermore the author shows how to equip the \(T^d_\theta\)-module \({\mathcal S}\) of Schwartz functions on \(\mathbb R^g\) with a complex structure corresponding to a Lagrangian submanifold \(\Omega\subseteq \mathbb R^d \oplus i \mathbb R^d\) (this module is denoted by \({\mathcal S}_\Omega\)) and proves that for positive \(\Omega\) (i.e. the imaginary part of the symplectic pairing with the symplectic structure corresponding to \(\theta\) of any non-zero vector in \(\Omega\) with its complex conjugate vector is positive) there is a unique one-dimensional subspace of holomorphic vectors in \({\mathcal S}_\Omega\) and the elements of this space are called theta vectors.

Specializing to the case of the canonical symplectic form, the author points out the relation between the theta vectors as introduced above and ordinary theta functions, which are holomorphic sections of line bundles (there are different such line bundles, since \(\theta\) resp. the corresponding symplectic form just determines the curvature and hence determines the line bundle up to torsion) over tori and can be represented as Schwartz functions on \(\mathbb R^g\) by \(\tau(x^1, \ldots, x^g)= \exp(i \pi x^\alpha\Omega_{\alpha\beta}x^\beta)\). This observation suggests defining generalized theta functions as holomorphic elements of projective modules over noncommutative tori.

In some concluding remarks the author points out that one can also consider \((T^d_\theta,T^d_{\widehat\theta})\)-bimodules that establish Morita equivalence between \(T^d_\theta\) and \(T^d_{\widehat\theta}\). Again this can also be formulated in the framework of formal deformation quantization [cf. H. Bursztyn, and S. Waldmann (loc. cit.) and B. Jurčo, P. Schupp, and J. Wess, Lett. Math. Phys. 61, 171-186 (2002; Zbl 1036.53070)].

Specializing to the case of the canonical symplectic form, the author points out the relation between the theta vectors as introduced above and ordinary theta functions, which are holomorphic sections of line bundles (there are different such line bundles, since \(\theta\) resp. the corresponding symplectic form just determines the curvature and hence determines the line bundle up to torsion) over tori and can be represented as Schwartz functions on \(\mathbb R^g\) by \(\tau(x^1, \ldots, x^g)= \exp(i \pi x^\alpha\Omega_{\alpha\beta}x^\beta)\). This observation suggests defining generalized theta functions as holomorphic elements of projective modules over noncommutative tori.

In some concluding remarks the author points out that one can also consider \((T^d_\theta,T^d_{\widehat\theta})\)-bimodules that establish Morita equivalence between \(T^d_\theta\) and \(T^d_{\widehat\theta}\). Again this can also be formulated in the framework of formal deformation quantization [cf. H. Bursztyn, and S. Waldmann (loc. cit.) and B. Jurčo, P. Schupp, and J. Wess, Lett. Math. Phys. 61, 171-186 (2002; Zbl 1036.53070)].

### MSC:

53D55 | Deformation quantization, star products |

58B34 | Noncommutative geometry (à la Connes) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |