# Theta functions on noncommutative T4

@article{Kim2003ThetaFO, title={Theta functions on noncommutative T4}, author={Hoil Kim and Chang-Yeong Lee}, journal={Journal of Mathematical Physics}, year={2003}, volume={45}, pages={461-474} }

We construct the so-called theta vectors on noncommutative T4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and then find the functions satisfying the holomorphic conditions, the theta vectors. The holomorphic structure in the noncommutative T4 case is given by a 2×2 complex matrix, and the consistency requires its off-diagonal elements to be the same. We also construct…

## 6 Citations

Quantum thetas on noncommutative {\bb T}^d with general embeddings

- Mathematics, Physics
- 2007

In this paper, we construct quantum theta functions over noncommutative T a with general embeddings. Manin has constructed quantum theta functions from the lattice embedding into vector space x…

Noncommutative Riemann Conditions

- Physics
- 2004

In this paper we study the holomorphic bundles over a noncommutative complex torus. We define a noncommutative abelian variety as a kind of deformation of abelian variety and we show that for a…

Quantum thetas on noncommutative {\bb T}^4 from embeddings into lattice

- Mathematics, Physics
- 2006

In this paper, we investigate the theta vector and quantum theta function over noncommutative from the embedding of . Manin has constructed the quantum theta functions from the lattice embedding into…

Categories of holomorphic line bundles on higher dimensional noncommutative complex tori

- Physics, Mathematics
- 2007

We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and…

Theta vectors and quantum theta functions

- Mathematics, Physics
- 2005

In this paper, we clarify the relation between Manin's quantum theta function and Schwarz's theta vector. We do this in comparison with the relation between the kq representation, which is equivalent…

Symmetry of quantum torus with crossed product algebra

- Physics, Mathematics
- 2006

In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and…

## References

SHOWING 1-10 OF 27 REFERENCES

Theta Functions on Noncommutative Tori

- Physics, Mathematics
- 2001

Ordinary theta functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta functions as holomorphic elements of projective modules…

Matrix theory compactification on noncommutative T4/Z2

- Physics, Mathematics
- 2001

In this article, we construct gauge bundles on a noncommutative toroidal orbifold Tθ4/Z2. First, we explicitly construct a bundle with constant curvature connections on a noncommutative Tθ4 following…

Differential and Complex Geometry of Two-Dimensional Noncommutative Tori

- Physics, Mathematics
- 2002

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules. We concentrate our attention on properties of holomorphic vectors in these…

Solitons on Noncommutative Orbifold T2/ZN

- Mathematics
- 2002

Following the construction of the projection operators on T2 presented by Gopakumar, Headrick and Spradlin, we construct a set of projection operators on the integral noncommutative orbifold…

Noncommutative Geometry and Matrix Theory: Compactification on Tori

- Physics
- 1997

We study toroidal compactification of Matrix theory, using ideas and results of noncommutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification…

Morita equivalence of multidimensional noncommutative tori

- Mathematics, Physics
- 1998

One can describe an n-dimensional noncommutative torus by means of an antisymmetric n×n matrix θ. We construct an action of the group SO(n,n|Z) on the space of n×n antisymmetric matrices and show…

Towards a noncommutative geometric approach to matrix compactification

- Physics
- 1998

In this paper we study generic M(atrix) theory compactifications that are specified by a set of quotient conditions. A procedure is proposed, which both associates an algebra to each compactification…

Gauge bundles and Born-Infeld on the noncommutative torus

- Physics
- 1998

In this paper, we describe non-abelian gauge bundles with magnetic and electric fluxes on higher-dimensional non-cummutative tori. We give an explicit construction of a large class of bundles with…

Noncommutative K3 surfaces

- Physics
- 2001

Abstract We consider deformations of a toroidal orbifold T 4 / Z 2 and an orbifold of quartic in CP 3 . In the T 4 / Z 2 case, we construct a family of noncommutative K3 surfaces obtained via both…

On noncommutative Calabi–Yau hypersurfaces

- Physics
- 2001

Using the algebraic geometry method of Berenstein et al. [hep-th/0005087], we reconsider the derivation of the noncommutative quintic algebra Anc(5) and derive new representations by choosing…