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**Categories of holomorphic vector bundles on noncommutative two-tori.**
*(English)*
Zbl 1033.58009

This paper is a careful presentation of a nice theory of noncommutative two-tori and holomorphic vector bundles on them.

As the paper develops a whole theory and there are no statements marked as theorems, it is the best first to quote the authors’ summary.

“In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.”

The framework of noncommutative two-tori gives a handle on certain modes of degenerations of ordinary elliptic curves. The major associated techniques include lattices in \(\mathbb C\), the group \(\mathrm{SL}_2(\mathbb Z)\), elliptic curves, theta functions, Fourier series in one and two variables, Fourier transforms, Fourier-Mukai transforms, the Gauss kernel \(e^{-x^2/2}\), Hermite polynomials, the operators of creation and annihilation, the Schwartz spaces \(\mathcal S(\mathbb R)\) and \(\mathcal S(\mathbb R\times\mathbb Z/m\mathbb Z)\), and analogs of the \(\overline\partial\)-operator on \(\mathbb C\).

The paper is very informative and pleasant to read.

As the paper develops a whole theory and there are no statements marked as theorems, it is the best first to quote the authors’ summary.

“In this paper we study the category of standard holomorphic vector bundles on a noncommutative two-torus. We construct a functor from the derived category of such bundles to the derived category of coherent sheaves on an elliptic curve and prove that it induces an equivalence with the subcategory of stable objects. By the homological mirror symmetry for elliptic curves this implies an equivalence between the derived category of holomorphic bundles on a noncommutative two-torus and the Fukaya category of the corresponding symplectic (commutative) torus.”

The framework of noncommutative two-tori gives a handle on certain modes of degenerations of ordinary elliptic curves. The major associated techniques include lattices in \(\mathbb C\), the group \(\mathrm{SL}_2(\mathbb Z)\), elliptic curves, theta functions, Fourier series in one and two variables, Fourier transforms, Fourier-Mukai transforms, the Gauss kernel \(e^{-x^2/2}\), Hermite polynomials, the operators of creation and annihilation, the Schwartz spaces \(\mathcal S(\mathbb R)\) and \(\mathcal S(\mathbb R\times\mathbb Z/m\mathbb Z)\), and analogs of the \(\overline\partial\)-operator on \(\mathbb C\).

The paper is very informative and pleasant to read.

Reviewer: Imre Patyi (La Jolla)

### MSC:

58B34 | Noncommutative geometry (à la Connes) |

53D35 | Global theory of symplectic and contact manifolds |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

32L05 | Holomorphic bundles and generalizations |

18E30 | Derived categories, triangulated categories (MSC2010) |