##
**Homological mirror symmetry for the 4-torus.**
*(English)*
Zbl 1195.14056

The homological mirror symmetry conjecture, due to M. Kontsevich [Proceedings of the international congress of mathematicians, ICM ’94, August 3–11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120–139 (1995; Zbl 0846.53021)], roughly states that given a pair of mirror Calabi-Yau manifolds \(X\) and \(Y\) there is an equivalence between the bounded derived category \({\text{D}}^{\text{b}}(X)\) of coherent sheaves on \(X\) and the Fukaya category \(\mathcal{F}(Y)\) of \(Y\), this latter category being constructed from the symplectic geometry of \(Y\). Despite being the focus of a lot of attention, the conjecture has been fully proved in only a handful of cases, prominent ones being the elliptic curve [A. Polishchuk and E. Zaslow, Adv. Theor. Math. Phys. 2, No. 2, 443–470 (1998; Zbl 0947.14017)] and the quartic K3 surface [P. Seidel, “Homological mirror symmetry for the quartic surface”, arxiv:math/0310414].

In the paper under review the authors prove the conjecture for the 4-torus \(T^4=\mathbb{R}^4 /\mathbb{Z}^4\). They then use the established equivalence to explore some consequences for the symplectic topology of the 4-torus by invoking certain results on sheaves on abelian varieties and transporting these results to the symplectic side. To be slightly more precise, the authors establish numerical restrictions on the intersections inside \(T^4\) of Lagrangian genus 2 surfaces of Maslov class zero with linear Lagrangian 2-tori.

The strategy of the proof is very roughly the following: One relates the Fukaya category \(\mathcal{F}(Y_1\times Y_2)\) of a product to the category of functors \({\text{Hom}}(\mathcal{F}(Y_1), \mathcal{F}(Y_2))\). If mirror symmetry is known for the factors, then the latter category is \({\text{Hom}}({\text{D}}^{\text{b}}_\infty(X_1),\text{D}^{\text{b}}_\infty(X_2))\) (where “\(\infty\)” denotes the dg-enhancement of the derived category) and by a result of Toën this category is equivalent to \({\text D}^{\text b}_\infty(X_1\times X_2)\). Applying this to \(Y_1=Y_2=T^2\) and using homological mirror symmetry for the \(2\)-torus (whose mirror is an elliptic curve) the argument then gives the result for \(T^4\). The main technical ingredient of the proof is the theory of pseudoholomorphic quilts, which is currently under development by Mau, Wehrheim and Woodward.

In the paper under review the authors prove the conjecture for the 4-torus \(T^4=\mathbb{R}^4 /\mathbb{Z}^4\). They then use the established equivalence to explore some consequences for the symplectic topology of the 4-torus by invoking certain results on sheaves on abelian varieties and transporting these results to the symplectic side. To be slightly more precise, the authors establish numerical restrictions on the intersections inside \(T^4\) of Lagrangian genus 2 surfaces of Maslov class zero with linear Lagrangian 2-tori.

The strategy of the proof is very roughly the following: One relates the Fukaya category \(\mathcal{F}(Y_1\times Y_2)\) of a product to the category of functors \({\text{Hom}}(\mathcal{F}(Y_1), \mathcal{F}(Y_2))\). If mirror symmetry is known for the factors, then the latter category is \({\text{Hom}}({\text{D}}^{\text{b}}_\infty(X_1),\text{D}^{\text{b}}_\infty(X_2))\) (where “\(\infty\)” denotes the dg-enhancement of the derived category) and by a result of Toën this category is equivalent to \({\text D}^{\text b}_\infty(X_1\times X_2)\). Applying this to \(Y_1=Y_2=T^2\) and using homological mirror symmetry for the \(2\)-torus (whose mirror is an elliptic curve) the argument then gives the result for \(T^4\). The main technical ingredient of the proof is the theory of pseudoholomorphic quilts, which is currently under development by Mau, Wehrheim and Woodward.

Reviewer: Pawel Sosna (Bonn)

### MSC:

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

53D40 | Symplectic aspects of Floer homology and cohomology |

### Keywords:

homological mirror symmetry; derived category; Calabi-Yau manifolds; Fleur (co)homology; pseudoholomorphic quilts
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\textit{M. Abouzaid} and \textit{I. Smith}, Duke Math. J. 152, No. 3, 373--440 (2010; Zbl 1195.14056)

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