Homological mirror symmetry and torus fibrations.

*(English)*Zbl 1072.14046
Fukaya, K. (ed.) et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific (ISBN 981-02-4714-1/hbk). 203-263 (2001).

The paper under review must be regarded as a novel, strategically pioneering and utmost important attempt to create a profound mathematical theory for the phenomenon of mirror symmetry in conformal field theory. In the 1990s, various approaches to this challenging mathematical problem have been developed, and today there are several ways to interpret physical mirror symmetry mathematically, among them being M. Kontsevich’s conceptual framework of homological mirror symmetry [in: Proc. ICM Zürich 1994, Vol. I, 120–139 (1995; Zbl 0846.53021)] and the program initiated by A. Strominger, S.-T. Yau and E. Zaslow [Nucl. Phys., B 479, No. 1–2, 243–259 (1996; Zbl 0896.14024)]. As to the latter, the so-called SYZ approach involves special Lagrangian submanifolds of Calabi-Yau manifolds, and the so-called SYZ conjecture predicts that Calabi-Yau manifolds “close to the large complex structure limit” admit special Lagrangian torus fibrations. According to the SYZ conjecture, mirror manifolds can be constructed, through a certain dualizing procedure, just from those Lagrangian torus fibrations.

In the present paper, the authors throw a bridge between the SYZ approach and the homological mirror symmetry approach. More precisely, as mirror symmetry is generally understood as a certain duality between manifolds in certain “degenerating” families, they develop a (mainly differential-geometric) model describing such degenerations, and show then that this model – together with a series of geometric conjectures related to it – can be applied to both the SYZ conjecture and Kontsevich’s “Homological Mirror Conjecture” proposed in 1994 [loc. cit.].

The paper as a whole consists of nine sections. After a brief motivating introduction, Section 2 discusses the underlying background from physics, i.e., from conformal field theory. In Section 3, the authors formulate and explain several conjectures about analytic and differential-geometric properties of Calabi-Yau manifolds at the large complex structure limit, i.e., in certain degenerating families. Section 4 provides a general framework of \(A_\infty\)-pre-categories adapted to the transversality problem in the definition of the Fukaya category. This is used, in the following Section 5, to define the Fukaya category of a symplectic manifold and to study its degeneration behavior. Section 6 deals with the \(A_\infty\)-category of smooth functions studied by K. Fukaya and Y.-G. Oh [Asian J. Math. 1, No. 1, 96–180 (1997; Zbl 0938.32009)]. The authors prove that this category is equivalent to the de Rham category of smooth functions on a compact Riemannian manifold. Section 7 discusses the analytic aspects of the “Homological Mirror Conjecture”. The authors show how a rigid analytic space can be assigned to the class of torus fibrations constructed in the course of Section 3.

The main result of the entire paper occurs in Section 8, where a rigorous proof of a variant of the first author’s “Homological Mirror Conjecture” is presented. This proof is heavily based on the methods developed in Sections 6 and 7, which in turn are built on the foregoing sections, and on the entire new approach outlined in this treatise. Apart from the use of rigid analytic geometry, the application of methods of non-Archimedean analysis to the study of the Fukaya category appears as another striking feature of the authors’ novel approach to mirror symmetry. Section 9 is meant as an appendix, in which the authors explain how their established non-Archimedean approach in Section 8 possibly could be carried over to the Archimedean case, i.e., to classical complex geometry.

In summary, it must be stated that the present paper is highly interesting, stimulating, and guiding with regard to further research in this direction. Although it is written in a merely explanatory style, without detailed proofs in many places, its outstanding significance for future developments in the field of mathematical mirror symmetry is absolutely evident. The ideas and methods developed by the authors provide an extremely rich source for further progress in understanding the fascinating interrelation of contemporary geometry and theoretical physics.

For the entire collection see [Zbl 0980.00036].

In the present paper, the authors throw a bridge between the SYZ approach and the homological mirror symmetry approach. More precisely, as mirror symmetry is generally understood as a certain duality between manifolds in certain “degenerating” families, they develop a (mainly differential-geometric) model describing such degenerations, and show then that this model – together with a series of geometric conjectures related to it – can be applied to both the SYZ conjecture and Kontsevich’s “Homological Mirror Conjecture” proposed in 1994 [loc. cit.].

The paper as a whole consists of nine sections. After a brief motivating introduction, Section 2 discusses the underlying background from physics, i.e., from conformal field theory. In Section 3, the authors formulate and explain several conjectures about analytic and differential-geometric properties of Calabi-Yau manifolds at the large complex structure limit, i.e., in certain degenerating families. Section 4 provides a general framework of \(A_\infty\)-pre-categories adapted to the transversality problem in the definition of the Fukaya category. This is used, in the following Section 5, to define the Fukaya category of a symplectic manifold and to study its degeneration behavior. Section 6 deals with the \(A_\infty\)-category of smooth functions studied by K. Fukaya and Y.-G. Oh [Asian J. Math. 1, No. 1, 96–180 (1997; Zbl 0938.32009)]. The authors prove that this category is equivalent to the de Rham category of smooth functions on a compact Riemannian manifold. Section 7 discusses the analytic aspects of the “Homological Mirror Conjecture”. The authors show how a rigid analytic space can be assigned to the class of torus fibrations constructed in the course of Section 3.

The main result of the entire paper occurs in Section 8, where a rigorous proof of a variant of the first author’s “Homological Mirror Conjecture” is presented. This proof is heavily based on the methods developed in Sections 6 and 7, which in turn are built on the foregoing sections, and on the entire new approach outlined in this treatise. Apart from the use of rigid analytic geometry, the application of methods of non-Archimedean analysis to the study of the Fukaya category appears as another striking feature of the authors’ novel approach to mirror symmetry. Section 9 is meant as an appendix, in which the authors explain how their established non-Archimedean approach in Section 8 possibly could be carried over to the Archimedean case, i.e., to classical complex geometry.

In summary, it must be stated that the present paper is highly interesting, stimulating, and guiding with regard to further research in this direction. Although it is written in a merely explanatory style, without detailed proofs in many places, its outstanding significance for future developments in the field of mathematical mirror symmetry is absolutely evident. The ideas and methods developed by the authors provide an extremely rich source for further progress in understanding the fascinating interrelation of contemporary geometry and theoretical physics.

For the entire collection see [Zbl 0980.00036].

Reviewer: Werner Kleinert (Berlin)

##### MSC:

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

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

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

32Q25 | Calabi-Yau theory (complex-analytic aspects) |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |