##
**Mirror symmetry.
(Symétrie miroir.)**
*(French)*
Zbl 0849.14001

Panoramas et Synthèses. 2. Paris: Société Mathématique de France. 148 p. (1996).

In two-dimensional conformal quantum field theory, there are various attempts by physicists to construct phenomenologically realistic (heterotic) superstring models in 10 dimensions. Most approaches assume a vacuum configuration of the form \(M_4 \times K_6\), where \(M_4\) is a four-dimensional maximally symmetric space-time and \(K_6\) is a compact complex Calabi-Yau threefold. In this context, mirror symmetry is the striking discovery that certain “mirror pairs” of Calabi-Yau threefolds lead to similar physical superstring theories which basically (and importantly) differ by the sign of a certain quantum number. – Good references, for both the physical background material and the algebro-geometric framework for modelling it, are provided by D. Morrison’s survey article “Mirror symmetry and the moduli spaces of superconformal field theories” [in: Proc. Intern. Congr. Math., Zürich 1994, Vol. II, 1904-1314 (1995; Zbl 0847.14025)] and by T. Hübsch’s monography “Calabi-Yau manifolds. A bestiary for physicists” (Singapore 1992; Zbl 0771.53002). The mirror symmetry phenomenon in physics has inspired (and challenged) complex geometers to investigate complex algebraic manifolds from a new classification-theoretic point of view, and mirror symmetry between certain classes of complex manifolds has become, over the recent five years, one of the most active areas of research in algebraic geometry. However, the rapidly increasing number of results and articles on this subject is candidly overwhelming, and makes its difficult for the non-specialist to keep up with this vast development, in its various aspects, or to maintain (at least) a panoramic overview. – The present booklet exactly aims at providing such a panoramic, mathematically systematic overview of the recent works on mirror symmetry of complex algebraic manifolds. Based on a course on this subject, which she held in the spring semester 1995 at the “Institut Henri Poincaré” in Paris, the author gives both a detailed introduction to the topic and a more specific explanation of the various directions in which it has expanded up to now.

After a brief, but very systematic and motivating introduction to the physical background and its complex-geometric framework, chapter 1 deals with the theory of Calabi-Yau manifolds. This includes the Kähler geometry and the algebraic geometry of Calabi-Yau manifolds, in particular the relevant metric, deformation-theoretic, and Hodge-theoretic aspects. Special emphasis is put on Calabi-Yau threefolds and related results around the corresponding mirror symmetry conjecture. Chapter 2 returns to the physical origin of the mirror symmetry conjecture. The author describes the bosonic sigma-model with \(N = 2\) supersymmetries, the quantification process, the famous Gepner conjecture on the correspondence between Calabi-Yau manifolds and \((N = 2)\)-superconformal field theories, the mirror symmetry from the physical viewpoint, and E. Witten’s interpretation of the mirror symmetry via Yukawa couplings [cf. E. Witten in: Essays on mirror manifolds, 120-158 (1992; Zbl 0834.58013)].

Chapter 3 discusses the recent work of P. Candelas, X. de la Ossa, P. Green and L. Parkes, “A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory” [Nucl. Physics, B 359, 21-74 (1991); see also: Essays on mirror manifolds, 31-95 (1992; Zbl 0826.32016)] from a rigorous mathematical viewpoint. The variation of Hodge structures of Calabi-Yau threefolds, the resulting explicit computation of Yukawa couplings, and the related theory of Picard-Fuchs equations are among the main results of this section.

Chapter 4 explains V. Batyrev’s work on mirror symmetry between hypersurfaces in toric varieties [cf. e.g.: V. Batyrev, “Hodge theory of hypersurfaces in toric varieties and recent developments in quantum physics” [Habilitationsschrift, Univ. Essen 1992]. This discussion comes with a brief introduction to toric varieties, Fano toric varieties, their desingularization, and their Hodge theory.

The concluding two chapters are linked by the concept of Floer cohomology, basically in its interpretation via quantum cohomology. Chapter 5, entitled “Quantum cohomology”, gives a description of the Gromov-Witten invariants after Kontsevich and Manin [cf.: M. Kontsevich and Yu. Manin, “Gromov-Witten classes, quantum cohomology and enumerative geometry,” Commun. Math. Phys. 164, No. 3, 525-562 (1994)], an explanation of their applications to mirror symmetry, and a discussion of the recent related work of Y. Ruan and G. Tian, “A mathematical theory of quantum cohomology” [J. Differ. Geom. 42, No. 2, 259-367 (1995)] and P. S. Aspinwall and D. R. Morrison [Commun. Math. Phys. 151, No. 2, 245-262 (1993; Zbl 0776.53043)].

Chapter 6 is devoted to a very recent construction by A. B. Givental, which leads to the same conclusions as the calculus of Candelas-Ossa-Green-Parkes (described in chapter 3). A. B. Givental’s construction [“Homological geometry and mirror symmetry” in: Proc. Intern. Congr. Math. Zürich 1994, Vol. I, 472-480 (1995)] is here discussed from the viewpoint of Floer cohomology and equivariant cohomology, both of which are introduced in the first sections of this concluding chapter.

Generally, the author has concentrated, throughout the text, on the basic definitions, methods of construction, and most significant results, above all with regard to their mathematical interrelation and physical significance. Technical details and complicated proofs have been widely omitted, and that for the benefit of a rather panoramic, strategic overview. The reader will find a very enlightening and systematic up-to-date survey on the very vivid subject of mirror symmetry, just as an extremely well-guided invitation to further studies in this direction. The many hints and references in the text, together with the carefully arranged bibliography, enhance this beautifully written text in a useful way. – Altogether, this booklet is a valuable source for interested mathematicians and physicists.

After a brief, but very systematic and motivating introduction to the physical background and its complex-geometric framework, chapter 1 deals with the theory of Calabi-Yau manifolds. This includes the Kähler geometry and the algebraic geometry of Calabi-Yau manifolds, in particular the relevant metric, deformation-theoretic, and Hodge-theoretic aspects. Special emphasis is put on Calabi-Yau threefolds and related results around the corresponding mirror symmetry conjecture. Chapter 2 returns to the physical origin of the mirror symmetry conjecture. The author describes the bosonic sigma-model with \(N = 2\) supersymmetries, the quantification process, the famous Gepner conjecture on the correspondence between Calabi-Yau manifolds and \((N = 2)\)-superconformal field theories, the mirror symmetry from the physical viewpoint, and E. Witten’s interpretation of the mirror symmetry via Yukawa couplings [cf. E. Witten in: Essays on mirror manifolds, 120-158 (1992; Zbl 0834.58013)].

Chapter 3 discusses the recent work of P. Candelas, X. de la Ossa, P. Green and L. Parkes, “A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory” [Nucl. Physics, B 359, 21-74 (1991); see also: Essays on mirror manifolds, 31-95 (1992; Zbl 0826.32016)] from a rigorous mathematical viewpoint. The variation of Hodge structures of Calabi-Yau threefolds, the resulting explicit computation of Yukawa couplings, and the related theory of Picard-Fuchs equations are among the main results of this section.

Chapter 4 explains V. Batyrev’s work on mirror symmetry between hypersurfaces in toric varieties [cf. e.g.: V. Batyrev, “Hodge theory of hypersurfaces in toric varieties and recent developments in quantum physics” [Habilitationsschrift, Univ. Essen 1992]. This discussion comes with a brief introduction to toric varieties, Fano toric varieties, their desingularization, and their Hodge theory.

The concluding two chapters are linked by the concept of Floer cohomology, basically in its interpretation via quantum cohomology. Chapter 5, entitled “Quantum cohomology”, gives a description of the Gromov-Witten invariants after Kontsevich and Manin [cf.: M. Kontsevich and Yu. Manin, “Gromov-Witten classes, quantum cohomology and enumerative geometry,” Commun. Math. Phys. 164, No. 3, 525-562 (1994)], an explanation of their applications to mirror symmetry, and a discussion of the recent related work of Y. Ruan and G. Tian, “A mathematical theory of quantum cohomology” [J. Differ. Geom. 42, No. 2, 259-367 (1995)] and P. S. Aspinwall and D. R. Morrison [Commun. Math. Phys. 151, No. 2, 245-262 (1993; Zbl 0776.53043)].

Chapter 6 is devoted to a very recent construction by A. B. Givental, which leads to the same conclusions as the calculus of Candelas-Ossa-Green-Parkes (described in chapter 3). A. B. Givental’s construction [“Homological geometry and mirror symmetry” in: Proc. Intern. Congr. Math. Zürich 1994, Vol. I, 472-480 (1995)] is here discussed from the viewpoint of Floer cohomology and equivariant cohomology, both of which are introduced in the first sections of this concluding chapter.

Generally, the author has concentrated, throughout the text, on the basic definitions, methods of construction, and most significant results, above all with regard to their mathematical interrelation and physical significance. Technical details and complicated proofs have been widely omitted, and that for the benefit of a rather panoramic, strategic overview. The reader will find a very enlightening and systematic up-to-date survey on the very vivid subject of mirror symmetry, just as an extremely well-guided invitation to further studies in this direction. The many hints and references in the text, together with the carefully arranged bibliography, enhance this beautifully written text in a useful way. – Altogether, this booklet is a valuable source for interested mathematicians and physicists.

Reviewer: W.Kleinert (Berlin)

### MSC:

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

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

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

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

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

14J45 | Fano varieties |

14J30 | \(3\)-folds |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |