Duality for toric Landau-Ginzburg models.

*(English)*Zbl 1386.81130In the publication the author generalises the mirror symmetry of pairs of Calabi-Yau manifolds to Kähler manifolds for which he uses a Landau-Ginzburg model as a mean to express this mirror symmetry. This Landau-Ginzburg model is imported to topology from the theory of superconductivity. Therefore, from the first sight on the title the reader might think that the publication deals with superconductivity. Only the adjective “toric” provides a first caveat. Actually, the subject is symplectic topology. In scanning the literature dealing with Landau-Ginzburg models in topology gives no hint on how this model is related to the treatment in superconductivity.

For a reader not being accustomed to the terms of symplectic topology (like me), the barrage of technical terms is frightening and keeps from reading this publication. The author uses the slang of his area of expertise which makes understanding quite difficult. For instance, he speaks of “Calabi-Yau” instead of “Calabi-Yau manifold” and “Gorenstein” as an adjective. Except for the general statement that a model can help to generalise a symmetry, the publication is not suitable for a broader audience. The expert in this field might get some deeper insight into articular considerations. However, the reader from outside with interest in topology will have difficulties to read this publication.

For a reader not being accustomed to the terms of symplectic topology (like me), the barrage of technical terms is frightening and keeps from reading this publication. The author uses the slang of his area of expertise which makes understanding quite difficult. For instance, he speaks of “Calabi-Yau” instead of “Calabi-Yau manifold” and “Gorenstein” as an adjective. Except for the general statement that a model can help to generalise a symmetry, the publication is not suitable for a broader audience. The expert in this field might get some deeper insight into articular considerations. However, the reader from outside with interest in topology will have difficulties to read this publication.

Reviewer: Stefan Groote (Tartu)

##### MSC:

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

81T60 | Supersymmetric field theories in quantum mechanics |

14J33 | Mirror symmetry (algebro-geometric aspects) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

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

32Q15 | Kähler manifolds |