Mirror duality of Landau-Ginzburg models via discrete Legendre transforms. (English) Zbl 1317.53117
Castano-Bernard, Ricardo (ed.) et al., Homological mirror symmetry and tropical geometry. Based on the workshop on mirror symmetry and tropical geometry, Cetraro, Italy, July 2–8, 2011. Cham: Springer (ISBN 978-3-319-06513-7/pbk; 978-3-319-06514-4/ebook). Lecture Notes of the Unione Matematica Italiana 15, 377-406 (2014).
Summary: We recall the semi-flat Strominger-Yau-Zaslow (SYZ) picture of mirror symmetry and discuss the transition from the Legendre transform to a discrete Legendre transform in the large complex structure limit. We recall the reconstruction problem of the singular Calabi-Yau fibres associated to a tropical manifold and review its solution in the toric setting. We discuss the monomial-divisor correspondence for discrete Legendre duals and use this to give a mirror duality for Landau Ginzburg models motivated from the SYZ perspective and Floer theory. We mention its application for the construction of mirror symmetry partners for varieties of general type and discuss the straightening of the boundary of a tropical manifold corresponding to a smoothing of the divisor in the complement of a special Lagrangian fibration.
For the entire collection see [Zbl 1300.14001].
For the entire collection see [Zbl 1300.14001].
MSC:
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14J10 | Families, moduli, classification: algebraic theory |
Keywords:
mirror duality; Landau-Ginzburg models; discrete Legendre transform; large complex structure limit; singular Calabi-Yau fibresReferences:
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