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Homological Mirror Symmetry and Tropical Geometry pp 377–406Cite as

Mirror Duality of Landau–Ginzburg Models via Discrete Legendre Transforms

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Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 15))

Abstract

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.

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Notes

  1. 1.

    This means \(\mathcal{X}\) is flat over the base such that \(X = \mathcal{X}_{t_{0}}\) for some t 0 ≠ 0 and \(T \in \mathrm{ End}(H^{\bullet }(X, \mathbb{Q}))\), the monodromy operator around the special fibre at t = 0, satisfies (T − id)n+1 = 0 and (T − id)n+1 ≠ 0 with n = dimX.

  2. 2.

    This means \(h^{\bullet }(X,\mathcal{O}_{X}) = h^{\bullet }(S^{n}, \mathbb{Q})\) for n = dimX.

  3. 3.

    Confusingly in the discrete world (e.g. [17]), for a piecewise affine function on a polyhedral complex the notion strictly convex is used for the property where the maximal cells coincide with non-extendable domains of linearity of the function. This is actually the type of function we want.

  4. 4.

    It is possible to choose them holomorphic, in fact there is a natural choice.

  5. 5.

    This means it doesn’t contain a non-trivial linear subspace.

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Acknowledgements

The author is indebted to Bernd Siebert, Denis Auroux and Mark Gross for what he learned from them.

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Authors and Affiliations

  1. JGU Mainz, Institut für Mathematik, Staudingerweg 9, 55099, Mainz, Germany

    Helge Ruddat

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  1. Helge Ruddat
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Correspondence to Helge Ruddat .

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Editors and Affiliations

  1. Mathematics Department, Kansas State University, Manhattan, Kansas, USA

    Ricardo Castano-Bernard

  2. Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

    Fabrizio Catanese

  3. Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France

    Maxim Kontsevich

  4. Mathematics Department, University of Pennsylvania, Philadelphia, Pennsylvania, USA

    Tony Pantev

  5. Department of Mathematics, Kansas State University, Manhattan, Kansas, USA

    Yan Soibelman

  6. Department of Mathematics, Kansas State University, Manhattan, Kansas, USA

    Ilia Zharkov

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Ruddat, H. (2014). Mirror Duality of Landau–Ginzburg Models via Discrete Legendre Transforms. In: Castano-Bernard, R., Catanese, F., Kontsevich, M., Pantev, T., Soibelman, Y., Zharkov, I. (eds) Homological Mirror Symmetry and Tropical Geometry. Lecture Notes of the Unione Matematica Italiana, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-06514-4_9

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