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Categorical Mirror Symmetry: The Elliptic Curve

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abstract

We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's category of Lagrangian submanifolds on $\widetilde{M}.$ We prove this equivalence when $M$ is an elliptic curve and $\widetilde{M}$ is its dual curve, exhibiting the dictionary in detail.

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representative citing papers

Landau-Ginzburg Paradigm of Topological Phases

cond-mat.str-el · 2025-06-05 · conditional · novelty 6.0

A modified string-net model turns anyon condensation into a lattice version of the Higgs mechanism, putting some topological phase transitions into the Landau-Ginzburg symmetry-breaking framework.

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  • Landau-Ginzburg Paradigm of Topological Phases cond-mat.str-el · 2025-06-05 · conditional · none · ref 65 · internal anchor

    A modified string-net model turns anyon condensation into a lattice version of the Higgs mechanism, putting some topological phase transitions into the Landau-Ginzburg symmetry-breaking framework.