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.
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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Landau-Ginzburg Paradigm of Topological Phases
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.