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The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties
TLDR
In this article, the authors describe recent results of the authors and David Nadler on microlocalization, the Fukaya category, and coherent sheaves on toric varieties, and present an expository article describing recent results.Abstract:
This is an expository article describing recent results of the authors and David Nadler on microlocalization, the Fukaya category, and coherent sheaves on toric varieties. The original papers are arXiv:math/0604379, arXiv:math/0612399 and arXiv:0811.1228v1.read more
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Journal ArticleDOI
Homological Mirror Symmetry for the genus two curve
TL;DR: In this paper, the Fukaya category of a genus two curve was shown to be equivalent to the category of Landau-Ginzburg branes on a singular rational surface.
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Homological mirror symmetry for the genus two curve
Abstract: Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. We prove a version of this conjecture in the simplest example, relating the Fukaya category of a genus two curve to the category of Landau-Ginzburg branes on a certain singular rational surface.
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Moment maps, nonlinear PDE, and stability in mirror symmetry
TL;DR: In this article, the deformed Hermitian-Yang-Mills (dHYM) equation is studied from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians.
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A categorification of Morelli's theorem
TL;DR: In this paper, a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves in a vector space was proved.
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Tropical Coamoeba and Torus-Equivariant Homological Mirror Symmetry for the Projective Space
Masahiro Futaki,Kazushi Ueda +1 more
TL;DR: The notion of a tropical coamoeba was introduced in this paper, which gives a combinatorial description of the Fukaya category of the mirror of a toric Fano stack.
References
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Book
Introduction to Toric Varieties.
TL;DR: In this article, a mini-course is presented to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications, concluding with Stanley's theorem characterizing the number of simplicies in each dimension in a convex simplicial polytope.
Journal ArticleDOI
Mirror symmetry is T duality
TL;DR: In this paper, it was argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles and that the moduli space of such cycles together with their flat connections is precisely the space Y.
Book ChapterDOI
Homological Algebra of Mirror Symmetry
TL;DR: Mirror symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros).
Book
Mirror Symmetry
Eric Zaslow,Ravi Vakil,Kentaro Hori,Richard P. Thomas,Cumrun Vafa,Albrecht Klemm,Rahul Pandharipande,Sheldon Katz +7 more
TL;DR: In this paper, the authors proved mirror symmetry for supersymmetric sigma models on Calabi-Yau manifolds in 1+1 dimensions and showed that the equivalence of the gauged linear sigma model embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type Standard R -> 1/R duality and dynamical generation of superpotential by vortices.
Journal ArticleDOI
The moment map and equivariant cohomology
TL;DR: In this article, the authors propose a solution to solve the problem of spamming, which is called spamming-based spamming.$$$/$/$/$/$$