Topic
Mirror symmetry
About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.
Papers published on a yearly basis
Papers
More filters
Posted Content•
TL;DR: In this paper, a refinement of homological mirror symmetry for the 2-torus has been explored, which relates exact symplectic topology to arithmetic algebraic geometry, and establishes a derived equivalence of the Fukaya category of the punctured torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]].
Abstract: This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y^2+xy=x^3 over Spec Z, the central fibre of the Tate curve; and, over the "punctured disc" Spec Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We also prove that the wrapped Fukaya category of the punctured torus is derived-equivalent over Z to bounded complexes of coherent sheaves on the central fiber of the Tate curve.
44 citations
TL;DR: The mirror symmetry conjecture of Hausel-Rodriguez-Villegas as discussed by the authors is related to the mirror symmetry conjectures of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan.
Abstract: The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.
43 citations
TL;DR: In this paper, the authors studied the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it, and derived an inhomogeneous Picard-Fuchs equation for the Abel-Jacobi map.
Abstract: We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.
43 citations
TL;DR: In this article, a Landau-Ginzburg model with the same data and symmetries as a Z 2 × Z 2 orbifold that corresponds to a class of realistic free-fermion models was constructed.
43 citations
TL;DR: Chan et al. as mentioned in this paper proved that the inverse of a mirror map for a toric Calabi-Yau manifold of the form K Y, where Y is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov-Witten invariants defined by Fukaya-Oh-Ohta-Ono.
43 citations