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Mirror symmetry
About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.
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TL;DR: In this article, a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic was shown to be equivalent to the Gromov-Witten theory of the Fermat quintic polynomial.
Abstract: We prove a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic. In particular we calculate the genus-zero FJRW theory for the pair (W, G) where W is the Fermat quintic polynomial and G = SL(W). We identify it with the Gromov-Witten theory of the mirror quintic three-fold via an explicit analytic continuation and symplectic transformation. In the process we prove a mirror theorem for the corresponding Landau-Ginzburg model (W,G).
15 citations
TL;DR: In this article, a new dynamical approach to the Blumberg's equation, a family of unimodal maps are presented, which are proportional to $Beta(p,q)$ probability densities functions.
Abstract: We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to $Beta(p,q)$ probability densities functions. Using the symmetry of the $Beta(p,q)$ distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.
15 citations
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TL;DR: The Calabi-Yau operators were introduced by Almkvist and Zudilin this paper and they conjecturally determine local systems that underly a $Sp(4)$-VHS with Hodge numbers, and in the best cases they make their appearance as Picard-Fuchs operators of families of Calabi--Yau threefolds.
Abstract: Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi--Yau operators, introduced by Almkvist and Zudilin. They conjecturally determine $Sp(4)$-local systems that underly a $\mathbb{Q}$-VHS with Hodge numbers \[h^{3 0}=h^{2 1}=h^{1 2}=h^{0 3}=1\] and in the best cases they make their appearance as Picard--Fuchs operators of families of Calabi--Yau threefolds with $h^{12}=1$ and encode the numbers of rational curves on a mirror manifold with $h^{11}=1$. We review some of the striking properties of this rich class of operators.
15 citations
01 Jan 2008
TL;DR: In this article, the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form e iJ and the holomorphic form.
Abstract: We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form e iJ and the holomorphic form . The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: e iJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly is closed in IIB. Modulo a different action of the B-field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.
15 citations
TL;DR: In this article, the authors established a version of homological mirror symmetry for punctured Riemann surfaces via the topological Fukaya category, which is equivalent to matrix factorizations of a certain mirror model (X, W).
Abstract: In this paper we establish a version of homological mirror symmetry for punctured
Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured
surface ? via the topological Fukaya category. We prove that the topological Fukaya
category of ? is equivalent to the category of matrix factorizations of a certain mirror LG
model (X, W). Along the way we establish new gluing results for the topological Fukaya
category of punctured surfaces which are of independent interest.
15 citations