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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, the Strominger-Yau-Zaslow (SYZ) conjecture is considered for affine hypersurfaces in toric varieties, and a Landau-Ginzburg model which is a SYZ mirror to the blowup of a hypersurface along a toric variety is constructed.
Abstract: We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.
22 citations
01 Apr 1995
TL;DR: In this paper, the authors review the mechanics of making these predictions, including a discussion of two conjectures which specify how the elusive ''constants of integration'' in the mirror map should be fixed.
Abstract: Given two Calabi--Yau threefolds which are believed to constitute a mirror pair, there are very precise predictions about the enumerative geometry of rational curves on one of the manifolds which can be made by performing calculations on the other. We review the mechanics of making these predictions, including a discussion of two conjectures which specify how the elusive ``constants of integration'' in the mirror map should be fixed. Such predictions can be useful for checking whether or not various conjectural constructions of mirror manifolds are producing reasonable answers.
22 citations
Book•
19 Apr 2006TL;DR: Warming up to enumerative geometry Enumerative geometry of lines Excess intersection Rational curves on the quintic threefold Mechanics Introduction to supersymmetry Introduction to string theory Topological quantum field theory Quantum cohomology as discussed by the authors.
Abstract: Warming up to enumerative geometry Enumerative geometry in the projective plane Stable maps and enumerative geometry Crash course in topology and manifolds Crash course in $C^\infty$ manifolds and cohomology Cellular decompositions and line bundles Enumerative geometry of lines Excess intersection Rational curves on the quintic threefold Mechanics Introduction to supersymmetry Introduction to string theory Topological quantum field theory Quantum cohomology and enumerative geometry Bibliography Index.
22 citations
01 Mar 1994
TL;DR: In this article, the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories.
Abstract: In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients locally of the form C^d/G where G is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case C^2/Z_n is analyzed explicitly. This is largely a review paper to appear in "Essays on Mirror Manifolds, II".
22 citations
TL;DR: In this article, an exotic monotone Lagrangian torus was constructed in CP^2 using techniques motivated by mirror symmetry, and it was shown that this exotic torus is not Hamiltonian isotopic to the known Clifford and Chekanov tori.
Abstract: We construct an exotic monotone Lagrangian torus in CP^2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian isotopic to the known Clifford and Chekanov tori.
22 citations