Topic
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 authors calculate the topological free energy for a number of Yang-Mills-Chern-Simons-matter theories at large N and fixed Chern-Simon levels, which are dual to a variety of Calabi-Yau four-fold singularities.
Abstract: In this paper, we calculate the topological free energy for a number of $$ \mathcal{N} $$
≥ 2 Yang-Mills-Chern-Simons-matter theories at large N and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on S
2 × S
1 with a topological A-twist along S
2 and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over various well-known homogeneous Sasaki-Einstein seven-manifolds, N 0,1,0, V 5,2, and Q
1,1,1. We check that the large N topological free energy can be matched for theories which are related by dualities, including mirror symmetry and $$ \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) $$
duality.
104 citations
TL;DR: In this article, the authors consider mirror symmetry for Fano manifolds and describe how one can recover the classification of 3D Fano manifold from the study of their mirror symmetry.
Abstract: We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.
104 citations
01 Jan 2002
TL;DR: In this article, the second part of a series of articles explaining applications of Floer homology to mirror symmetry and D-brane is presented, independent of part I [Fu9].
Abstract: This is the second part of a series of articles explaining applications of Floer homology to mirror symmetry and D-brane. This article is independent of part I [Fu9]. We will associate an $A_{\infty}$ category to a symplectic manifold. This is an improved version of previous ones [Fu1], [Fu4] in which there were some flaw. The correction is based on a book [FOOO] written jointly with Oh, Ohta, Ono. While correcting the flaw, we find various interesting new phenomena which are related to mirror symmetry. We also discuss homological algebra of $A_{\infty}$ category in this article. This article is a survey article. So most of the material written here are minor modifications of the results which are already known to somebody. However it is rather hard to find a reference of them.
104 citations
TL;DR: In this article, the authors give geometric explanations and proofs of various mirror symmetry conjectures for T -invariant Calabi-Yau manifolds when instanton corrections are absent.
Abstract: We give geometric explanations and proofs of various mirror symmetry conjectures for T -invariant Calabi-Yau manifolds when instanton corrections are absent. This uses a fiberwise Fourier transformation together with a base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex structures and complexified symplectic structures, H’s, Yukawa couplings; (ii) sl (2)× sl (2)-actions; (iii) holomorphic and symplectic automorphisms and (iv) Aand B-connections, supersymmetric Aand B-cycles, correlation functions. We also study (ii) for T -invariant hyperkahler manifolds.
103 citations
TL;DR: In this article, the connection between topological strings and contact homology was studied in the context of knots invariants, and the relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homologies algebra was established.
Abstract: We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the $Q$-deformed $A$-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the “D-model”. In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large $N$ limit of the colored HOMFLY, which we conjecture to be related to a $Q$-deformation of the extension of $A$-polynomials associated with the link complement.
103 citations