Related papers: Some integrality properties in local mirror symmet…
As a continuation of \lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a…
We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a…
Using mirror symmetry, we show that Chern-Simons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more…
In this paper, we show that if $(X,g)$ is an oriented four dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in $X$ of appropriate spin enjoy the Calabi-Yau property, meaning that every immersed…
The aim of this article is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds. We first construct four new smooth non-complete intersection Calabi-Yau threefolds with…
The B-side of Kontsevich's Homological Mirror Symmetry Conjecture is discussed. We give first a self-contained study of derived categories and their homological algebra, and later restrict to the bounded derived category of schemes and…
We study the differential equations governing mirror symmetry of elliptic curves, and obtain a characterization of the ODEs which give rise to the integral ${\bf q}$-expansion of mirror maps. Through theta function representation of the…
We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We emphasize the simplifying features of dimension three and supply more robust methods that do not rely on such special characteristics and…
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate…
We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of…
We examine the physical significance of torsion co-cycles in the cohomology of a projective Calabi-Yau three-fold for the (2,2) superconformal field theory (SCFT) associated to the non-linear sigma model with such a manifold as a target…
We show that there are two different dualities of two dimensional gauge theories with N=(2,2) supersymmetry. One is basically a consequence of 3d mirror symmetry. The non-linear sigma model with Calabi-Yau target space on the Higgs branch…
We relate a coherent sheaf supported on a holomorphic curve with its mirror Langrangian submanifold in local mirror symmetry through a tropical curve by interpreting their central charges using the combinatorial information of the tropical…
Hodge-theoretic mirror symmetry for a Calabi-Yau mirror pair says that the variation of Hodge structure arising from quantum cohomology of a Calabi-Yau manifold and that arising from deformation of complex structures on the dual Calabi-Yau…
We revisit the backgrounds of type IIB on manifolds with $SU(4)$-structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex…
We survey recent developments in the study of SYZ mirror symmetry for compact toric and toric Calabi-Yau varieties, with a special emphasis on works of the author and his collaborators.
We give a survey on results related to the Berglund-H\"ubsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
This note is a survey of the enumerative geometry of rational curves on Calabi-Yau threefolds, based on a talk given by the author at the May 1991 Workshop on Mirror Symmetry at MSRI. An earlier version appeared in "Essays on Mirror…
We prove homological mirror symmetry for the quintic Calabi-Yau 3-fold. The proof follows that for the quartic surface by Seidel (arXiv:math/0310414) closely, and uses a result of Sheridan (arXiv:1012.3238). In contrast to Sheridan's…
We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories…