Related papers: Mirror symmetry and self-dual manifolds
In this paper we outline a setup for Homological Mirror Symmetry for manifolds of general type. Both Physics and Categorical perspectives are considered.
Ebeling and Ploog \cite{EbelingPloog} studied a duality of bimodular singularities which is part of the Berglund--H$\ddot{\textnormal{u}}$bsch mirror symmetry. Mase and Ueda \cite{MU} showed that this duality leads to a polytope mirror…
We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian…
We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n…
In this paper, we extend our geometrical derivation of expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two K\"ahler forms. Especially, we take Hirzebruch surfaces F_{0}, F_{3} and…
We clarify how mirror symmetry acts on 3d theories with N=2,3 or 4 supersymmetries and non-abelian Chern-Simons terms and then construct many new examples. We identify a new duality, geometric duality, that allows us to generate large…
We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold…
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces.…
We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of…
Mirror symmetry is a type of infrared duality in 3D quantum field theory that relates the low-energy dynamics of two distinct ultraviolet descriptions. Though first discovered in the supersymmetric context, it has far-reaching implications…
We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential…
We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-H\"ubsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped…
We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a…
Gross and Siebert identified a class of singular Lagrangian torus fibrations which arise when smoothing toroidal degenerations, and which come in pairs that are related by mirror symmetry. We identify an immersed Lagrangian in each of these…
We study the group properties and the similarity solutions for the constraint conditions of anti-self-dual null K\"{a}hler four-dimensional manifolds with at least a Killing symmetry vector. Specifically we apply the theory of Lie…
These notes are aimed at mathematicians working on topics related to mirror symmetry, but are unfamiliar with the physical origins of this subject. We explain the physical concepts that enable this surprising duality to exist, using the…
The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and K\"ahler rigid structures. Inspired by the works of Dolgachev, Aspinwall-Morrison and Huybrechts, we introduce a…
We explicitly apply localization results to study the interpolation between three and two dimensional mirror symmetry for Abelian gauge theories with four supercharges. We first use the ellipsoid S_b^3 partition functions to verify the…
The type $A_n$-singularity $\mathbb{C}^2/\mathbb{Z}_{n+1}$ can be resolved by hyper-K\"ahler manifolds $X_{\zeta}$ with underlying smooth manifolds diffeomorphic to the resolution of singularities $X_{\text{res}}$, whose hyper-K\"ahler…
In this article, we establish homological Berglund--H\"ubsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--H\"ubsch--Henningson mirror symmetry, including for…