Related papers: Deformed Mirror Symmetry for Punctured Surfaces
We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove…
The general quintic hypersurface in ${\mathbb P}^4$ is the most famous example of a Calabi--Yau threefold for which mirror symmetry has been investigated in detail. There is a description of the mirror as a hypersurface in a certain…
We study \Omega-deformation of B-twisted gauge theories in two dimensions. As an application, we construct an \Omega-deformed, topologically twisted five-dimensional maximally supersymmetric Yang-Mills theory on the product of a Riemann…
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…
In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric variety. We calculate a part of the prepotential of the A-model Yukawa couplings of the…
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…
Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched…
After the appearance of my preprint [T3] (Special Lagrangian geometry and slightly deformed algebraic geometry (spLag and sdAG), Warwick preprint 22/1998, alg-geom/9806006, 54 pp.). I received an e-mail from Cumrun Vafa, who recognized that…
It is shown that a superconformal surface with arbitrary codimension in flat Euclidean space has a (necessarily unique) dual superconformal surface if and only if the surface is S-Willmore, the latter a well-known necessary condition to…
We discuss homological mirror symmetry for the conifold from the point of view of the Strominger-Yau-Zaslow conjecture.
This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an algebraically closed field of characteristic…
We consider fibrations by abelian surfaces and K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain dual fibrations that are derived equivalent to the original fibration. Finally, we relate the problem to mirror…
This paper is devoted to systematically extend $f$-mirror symmetry between families of hypersurfaces in complete toric varieties, as introduced in \cite{R-fTV}, to families of complete intersections subvarieties. Namely, $f$-mirror symmetry…
We study homological mirror symmetry for Hirzebruch surface $\mathbb{F}_k$ as a complex manifold by using the Strominger-Yau-Zaslow construction of mirror pair and Morse homotopy. For the toric Fano surfaces, Futaki-Kajiura and the author…
When studying mirror symmetry in the context of K3 surfaces, the hyperkaehler structure of K3 makes the notion of exchanging Kaehler and complex moduli ambiguous. On the other hand, the metric is not renormalized due to the higher amount of…
In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the…
We prove one direction of homological mirror symmetry for complete intersections in algebraic tori, in all dimensions. The mirror geometry is not a space but a LG model, i.e. a pair given by a space and a regular function. We show that the…
The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact K\"ahler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved…
A simple way to accommodate dark matter is to postulate the existence of a hidden sector. That is, a set of new particles and forces interacting with the known particles predominantly via gravity. In general this leads to a large set of…
We introduce a notion of $Q$-algebra that can be considered as a generalization of the notion of $Q$-manifold (a supermanifold equipped with an odd vector field obeying $\{Q,Q\} =0$). We develop the theory of connections on modules over…