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Mirror symmetry suggests unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite…
We study open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces in weighted projective space. We identify mirror pairs of D-brane configurations, derive the corresponding inhomogeneous Picard-Fuchs equations, and solve for…
The Kahler moduli space of a particular non-simply-connected Calabi-Yau manifold is mapped out using mirror symmetry. It is found that, for the model considered, the chiral ring may be identical for different associated conformal field…
By normalizing the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkahler orbifolds which satisfy the conditions to be mirror partners in…
We continue our study on the pairs of singular Calabi--Yau varieties arising from double covers over semi-Fano toric manifolds. In this paper, we first investigate singular CY double covers of \(\mathbb{P}^{3}\) branched along (1) a union…
We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary…
For each Fano threefold, we construct a family of Landau-Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi-Yau varieties with proper potential maps; they admit open…
We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with…
We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N=(4,4) superconformal field theories with central charge c=6. Our approach enables us to determine the action of mirror symmetry on (non-stable)…
We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational…
Calabi-Yau manifolds are important objects in algebraic geometry and in theoretical physics. A hypothesis of mirror symmetry is that Calabi-Yau manifolds of dimension 3 come in pairs, with the Hodge numbers of one manifold mirroring the…
The construction of mirror symmetry in the heterotic string is reviewed in the context of Calabi-Yau and Landau-Ginzburg compactifications. This framework has the virtue of providing a large subspace of the configuration space of the…
The recent classification of Landau--Ginzburg potentials and their abelian symmetries focuses attention on a number of models with large positive Euler number for which no mirror partner is known. All of these models are related to…
We study black hole attractor equations for one-(complex structure)modulus Calabi-Yau spaces which are the mirror dual of Fermat Calabi-Yau threefolds (CY_{3}s). When exploring non-degenerate solutions near the Landau-Ginzburg point of the…
We study mirror symmetry of a family of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces with Euler characteristic zero. By describing the parameter space globally, we find all expected boundary points (LCSLs), including…
Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. The tension of the domainwall between the two vacua on the brane, which is the generating function for the open…
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…
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…
Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef.…
Using mirror pairs (M_3, W_3) in type II superstring compactifications on Calabi-Yau threefolds, we study, geometrically, F-theory duals of M-theory on seven manifolds with G_2 holonomy. We first develop a way for getting Landau Ginzburg…