Related papers: Towards the Doran-Harder-Thompson conjecture via t…
We give an example of the recent proposed mirror construction of Strominger, Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper first considers mirror symmetry for K3 surfaces in light of this construction. We…
Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration $f : Y \rightarrow \mathbb{P}^1$ with $-K_Y = f^*\mathcal{O}(1)$. We show that the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces…
From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mari\~no, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special…
Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve counting on a Calabi--Yau…
In this paper we analyze four examples of birational transformations between local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero…
Superconformal sigma models with Calabi--Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror…
We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry…
Motivated by SU(3) structure compactifications, we show explicitly how to construct half--flat topological mirrors to Calabi--Yau manifolds with NS fluxes. Units of flux are exchanged with torsion factors in the cohomology of the mirror;…
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…
In this work we explore the physics associated to Calabi-Yau (CY) n-folds that can be described as a fibration in more than one way. Beginning with F-theory vacua in various dimensions, we consider limits/dualities with M-theory, type IIA,…
We describe the tropical mirror for complex toric surfaces. In particular we provide an explicit expression for the mirror states and show that they can be written in enumerative form. Their holomorphic germs give an explicit form of good…
In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK…
Standard toric geometry methods used to construct Calabi-Yau varieties may be extended to complete intersections in non-Fano varieties encoded by star triangulating non-convex polytopes. Similarly, mirror symmetry is conjectured to hold in…
The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in P^n and P^n/G, where G is a finite group that acts on P^n and preserves the Fermat hypersurface. We generalise this to hypersurfaces in Grassmannians,…
We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case.…
This work establishes a subtle connection between mirror symmetry for Calabi-Yau threefolds and that of curves of higher genus. The linking structure is what we call a perverse curve. We show how to obtain such from Calabi-Yau threefolds in…
Fibrations of flux backgrounds by supersymmetric cycles are investigated. For an internal six-manifold M with static SU(2) structure and mirror \hat{M}, it is argued that the product M x \hat{M} is doubly fibered by supersymmetric…
We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use…
In this paper, we establish the convergence for Gromov-Witten invariant of elliptic orbifold $\mathbb{P}^1$ with type $(3,3,3), (4,4,2)$ and $(6,3,2)$. We also prove the mirror theorems of Gromov-Witten theory for those orbifolds and FJRW…
For a large class of $N=2$ SCFTs, which includes minimal models and many $\s$ models on Calabi-Yau manifolds, the mirror theory can be obtained as an orbifold. We show that in such a situation the construction of the mirror can be extended…