Related papers: Mirror Symmetry for Five-Parameter Hulek-Verrill M…
This paper explores the relationship between mirror symmetry for P^2, at the level of big quantum cohomology, and tropical geometry. The mirror of P^2 is typically taken to be ((C^*)^2,W), where W is a Landau-Ginzburg potential of the form…
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…
We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_{\Sigma}$…
We study the gauging of a discrete $\mathbb{Z}_3$ symmetry in the five-dimensional superconformal $T_N$ theories. We argue that this leads to an infinite sequence of five-dimensional superconformal theories with either $E_6 \times SU(N)$ or…
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it…
Applying tropical geometry a framework for mirror symmetry, including a mirror construction for Calabi-Yau varieties, was proposed by the author. We discuss the conceptual foundations of this construction based on a natural mirror map…
Mirror symmetry for a semi-stable degeneration of a Calabi-Yau manifold was first investigated by Doran-Harder-Thompson when the degeneration fiber is a union of two (quasi)-Fano manifolds. They propose a topological construction of a…
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…
Motivated by the classical statements of Mirror Symmetry, we study certain Kahler metrics on the complexified Kahler cone of a Calabi-Yau threefold, conjecturally corresponding to approximations to the Weil-Petersson metric near large…
We propose and give strong evidence for a duality relating Type II theories on Calabi-Yau spaces and heterotic strings on $K3 \times T^2$, both of which have $N=2$ spacetime supersymmetry. Entries in the dictionary relating the dual…
In this paper, the numbers of rational curves on general complete intersection Calabi-Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror…
By analyzing qualitative aspects of NS5-brane instanton corrections, we determine the topology of the hypermultiplet moduli space M_H in Calabi-Yau compactifications of type II string theories at fixed value of the dilaton and of the…
In type IIB string compactifications on a Calabi-Yau threefold, the hypermultiplet moduli space $M_H$ must carry an isometric action of the modular group SL(2,Z), inherited from the S-duality symmetry of type IIB string theory in ten…
The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to better-known approaches using moduli spaces of stable maps and special Lagrangian fibrations, this…
We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for…
We study the cohomology of a 1-parameter family Y_t of Calabi-Yau 3-folds introduced by Aspinwall and Morrison, related to the mirror quintic family. Szendr\H{o}i proved that Y_t, Y_{xi t}, ..., Y_{xi^4 t}, where xi is a fifth root of…
In this paper we explore contributions to non-perturbative superpotentials arising from instantons wrapping effective divisors in smooth Calabi-Yau four-folds. We concentrate on the case of manifolds constructed as complete intersections in…
We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups $G_2$ and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to…
Recently, at least 50 million of novel examples of compact $G_2$ holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry…
For any positive integer $g$, we introduce the moduli space $\mathcal{A}^F_g =[\mathcal{H}_g/P_g(\mathbb{Z})]$ parametrizing $g$-dimensional principally polarized abelian varieties $V_\tau$ together with a Strominger-Yau-Zalsow (SYZ)…