Related papers: Crepant resolutions and brane tilings I: Toric rea…

Reflexive polygons have attracted great interest both in mathematics and in physics. This paper discusses a new aspect of the existing study in the context of quiver gauge theories. These theories are 4d supersymmetric worldvolume theories…

We propose expressions for refined open topological string partition function on certain non-compact Calabi Yau 3-folds with topological branes wrapped on the special lagrangian submanifolds. The corresponding web diagrams are partially…

Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan-Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic…

We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations,…

We study pairs of 4d N=1 supersymmetric gauge theories that share the same vacuum moduli space and the same chiral field content, encoded by a common quiver, but differ in their superpotentials. These theories arise as worldvolume theories…

Given a rational convex polyhedral Gorenstein cone constructed as cone over a lattice polytope P, we establish that toric non-commutative crepant resolutions (NCCRs) of its associated toric algebra descend to toric NCCRs of the algebras…

It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that…

For a toric Calabi-Yau 3-orbifold relative to s Aganagic-Vafa outer branes, we prove a correspondence among the genus-zero open Gromov-Witten invariants with maximal winding at each brane and: (i) closed invariants of a toric Calabi-Yau…

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces $\Bbb{C}^{r}/G$ in dimensions $r\geq 4$ would primarily demand the existence of projective, crepant, full desingularizations. Since this is not…

The connections amongst (1) quivers whose representation varieties are Calabi-Yau, (2) the combinatorics of bipartite graphs on Riemann surfaces, and (3) the geometry of mirror symmetry have engendered a rich subject at whose heart is the…

We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping…

The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. This work introduces the mathematical machinery to derive the complete moduli dependence of…

We present the first explicit examples of a rational threefold and a Calabi-Yau threefold, admitting biregular automorphisms of positive entropy not preserving any dominant rational maps to lower positive dimensional varieties. The most…

We review recent developments in the theory of brane tilings and four-dimensional N=1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical…

We analyze the D-branes of a type IIB string theory on an orbifold singularity including the possibility of discrete torsion following the work of Douglas et al. First we prove some general results about the moduli space of a point…

We study the recently proposed crystal model for three dimensional superconformal field theories arising from M2-branes probing toric Calabi-Yau four-fold singularities. We explain the algorithms mapping a toric Calabi-Yau to a crystal and…

Let X be the toric variety (P^1)^4 associated with its four-dimensional polytope. Consider a resolution of the singular Fano variety associated with the dual polytope of X. Generically, anti-canonical sections Y of X and anticanonical…

In this paper, we construct certain algebraic correspondences between genus three curves and certain type of Calabi-Yau threefolds which is double coverings of three dimensional projective space. Via this correspondences, the first…

We introduce new techniques based on brane tilings to investigate D3-branes probing orientifolds of toric Calabi-Yau singularities. With these new tools, one can write down many orientifold models and derive the resulting low-energy gauge…

We construct curve counting invariants for a Calabi-Yau threefold $Y$ equipped with a dominant birational morphism $\pi:Y \to X$. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when…