Kazushi Ueda

Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories

We propose a new method to find gravity duals to a large class of three-dimensional Chern-Simons-matter theories, using techniques from dimer models. The gravity dual is given by M-theory on AdS4 × Y7, where Y7 is an arbitrary seven-dimensional toric Sasaki-Einstein manifold. The cone of Y7 is a toric Calabi-Yau 4-fold, which coincides with a branch of the vacuum moduli space of Chern-Simons-matter theories.

Dimer models and the special McKay correspondence

We study the behavior of a dimer model under the operation of removing a corner from the lattice polygon and taking the convex hull of the rest. This refines an operation of Gulotta, and the special McKay correspondence plays an essential role in this refinement. As a corollary, we show that for any lattice polygon, there is a dimer model such that the derived category of finitely-generated modules over the path algebra of the corresponding quiver with relations is equivalent to the derived category of coherent sheaves on a toric Calabi-Yau 3-fold determined by the lattice polygon. Our proof is based on a detailed study of relationship between combinatorics of dimer models and geometry of moduli spaces, and does not depend on the result of math/9908027.

Homological mirror symmetry for toric orbifolds of toric del Pezzo surfaces

Abstract 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 this conjecture in some cases, and obtain homological mirror symmetry for quotient stacks of toric del Pezzo surfaces by finite subgroups of the torus as a corollary.

Stokes matrices for the quantum cohomologies of Grassmannians

We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of the Grassmannian. The proof is based on the relation between the quantum cohomology of the Grassmannian and that of the projective space.

Lagrangian Torus Fibrations and Homological Mirror Symmetry for the Conifold

We discuss homological mirror symmetry for the conifold from the point of view of the Strominger–Yau–Zaslow conjecture.

Homological mirror symmetry for the quintic 3-fold

We prove homological mirror symmetry for the quintic Calabi-Yau 3-fold. Essential ingredients of the proof already exist in the literature, and this note only fills the remaining gap.

Dual torus fibrations and homological mirror symmetry for

We study homological mirror symmetry for not necessarily compactly supported coherent sheaves on the minimal resolutions of A_n-singularities. An emphasis is put on the relation with the Strominger-Yau-Zaslow conjecture.

A_n

A bstractWe compute genus-zero Gromov-Witten invariants of Calabi-Yau complete intersection 3-folds in Grassmannians using supersymmetric localization in A-twisted nonAbelian gauged linear sigma models. We also discuss a Seiberg-like duality interchanging Gr(n, m) and Gr(m − n, m).

-singularities

We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

Equivariant A-twisted GLSM and Gromov-Witten invariants of CY 3-folds in Grassmannians

We prove there is an equivalence of derived categories between Orlovs triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations, which is obtained from a McKay quiver by removing one vertex and half of the arrows. This result produces examples of distinct quivers with relations which have equivalent derived categories of representations.