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William Fulton

Bio: William Fulton is an academic researcher. The author has contributed to research in topics: Toric variety. The author has an hindex of 1, co-authored 1 publications receiving 1297 citations.
Topics: Toric variety

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31 Jan 1993

1,389 citations


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Journal ArticleDOI
TL;DR: In this paper, a brane tiling is constructed for non-compact 3-dimensional toric Calabi-Yau manifolds, which can be represented as a periodic tiling of the plane.
Abstract: We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

554 citations

Journal ArticleDOI
TL;DR: In this paper, a brane tiling is constructed for non-compact 3-dimensional toric Calabi-Yau manifolds, which can be represented as a periodic tiling of the plane.
Abstract: We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of = 1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Yp,q family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

523 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied quivers with potentials and their representation in cluster algebras and showed that the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a class of integer polynomials called F-polynomials.
Abstract: We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit.

444 citations

Journal ArticleDOI
TL;DR: In this article, Chen and Ruan developed the theory of toric Deligne-Mumford stacks, which corresponds to a combinatorial object called a stacky fan.
Abstract: The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring in troduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge num bers, of the underlying variety. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [28], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and G?ttsche [14] and Uribe [25] verify this conjecture when the orbifold is Symn(5) where 5 is a smooth projective surface with Ks = 0 and the resolution is Hilbn(?>). The initial motivation for this project was to compare the orbifold Chow ring of a simplicial toric variety with the Chow ring of a cr?pant resolution. To achieve this goal, we first develop the theory of toric Deligne-Mumford stacks. Modeled on simplicial toric varieties, a toric Deligne-Mumford stack corresponds to a combinatorial object called a stacky fan. As a first approximation, this object is a simplicial fan with a distinguished lattice point on each ray in the fan. More precisely, a stacky fan S is a triple consisting of a finitely generated abelian group N, a simplicial fan E in Q z N with n rays, and a map ?: Zn ?> N where the image of the standard basis in Zn generates the rays in E. A rational simplicial fan E produces a canonical stacky fan S := (N, E, ?) where N is the distinguished lattice and ? is the map defined by the minimal lattice points on the rays. Hence, there is a natural toric Deligne-Mumford stack associated to every simplicial toric variety. A stacky fan ? encodes a group action on a quasi-affine variety and the toric Deligne-Mumford stack #(?) is the quotient. If E corresponds to a smooth toric variety X?E) and S is the canonical stacky fan associated to E, then we simply have #(!?) = X?Z). We show that many of the basic concepts, such as open and closed toric substacks, line bundles, and maps between toric Deligne Mumford stacks, correspond to combinatorial notions. We expect that many more results about toric varieties lift to the realm of stacks and hope that toric Deligne Mumford stacks will serve as a useful testing ground for general theories.

378 citations

Journal ArticleDOI
TL;DR: Mixed subdivisions of Newton polytopes are introduced, and they are applied to give a new proof and algorithm for Bernstein's theorem on the expected number of roots, which results in a numerical homotopy with the optimal number of paths to be followed.
Abstract: A continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations. We introduce mixed subdivisions of Newton polytopes, and we apply them to give a new proof and algorithm for Bernstein's theorem on the expected number of roots. This results in a numerical homotopy with the optimal number of paths to be followed. In this homotopy there is one starting system for each cell of the mixed subdivision, and the roots of these starting systems are obtained by an easy combinatorial construction.

375 citations