Showing papers on "Toric variety published in 2006"
TL;DR: In this paper, a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity is provided, combining information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings.
Abstract: We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki-Einstein manifolds La,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily La,b,a, whose smallest member is the Suspended Pinch Point.
550 citations
TL;DR: In this article, it was shown that the Reeb vector and the volume of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n can be computed by minimising a function Z on the toric data that defines the singularity.
Abstract: We show that the Reeb vector, and hence in particular the volume, of a Sasaki–Einstein metric on the base of a toric Calabi–Yau cone of complex dimension n may be computed by minimising a function Z on
$$\mathbb {R}^{n}$$
which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki–Einstein manifold without finding the metric explicitly. For complex dimension n = 3 the Reeb vector and the volume correspond to the R–symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a–maximisation. We illustrate our results with some examples, including the Y
p,q
singularities and the complex cone over the second del Pezzo surface.
440 citations
TL;DR: In this article, it was shown that the Lagrangian of the toric diagram for the complex cone over the first del Pezzo surface is a Kahler quotient, which is equivalent to the vacua of gauge models with charges (p,p, −p+q,−p−q), and that these can be embedded in toric diagrams for the orbifold.
Abstract: Recently an infinite family of explicit Sasaki–Einstein metrics Y
p,q
on S
2×S
3 has been discovered, where p and q are two coprime positive integers, with q
355 citations
TL;DR: In this paper, it was shown that rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincide with the counting of certain tropical curves.
Abstract: We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory
240 citations
216 citations
TL;DR: In this article, a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties is provided.
Abstract: We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.
182 citations
TL;DR: In this article, the derived McKay correspondence holds for finite abelian groups and subgroups of GL(2,C) groups, and it is shown that K-equivalent toric birational maps are decomposed into toric flops.
Abstract: We prove that the derived McKay correspondence holds for the cases of finite abelian groups and subgroups of GL(2,C) .W e also prove thatK-equivalent toric birational maps are decomposed into toric flops.
155 citations
TL;DR: In this paper, the authors describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks and check that physical predictions of those gauged lasso models exactly match the corresponding stacks.
Abstract: In this paper, we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) C × quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also see in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFTs to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using Morrison– Plesser–Hori–Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev’s mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFTs, involving fields valued in roots of unity.
130 citations
TL;DR: In this article, it was shown that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan.
Abstract: We show that the equivariant Chow cohomology ring of a toric vari- ety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which lo- calization holds in equivariant Chow cohomology with integer coe!cients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions. If X = X(!) is a smooth, complete complex toric variety then the follow- ing rings are canonically isomorphic: the equivariant singular cohomology ring H ! T (X), the equivariant Chow cohomology ring A ! (X), the Stanley-Riesner ring SR(!), and the ring of integral piecewise polynomial functions PP ! (!). If X is simplicial but not smooth then H ! (X) may have torsion and the natural map from SR(!) takes monomial generators to piecewise linear functions with ra- tional, but not necessarily integral, coe"cients. In such cases, these rings are not isomorphic, but they become isomorphic after tensoring with Q. When X is not simplicial, there are still natural maps between these rings, for instance from A ! (X)Q to H ! (X)Q and from H ! T (X) to PP ! (!), but these maps are far
95 citations
TL;DR: In this article, Kontsevich's homological mirror symmetry conjecture was shown to hold in the context of toric varieties, where the origin point of a toric variety is a convex hull of the primitive vertices of the 1-cones of a simplicial rational polyhedral fan.
Abstract: In this paper we give some evidence for M Kontsevich’s homological mirror symmetry conjecture [13] in the context of toric varieties. Recall that a smooth complete toric variety is given by a simplicial rational polyhedral fan such that jj D R and all maximal cones are non-singular (Fulton [10, Section 2.1]). The convex hull of the primitive vertices of the 1–cones of is a convex polytope which we denote by P , containing the origin as an interior point, and may be thought of as the Newton polytope of a Laurent polynomial W W .C/ ! C. This Laurent polynomial is the Landau–Ginzburg mirror of X .
90 citations
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TL;DR: A-infinity Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of a smooth projective toric variety X were constructed in this paper.
Abstract: Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric varieties.
TL;DR: In this paper, the Hirzebruch surface is iteratively blown up three times, and it is shown that there exist no strongly exceptional sequences of length 7 which consist of line bundles.
Abstract: King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7 which consist of line bundles.
TL;DR: In this article, the precise distribution of R-charges in the quiver gauge theory using dimers technology has been found for all toric singularities, and the results are related to the recently discovered map between toric and tilings.
Abstract: AdS/CFT predicts a precise relation between the central charge a, the scaling dimensions of some operators in the CFT on D3-branes at conical singularities and the volumes of the horizon and of certain cycles in the supergravity dual. We review how this quantitative check can be performed for all toric singularities. We discuss how these results are related to the recently discovered map between toric singularities and tilings; in particular, we discuss how to find the precise distribution of R-charges in the quiver gauge theory using dimers technology.
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TL;DR: In this article, a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties was proposed, based on the notion of a "proper polyhedral divisor" introduced in earlier work.
Abstract: Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a ``proper polyhedral divisor'' introduced in earlier work, we develop the concept of a ``divisorial fan'' and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (non-split) vector bundles over toric varieties.
TL;DR: In this paper, the homological mirror conjecture for toric del Pezzo surfaces was shown to coincide with the derived Fukaya category of coherent sheaves on the original manifold, where the mirror object is a regular function on an algebraic torus.
Abstract: We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus Open image in new window We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.
TL;DR: Upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset {\mathbb R}^2$ are obtained by examining Minkowski sum decompositions of subpolygons of $P$.
Abstract: Toric codes are evaluation codes obtained from an integral convex polytope $P \subset {\mathbb R}^n$ and finite field ${\mathbb F}_q$. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently in [V. Diaz, C. Guevara, and M. Vath, Proceedings of Simu Summer Institute, 2001], [J. Hansen, Appl. Algebra Engrg. Comm. Comput., 13 (2002), pp. 289-300; Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, pp. 132-142], and [D. Joyner, Appl. Algebra Engrg. Comm. Comput., 15 (2004), pp. 63-79]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset {\mathbb R}^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds in Hansen’s work and empirical results of Joyner; they also apply to previously unknown cases.
TL;DR: In this paper, the authors studied the equations defining a projective embedding of a variety X using multigraded Castelnuovo-Mumford regularity, and gave conditions on the mi which guarantee that the ideal of X in P(H 0 (X, L) � ) is generated by quadrics and the first p syzygies are linear.
Abstract: Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1, . . . , Bl on X and m1, . . . , ml ∈ N, consider the line bundle L := B m 1 1 ⊗ � � � ⊗ B ml l . We give conditions on the mi which guarantee that the ideal of X in P(H 0 (X, L) � ) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
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TL;DR: The q-Eulerian polynomials as discussed by the authors are the enumerators for the joint distribution of the excedance statistic and the major index, which is a special case of the Eulerian permutation statistics.
Abstract: In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Bj\"orner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.
01 Jan 2006
TL;DR: In this article, it was shown that the integral cohomology of a smooth toric variety X Σ is determined by the Stanley-Reisner ring of Σ and that the cycle map from Chow groups to Borel-Moore homology is split injective.
Abstract: We prove that the integral cohomology of a smooth, not necessarily compact, toric variety XΣ is determined by the Stanley-Reisner ring of Σ. This follows from a formality result for singular cochains on the Borel construction of XΣ. As a consequence, we show that the cycle map from Chow groups to Borel-Moore homology is split injective.
TL;DR: This theory shows how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions and gives lower bounds in the real Schubert calculus through the sagbi degeneration of the Grassmannian to a toric variety.
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TL;DR: In this article, it was shown that integral F1-schemes of finite type are essentially the same as toric varieties, and a description of the F 1-zeta function in terms of toric geometry is given.
Abstract: This paper contains a loose collection of remarks on F1-schemes. Etale morphisms and universal coverings are introduced. The relation to toric varieties, at least for integral schemes, is clarified.In this paper it is shown that integral F1-schemes of finite type are essentially the same as toric varieties. A description of the F1-zeta function in terms of toric geometry is given. Etale morphisms and universal coverings are introduced.
TL;DR: In this article, the class of Delzant integral polytopes for which a combinatorial invariant vanishes is defined, called defect polytope, and the structure of such polytopes is described.
Abstract: Non-singular toric embeddings with dual defect are classified. The associated polytopes, called defect polytopes, are proven to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes. The structure of a defect polytope is described.
TL;DR: In this paper, a conjectural ring structure on the intersection cohomology of a hypertoric variety is introduced, which is a quaternionic analogue of a toric variety, and it is shown that the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids.
Abstract: A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.
TL;DR: In this paper, an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories is proposed, namely the twisted sigma model of a toric variety in the infinite volume limit (the A-model) and the intermediate model, which is called the I-model.
Abstract: We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.
TL;DR: In this paper, it was shown that the automorphism group of a complete toric Fano variety is reductive if the barycenter of the associated reflexive polytope is zero.
Abstract: We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.
TL;DR: In this article, the authors classified all toric Fano 3-folds with terminal singularities by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point.
Abstract: This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point.
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TL;DR: In this paper, the authors apply this combinatorial geometric technique to investigate the existence of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground.
Abstract: We associate to each toric vector bundle on a toric variety X(Delta) a "branched cover" of the fan Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently.
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TL;DR: In this paper, the authors give a characterisation of the amoeba based on the triangle inequality, which they call testing for lopsidedness, and show that if a point is outside the amobaba of an affine algebraic variety V, there is an element of the defining ideal which witnesses this fact by being Lopsided.
Abstract: The amoeba of an affine algebraic variety V in (C^*)^r is the image of V under the map (z_1, ..., z_r) -> (log|z_1|, ..., log|z_r|). We give a characterisation of the amoeba based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of V, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension, as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hypersurface amoebas and their spines by systems of linear inequalities. Finally, we remark that our main result can be seen a precise analogue of a Nullstellensatz statement for tropical varieties.
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TL;DR: A brief introduction to the construction of toric Calabi-Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations can be found in this article.
Abstract: These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues and topological transitions.
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TL;DR: In this paper, the authors studied the geometry of cut ideals and the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal.
Abstract: Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal. Cut ideals generalize toric ideals arising in phylogenetics and the study of contingency tables.