Canonical Dual Research Articles

Relating conifold geometries to NS5-branes

We construct the first known example of a near horizon supergravity solution for a pair of NS5-branes, intersecting on R 1 , 3 and localised in all directions except a single transverse circle. We do this by establishing an explicit map between the conifold metric and the near horizon geometry of two intersecting NS5-branes, clarifying and correcting a number of open issues in the literature en route. Our technique is general in nature and may be applied to a whole class of 1/4-BPS five-brane webs and their geometric duals. These 1/4-BPS solutions may have an interesting holographic interpretation in terms of little string theory.

4d-polytopes described by Coxeter diagrams and quaternions

4D-polytopes and their dual polytopes can be described as the orbits of the rank-4 Coxeter-Weyl groups. Their symmetries follow from the quaternionic descriptions of the rank-4 Coxeter-Dynkin diagrams. There exists a one to one correspondence between the finite subgroups of quaternions and the rank-4 Coxeter-Weyl groups.

On duality in multiple objective linear programming

In this paper we present two approaches to duality in multiple objective linear programming. The first approach is based on a duality relation between maximal elements of a set and minimal elements of its complement. It offers a general duality scheme which unifies a number of known dual constructions and improves several existing duality relations. The second approach utilizes polarity between a convex polyhedral set and the epigraph of its support function. It leads to a parametric dual problem and yields strong duality relations, including those of geometric duality.

Gauge theories from D7-branes over vanishing 4-cycles

We study quiver gauge theories on D7-branes wrapped over vanishing holomorphic 4-cycles. We investigate how to incorporate O7-planes and/or flavor D7-branes, which are necessary to cancel anomalies. These theories are chiral, preserve four supercharges and exhibit very rich infrared dynamics. Geometric transitions and duality in the presence of O-planes are analyzed. We study the Higgs branch of these quiver theories, showing the emergence of fuzzy internal dimensions. This branch is related to noncommutative instantons on the divisor wrapped by the seven-branes. Our results have a natural application to the recently introduced F(uzz) limit of F-theory.

Quaternionic representation of snub 24-cell and its dual polytope derived from [formula omitted] root system

Vertices of the 4-dimensional semi-regular polytope, snub 24-cell and its symmetry group ( W ( D 4 ) / C 2 ) : S 3 of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of E 8 root system. A simple method is employed to construct the E 8 root system in terms of icosians which decomposes into two copies of the quaternionic root system of the Coxeter group W ( H 4 ) , while one set is the elements of the binary icosahedral group the other set is a scaled copy of the first. The quaternionic root system of H 4 splits as the vertices of 24-cell and the snub 24-cell under the symmetry group of the snub 24-cell which is one of the maximal subgroups of the group W ( H 4 ) as well as W ( F 4 ) . It is noted that the group is isomorphic to the semi-direct product of the proper rotation subgroup of the Weyl group of D 4 with symmetric group of order 3 denoted by ( W ( D 4 ) / C 2 ) : S 3 , the Coxeter notation for which is [ 3 , 4 , 3 + ] . We analyze the vertex structure of the snub 24-cell and decompose the orbits of W ( H 4 ) under the orbits of ( W ( D 4 ) / C 2 ) : S 3 . The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group ( W ( D 4 ) / C 2 ) : S 3 . In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120 = 24 + 96 and 600 = 24 + 96 + 192 + 288 respectively under the group ( W ( D 4 ) / C 2 ) : S 3 . The dual polytope of the snub 24-cell is explicitly constructed. Decompositions of the Archimedean W ( H 4 ) polytopes under the symmetry of the group ( W ( D 4 ) / C 2 ) : S 3 are given in the appendix.

Lectures on generalized complex geometry for physicists

AbstractIn these lectures we review Generalized Complex Geometry and discuss two main applications to string theory: the description of supersymmetric flux compactifications and the supersymmetric embedding of D‐branes. We start by reviewing G‐structures, and in particular SU(3)‐structure and its torsion classes, before extending to Generalized Complex Geometry. We then discuss the supersymmetry conditions of type II supergravity in terms of differential conditions on pure spinors, and finally introduce generalized calibrations to describe D‐branes. As examples we discuss in some detail AdS4 compactifications, which play a role as the geometric duals in the AdS4/CFT3‐correspondence.

Finding Gale Strings

Abstract The problem 2- Nash of finding a Nash equilibrium of a bimatrix game belongs to the complexity class PPAD. This class comprises computational problems that are known to have a solution by means of a path-following argument. For bimatrix games, this argument is provided by the Lemke–Howson algorithm. It has been shown that this algorithm is worst-case exponential with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterize the vertices of these polytopes. We define the combinatorial problem Another completely labeled Gale string whose solutions define the Nash equilibria of games defined by cyclic polytopes, including games where the Lemke–Howson algorithm takes exponential time. If this problem was PPAD-complete, this would imply that 2- Nash is PPAD-complete, in a much simpler way than the currently known proofs, including the original proof by Chen and Deng [Chen, X., and X. Deng (2006). Settling the complexity of two-player Nash equilibrium. Proc. 47th FOCS , 261–272]. However, we show that Another completely labeled Gale string is solvable in polynomial time by a simple reduction to Perfect matching in graphs, making it unlikely to be PPAD-complete. Although this result is negative, we hope that it stimulates research into combinatorially defined problems that are PPAD-complete and imply this property for 2- Nash .

Specification of the Geometric Regularity Model for FuzzyIf-ThenRule-Based Deinterlacing

A fuzzy if-then rule-based intra-field deinterlacing method using geometric duality is presented in this paper. The proposed method is a content-based hybrid scheme switching between the well-known edge-based linear average method and the proposed geometric duality-based deinterlacing method. Conventional deinterlacing methods usually employ edge-based interpolation techniques within pixel-based estimations. However, they are somewhat sensitive to noise and intensity variations in the image. Moreover, their performance is visually unacceptable due to their failure to estimate edge direction. To reduce this sensitivity, the proposed algorithm investigates features from low-resolution images, and applies them to high-resolution images to calculate the missing pixels. We analyzed properties of the missing pixels and modeled them using geometric regularity. Depending on the features of the region, the missing pixels were interpolated in different ways. The proposed algorithm is computationally feasible and promises to be a good candidate for a low-cost hardware interpolator.

EFFICIENT NON-INTERSECTION QUERIES ON AGGREGATED GEOMETRIC DATA

Geometric intersection searching problems are a well-studied class of query-retrieval problems with many applications. The goal here is to preprocess a set of geometric objects so that the ones that are intersected by a query object can be reported efficiently. Often, a more general version of the problem arises, where the data comes aggregated in disjoint groups and of interest are the groups, not the individual objects, that are intersected by the query object. One approach to a generalized problem is to ignore the grouping, solve the corresponding classical problem, and then infer the groups from the reported answer. However, this is not efficient, since the number of objects intersected can be much larger than the number of groups (i.e., the output size). The problem of designing efficient, output-sensitive query algorithms for generalized intersection searching has received much attention in recent years, and such solutions have been developed for several problems. This paper considers a new class of generalized query-retrieval problems. Specifically, given aggregated geometric data the goal is to report the distinct groups such that no objects from those groups are intersected by the query. Of interest in these generalized non-intersection searching problems are solutions where the query time is sensitive to the output size, i.e., the number of groups reported. Unfortunately, the obvious approaches of (i) solving the corresponding generalized intersection searching problem and reporting the complement, or (ii) solving a generalized intersection searching problem with the complement of the query are either inefficient or incorrect. This paper provides efficient, output-sensitive solutions to several generalized non-intersection searching problems, using techniques such as geometric duality, sparsification, persistence, filtering search, and pruning.

Grand antiprism and quaternions

Vertices of the four-dimensional (4D) semi-regular polytope, the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the E8 root system which decomposes into two copies of the root system of H4. The symmetry of the grand antiprism is a maximal subgroup of the Coxeter group W(H4). It is the group Aut(H2 ⊕ H′2) which is constructed in terms of 20 quaternionic roots of the Coxeter diagram H2 ⊕ H′2. The root system of H4 represented by the binary icosahedral group I of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram H2 ⊕ H′2 are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the grand antiprism. We give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions. The dual polytope of the grand antiprism has also been constructed.

The k-Centrum Straight-line Location Problem

A line is sought in the plane which minimizes the sum of the k largest (Euclidean) weighted distances from n given points. This problem generalizes the known straight-line center and median problems and, as far as the authors are aware, has not been tackled up to now. By way of geometric duality it is shown that such a line may always be found which either passes through two of the given points or lying at equal weighted distance from three of these. This allows construction of an algorithm to find all t-centrum lines for 1 ≤ t ≤ k in O((k + logn)n3). Finally it is shown that both, the characterization of an optimal line and the algorithm, can be extended to any smooth norm.

Gravity and hydrodynamics: lectures on the fluid-gravity correspondence

We discuss recent developments in the hydrodynamic description of strongly coupled conformal field theories using the AdS/CFT correspondence. In particular, we review aspects of the fluid-gravity correspondence which provides a map between a class of inhomogeneous, dynamical, black hole solutions in asymptotically AdS spacetimes and arbitrary fluid flows in the strongly interacting boundary field theory. We explain how the geometric duals to the fluid dynamics are constructed in a boundary derivative expansion and use the construction to extract the hydrodynamic transport coefficients. In addition, we also describe the recent developments extending the correspondence to incorporate matter fields and to non-relativistic systems. Based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge Theories, Geneva, Switzerland (February 2009).

ABJM mirrors and a duality of dualities

We clarify how mirror symmetry acts on 3d theories with = 2,3 or 4 supersymmetries and non-abelian Chern-Simons terms and then construct many new examples. We identify a new duality, geometric duality, that allows us to generate large families of gauge theories, with and without Chern-Simons term, that all flow to the same conformal field theory in the infrared. In particular, we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be understood as the simple result that all of these theories are different realizations of one and the same system in M-theory.

Tight spans of distances and the dual fractionality of undirected multiflow problems

In this paper, we give a complete characterization of the class of weighted maximum multiflow problems whose dual polyhedra have bounded fractionality. This is a common generalization of two fundamental results of Karzanov. The first one is a characterization of commodity graphs H for which the dual of maximum multiflow problem with respect to H has bounded fractionality, and the second one is a characterization of metrics d on terminals for which the dual of metric-weighed maximum multiflow problem has bounded fractionality. A key ingredient of the present paper is a nonmetric generalization of the tight span, which was originally introduced for metrics by Isbell and Dress. A theory of nonmetric tight spans provides a unified duality framework to the weighted maximum multiflow problems, and gives a unified interpretation of combinatorial dual solutions of several known min–max theorems in the multiflow theory.

Contour approximation of data: A duality theory

Given a dataset D partitioned in clusters, the joint distance function (JDF) J ( x ) at any point x is the harmonic mean of the distances between x and the cluster centers. The JDF is a continuous function, capturing the data points in its lower level sets (a property called contour approximation), and is a useful concept in probabilistic clustering and data analysis. In particular, contour approximation allows a compact representation of the data: for a dataset in R n with N points organized in K clusters, the JDF requires K centers and covariances (if Mahalanobis distances are used), for a total of Kn ( n + 3 ) / 2 parameters, and a considerable reduction of storage if N ≫ K , n . The JDF of the whole dataset, J ( D ) ≔ ∑ { J ( x ) : x ∈ D } , is a measure of the classifiability of the data, and can be used to determine the “right” number of clusters for D . A duality theory for the JDF J ( D ) is given, in analogy with Kuhn’s geometric duality theory for the Fermat–Weber location problem. The JDF J ( D ) is the optimal value of a primal problem (P), for which a dual problem (D) is given, with a sharp lower bound on J ( D ) .

Cellular Resolutions of Cohen-Macaulay Monomial Ideals

We investigate monomial labellings on cell complexes, giving a minimal cellular resolution of the ideal generated by these monomials, and such that the associated quotient ring is Cohen-Macaulay. We introduce a notion of such a labelling being maximal. There is only a finite number of maximal labellings for each cell complex, and we classify these for trees, partly for subdivisions of polygons, and for some classes of selfdual polytopes.

N-Point Hough transform for line detection

In this paper, we introduce the N-point Hough transform for the detection of a large number of planar lines in a noisy image. The N-point Hough transform yields the randomized Hough transform and the three-point Hough transform if we set N = 2 and N = 3 , respectively. From the viewpoint of the number of sample points required for the computation of lines, the N-point Hough transform is a generalization of the usual randomized Hough transform. The three-point Hough transform is introduced to increase the speed of the randomized Hough transform; the third point is used to avoid the selection of meaningless first and second sample points, which are used for the computation of the parameters of a line. This additional sample point guarantees the accuracy and robustness of a line detected using the first and second sample points. The N-point Hough transform evaluates the accuracy and robustness of a computed line using additional ( N - 2 ) points for each line. The evaluation in the N-point Hough transform is achieved by counting the cardinality of sample points in the neighborhood of this line as the support of the sample points for the acceptance of this line. First, to define the neighborhood of a line mathematically, in this paper we clarify the relationship between a line and a set of parameters in the voting space using geometric duality. This relationship allows us to define a metric in the voting space. The metric is used for the clustering of bins in the spherical voting space to guarantee the accurate and robust computation of lines. Finally, we evaluate the performance of the N-point Hough transform by comparing it with the randomized Hough transform, which is the two-point Hough transform in our framework of the voting method. This comparative study shows the geometric and computational advantages of the N-point Hough transform.

Approximation of the curvature of Alexandrov surfaces using dual polyhedra

It is shown that a sequence of pairs of locally polar polyhedra makes it possible to construct piecewise-affine approximation to the spherical Gauss map and to construct convergent pointwise approximations to mean and Gauss curvatures of regular surfaces. Any DC surface (representable as a difference of convex functions, A. D. Alexandrov, 1949) possesses a second differential and a tangent paraboloid almost everywhere, hence approximation by dual polyhedra allows us to approximate the curvature of such surfaces.

Dual Poisson-Disk Tiling: An Efficient Method for Distributing Features on Arbitrary Surfaces

This paper introduces a novel surface-modeling method to stochastically distribute features on arbitrary topological surfaces. The generated distribution of features follows the Poisson disk distribution, so we can have a minimum separation guarantee between features and avoid feature overlap. With the proposed method, we not only can interactively adjust and edit features with the help of the proposed Poisson disk map, but can also efficiently re-distribute features on object surfaces. The underlying mechanism is our dual tiling scheme, known as the Dual Poisson-Disk Tiling. First, we compute the dual of a given surface parameterization, and tile the dual surface by our specially-designed dual tiles; during the pre-processing, the Poisson disk distribution has been pre-generated on these tiles. By dual tiling, we can nicely avoid the problem of corner heterogeneity when tiling arbitrary parameterized surfaces, and can also reduce the tile set complexity. Furthermore, the dual tiling scheme is non-periodic, and we can also maintain a manageable tile set. To demonstrate the applicability of this technique, we explore a number of surface-modeling applications: pattern and shape distribution, bump-mapping, illustrative rendering, mold simulation, the modeling of separable features in texture and BTF, and the distribution of geometric textures in shell space.

``Anomaly'' inn= ∞ Alday-Maldacena duality for wavy circle

If the Alday-Maldacena version of string/gauge duality is formulated as an equivalence between double loop and area integrals a la arXiv: 0708.1625, then this pure geometric relation can be tested for various choices of n-side polygons. The simplest possibility arises at n = ∞, with polygon substituted by an arbitrary continuous curve. If the curve is a circle, the minimal surface problem is exactly solvable. If it infinitesimally deviates from a circle, then the duality relation can be studied by expanding in powers of a small parameter. In the first approximation the Nambu-Goto (NG) equations can be linearized, and the peculiar NG Laplacian ΔNG = Δ0−2+ plays the central role. Making use of explicit zero-modes of this operator (NG-harmonic functions), we investigate the geometric duality in the lowest orders for small deformations of arbitrary shape lying in the plane of the original circle. We find a surprisingly strong dependence of the minimal area on regularization procedure affecting the boundary terms in minimal area. If these terms are totally omitted, the remaining piece is regularization independent, but still differs by simple numerical factors like 4 from the double-loop integral which represents the BDS formula so that we stop short from the first non-trivial confirmation of the Alday-Maldacena duality. This confirms the earlier-found discrepancy for two parallel lines at n = ∞, but demonstrates that it actually affects only a finite number (out of infinitely many) of parameters in the functional dependence on the shape of the boundary, and the duality is only slightly violated, which allows one to call this violation an anomaly.