Higher Dimensional Generalization Research Articles

Effective bounds of linear series on algebraic varieties and arithmetic varieties

Abstract In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert–Samuel formula and the arithmetic Hilbert–Samuel formula. The treatments are high-dimensional generalizations of [25] and [26]. Similar results are obtained independently by Huayi Chen [7] with less explicit error terms.

Helical cosmological magnetic fields from extra-dimensions

We study the inflationary generation of helical cosmological magnetic fields in a higher-dimensional generalization of the electromagnetic theory. For this purpose, we also include a parity breaking piece to the electromagnetic action. The evolution of extra-dimensional scale factor allows the breaking of conformal invariance of the effective electromagnetic action in $1+3$ dimensions required for such generation. Analytical solutions for the vector potential can be obtained in terms of Coulomb wave-functions for some special cases. We also present numerical solutions for the vector potential evolution in more general cases. In the presence of a higher-dimensional cosmological constant there exist solutions for the scale factors in which both normal and extra dimensional space either inflate or deflate simultaneously with the same rate. In such a scenario, with the number of extra dimensions $D=4$, a scale invariant spectrum of helical magnetic field is obtained. The net helicity arises, as one helical mode comes to dominate over the other at the superhorizon scales. A magnetic field strength of the order of $10^{-9}$ $G$ can be obtained for the inflationary scale $H\simeq 10^{-3}$ $M_{pl}$. Weaker fields will be generated for lower scales of inflation. Magnetic fields generated in this model respects the bounds on magnetic fields by Planck and $\gamma$-ray observations (i.e. $10^{-16}$ $G$ $<$ $B_{obs}<3.4\times 10^{-9}$ $G$).

Limit groups over partially commutative groups and group actions on real cubings

The study of limit groups, that is, finitely generated fully residually free groups, was a key first step towards the understanding of the elementary theory of a free group. In this paper we conduct a systematic study of the class U of finitely generated fully residually partially commutative groups. Our first main goal is to give an algebraic characterisation of the class U: a finitely generated group G is fully residually partially commutative if and only if it is a subgroup of a graph tower (a group built hierarchically using partially commutative groups and (nonexceptional) surfaces.) Furthermore, if the group G is given by its finite radical presentation, then the graph tower and the embedding can be effectively constructed. This result generalises the work of Kharlampovich and Miasnikov on fully residually free groups. Following Sela’s approach to limit groups, the second goal of the paper is to provide a dynamical characterisation of the class U. We introduce a class of spaces, called real cubings, as higher-dimensional generalisations of real trees and show that a specific type of action on these spaces characterises the class U: a finitely generated group acts freely cospecially on a real cubing if and only if it is fully residually partially commutative. As a corollary we get that (geometric) limit groups over partially commutative groups are fully residually partially commutative. This result generalises the work of Sela on limit groups over free groups. 20F65, 20F67; 20F70, 20E08

Geometric extensions of many-particle Hardy inequalities

Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of . This includes geometric extensions of the standard Hardy inequalities to involve volumes of simplices spanned by a subset of points. Clifford/multilinear algebra is employed to simplify geometric computations. These results and the techniques involved are relevant for classes of exactly solvable quantum systems such as the Calogero–Sutherland models and their higher-dimensional generalizations, as well as for membrane matrix models, and models of more complicated particle interactions of geometric character.

On the examples of Nill and Paffenholz

In 1999, Batyrev and Selivanova introduced the notion of "symmetric toric Fano manifolds" and proved that every symmetric toric Fano manifold admits an Einstein–Kähler metric ([1]). For n ≦ 4, every Einstein–Kähler toric Fano n-fold is symmetric. Hence, it was conjectured that every Einstein–Kähler toric Fano manifold is symmetric. However, in [6], Nill and Paffenholz constructed the counter-examples of this conjecture. In this paper, we shall construct higher-dimensional generalizations of their examples and also discuss the Chow unstability of these generalizations.

Dirac and Laplace operators on some non-orientable conformally flat orbifolds

In this paper we present an explicit construction of the fundamental solution to the Dirac and Laplace operator on some non-orientable conformally flat manifolds and special orbifolds. We first treat a class of projective cylinders and tori where we can study monogenic sections with values in different pin bundles. Then we discuss the Möbius strip, the Klein bottle and higher dimensional generalizations of them. We present integral representation formulas together with some elementary tools of harmonic analysis on these manifolds resp. orbifolds.

Poincaré linearizers in higher dimensions

It is well-known that a holomorphic function near a repelling fixed point may be conjugated to a linear function. The function which conjugates is called a Poincar\'e linearizer and may be extended to a transcendental entire function in the plane. In this paper, we study the dynamics of a higher dimensional generalization of Poincar\'e linearizers. These arise by conjugating a uniformly quasiregular mapping in $\R^m$ near a repelling fixed point to the mapping $x\mapsto 2x$. In particular, we show that the fast escaping set of such a linearizer has a spider's web structure.

Bosonic short-range entangled states beyond group cohomology classification

We explore and construct a class of bosonic short-range entangled (BSRE) states in all $4k+2$ spatial dimensions, which are higher dimensional generalizations of the well-known Kitaev's ${E}_{8}$ state in $2d$ [Ann. Phys. (N.Y.) 321, 2 (2006); http://online.kitp.ucsb.edu/online/topomat11/kitaev]. These BSRE states share the following properties: (1) their bulk is fully gapped and nondegenerate; (2) their $(4k+1)d$ boundary is described by a ``self-dual'' rank-$2k$ antisymmetric tensor gauge field, and it is guaranteed to be gapless without assuming any symmetry; (3) their $(4k+1)d$ boundary has intrinsic gravitational anomaly once coupled to the gravitational field; (4) their bulk is described by an effective Chern-Simons field theory with rank-$(2k+1)$ antisymmetric tensor fields, whose ${K}^{IJ}$ matrix is identical to that of the ${E}_{8}$ state in $2d$; (5) the existence of these BSRE states leads to various bosonic symmetry protected topological (BSPT) states as their descendants in other dimensions; (6) these BSRE states can be constructed by confining fermionic degrees of freedom from eight copies of $(4k+2)d$ SRE states with fermionic $2k\text{\ensuremath{-}}\mathrm{branes}$; (7) after compactifying the $(4k+2)d$ BSRE state on a closed $4k$ dimensional manifold, depending on the topology of the compact $4k$ manifold, the system could reduce to nontrivial $2d$ BSRE states.

On a higher dimensional generalization of Seifert fibrations

The notion of generalized Seifert fibration is introduced; it is shown that the projections of certain Eschenburg 7-manifolds \(W_{\bar n}^7 \) onto ℂP2 define such fibrations; and their characteristic classes corresponding to the generators of H2(B(U(2)/ℤ2n);ℤ) are defined.

Curvatures and anisometry of maps

We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The proofs relate to a conjectural isoperimetric inequality for manifolds whose curvature is bounded above, and to a higher-dimensional generalization of the Schwarz-Ahlfors lemma.

Characterization of the split closure via geometric lifting

We analyze split cuts from the perspective of cut generating functions via geometric lifting. We show that $\alpha$-cuts, a natural higher-dimensional generalization of the $k$-cuts of Cornu\'{e}jols et al., gives all the split cuts for the mixed-integer corner relaxation. As an immediate consequence we obtain that the $k$-cuts are equivalent to split cuts for the 1-row mixed-integer relaxation. Further, we show that split cuts for finite-dimensional corner relaxations are restrictions of split cuts for the infinite-dimensional relaxation. In a final application of this equivalence, we exhibit a family of pure-integer programs whose split closures have arbitrarily bad integrality gap. This complements the mixed-integer example provided by Basu et al [On the relative strength of split, triangle and quadrilateral cuts, Math. Program. 126(2):281--314, 2011].

Greybody factors for Myers–Perry black holes

The Myers–Perry black holes are higher-dimensional generalizations of the usual (3+1)-dimensional rotating Kerr black hole. They are of considerable interest in Kaluza–Klein models, specifically within the context of brane-world versions thereof. In the present article, we shall consider the greybody factors associated with scalar field excitations of the Myers–Perry spacetimes, and develop some rigorous bounds on these greybody factors. These bounds are of relevance for characterizing both the higher-dimensional Hawking radiation, and the super-radiance, that is expected for these spacetimes.

Twistor/ambitwistor strings and null-superstrings in spacetime of D=4, 10 and 11 dimensions

We show that, at the classical level, the recently proposed `ambitwistor string' model is equivalent to the spinor moving frame formulation of null-supersting, which in its turn is equivalent to Siegel's formulation of closed twistor string or to its higher dimensional generalizations. Although the null-(super)string is usually considered as describing the tensionless limit of (super)string, we show that its action can be derived from the spinor moving frame formulation of superstring also in the infinite tension limit. This observation allows us to argue on the absence of critical dimensions and suggests that the (ambi)twistor string based technique(s) to calculate field theory amplitudes can be developed not only in D=10 or 26, but also in D=11 and other dimensions. The D=11 and D=10 twistor strings are described in some details.

Non-integrability vs. integrability in pentagram maps

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.

Weighting bubbles in group field theory

Group field theories (GFT) are higher dimensional generalizations of matrix models whose Feynman diagrams are dual to triangulations. Here we propose a modification of GFT models that includes extra field indices keeping track of the bubbles of the graphs in the Feynman evaluations. In dimension three, our model exhibits new symmetries, interpreted as the action of the vertex translations of the triangulation. The extra field indices have an elegant algebraic interpretation: they encode the structure of a semi-simple algebra. Remarkably, when the algebra is chosen to be associative, the new structure contributes a topological invariant from each bubble of the graph to the Feynman amplitudes.

General Wahlquist metrics in all dimensions

It is shown that the Wahlquist metric, which is a stationary, axially symmetric perfect fluid solution with $\rho+3p=\text{const.}$, admits a rank-2 generalized closed conformal Killing-Yano tensor with a skew-symmetric torsion. Taking advantage of the presence of such a tensor, we obtain a higher-dimensional generalization of the Wahlquist metric in arbitrary dimensions, including a family of vacuum black hole solutions with spherical horizon topology such as Schwarzschild-Tangherlini, Myers-Perry and higher-dimensional Kerr-NUT-(A)dS metrics and a family of static, spherically symmetric perfect fluid solutions in higher dimensions.

From a projective invariant to some new properties of algebraic hypersurfaces

Projective invariants are not only important objects in mathematics especially in geometry, but also widely used in many practical applications such as in computer vision and object recognition. In this work, we show a projective invariant named as characteristic number, from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property, two high-dimensional generalizations of Pascal’s theorem are given, one establishing the connection of hypersurfaces of distinct degrees, and the other concerned with the intersections of a hypersurface and a simplex.

Uniform Correlation Mixture of Bivariate Normal Distributions and Hypercubically Contoured Densities That Are Marginally Normal

The bivariate normal density with unit variance and correlation ρ is well known. We show that by integrating out ρ, the result is a function of the maximum norm. The Bayesian interpretation of this result is that if we put a uniform prior over ρ, then the marginal bivariate density depends only on the maximal magnitude of the variables. The square-shaped isodensity contour of this resulting marginal bivariate density can also be regarded as the equally weighted mixture of bivariate normal distributions over all possible correlation coefficients. This density links to the Khintchine mixture method of generating random variables. We use this method to construct the higher dimensional generalizations of this distribution. We further show that for each dimension, there is a unique multivariate density that is a differentiable function of the maximum norm and is marginally normal, and the bivariate density from the integral over ρ is its special case in two dimensions.

Fast decoders for qudit topological codes

Qudit toric codes are a natural higher-dimensional generalization of the well-studied qubit toric code. However, standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyse their error correction thresholds and efficiency. The first decoder is a generalization of a ‘hard-decisions’ decoder due to Bravyi and Haah (arXiv:1112.3252). We modify this decoder to overcome a percolation effect which limits its threshold performance for many-level quantum systems. The second decoder is a generalization of a ‘soft-decisions’ decoder due to Poulin and Duclos-Cianci (2010 Phys. Rev. Lett. 104 050504), with a small cell size to optimize the efficiency of implementation in the high dimensional case. In each case, we estimate thresholds for the uncorrelated bit-flip error model and provide a comparative analysis of the performance of both these approaches to error correction of qudit toric codes.

Motion fields for interactive character locomotion

We propose a novel representation of motion data and control of virtual characters that gives highly agile responses to user input and allows a natural handling of arbitrary external disturbances. In contrast to traditional approaches based on replaying segments of motion data directly, our representation organizes samples of motion data into a high-dimensional generalization of a vector field that we call a motion field . Our runtime motion synthesis mechanism freely flows through the motion field in response to user commands. The motions we create appear natural, are highly responsive to real-time user input, and are not explicitly specified in the data.