Dimension 2k Research Articles

De-projecting the EFThedron

The space of Wilson coefficients of EFT that can be UV completed into consistent theories was recently shown to be described analytically by a positive geometry, termed the EFThedron. However, this geometry, as well as complementary numerical methods of semi-definite programming, have so far focused on the positivity of the partial wave expansion, which allows bounding only ratios of couplings. In this paper we describe how the unitarity upper bound of the partial waves can be incorporated. This new problem can be formulated in terms of the well known L-moment problem, which we generalize and solve from a geometrical perspective. We find the non-projective generalization of the EFThedron has an infinite number of non-linear facets, which in some cases have remarkably simple descriptions. We use these results to derive bounds on single couplings, finding that the leading derivative operators are bounded by unity, when normalized by the cut-off scale and loop factors. For general operators of mass dimension 2k we find the upper bound is heavily suppressed at large k, with an 1/k fall-off.

Universal scaling of higher-order spacing ratios in Gaussian random matrices.

Higher-order spacing ratios are investigated analytically using a Wigner-like surmise for Gaussian ensembles of random matrices. For a kth order spacing ratio (r^{(k)},k>1) the matrix of dimension 2k+1 is considered. A universal scaling relation for this ratio, known from earlier numerical studies, is proved in the asymptotic limits of r^{(k)}→0 and r^{(k)}→∞.

Moduli space of nonnegatively curved metrics on manifolds of dimension 4k + 1

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

A method of construction of differentially 4-uniform permutations overVmfor evenm

AbstractA generalization of the method of C. Carlet for constructing differentially 4-uniform permutations of binary vector spaces in even dimension 2kis suggested. It consists in restricting APN-functions in 2k+1 variables to a linear manifold of dimension 2k. The general construction of the method is proposed and a criterion for its applicability is established. Power permutations to which this construction is applicable are completely described and a class of suitable not one-to-one functions is presented.

Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry

We study Lorentzian manifolds (M,g) of dimension n≥4, equipped with a maximally twisting shearfree null vector field p, for which the leaf space S=M/{exp⁡tp} is a smooth manifold. If n=2k, the quotient S=M/{exp⁡tp} is naturally equipped with a subconformal structure of contact type and, in the most interesting cases, it is a regular Sasaki manifold projecting onto a quantisable Kähler manifold of real dimension 2k−2. Going backwards through this line of ideas, for any quantisable Kähler manifold with associated Sasaki manifold S, we give the local description of all Lorentzian metrics g on the total spaces M of A-bundles π:M→S, A=S1,R, such that the generator of the group action is a maximally twisting shearfree g-null vector field p. We also prove that on any such Lorentzian manifold (M,g) there exists a non-trivial generalised electromagnetic plane wave having p as propagating direction field, a result that can be considered as a generalisation of the classical 4-dimensional Robinson Theorem. We finally construct a 2-parametric family of Einstein metrics on a trivial bundle M=R×S for any prescribed value of the Einstein constant. If dim⁡M=4, the Ricci flat metrics obtained in this way are the well-known Taub-NUT metrics.

The first Pontrjagin classes of homotopy complex projective spaces

Let M2n be an oriented closed smooth manifold homotopy equivalent to the complex projective space CP(n). The main purpose of this paper is to show that when n is even, the difference between the first Pontrjagin class of M2n and that of CP(n) is divisible by 16. As a geometric application of this result, we prove that the Kervaire sphere of dimension 4k+1 does not admit any free circle group action if k+1 is not a power of 2.

Eulerian orientations and vertex-connectivity

It is well-known that every Eulerian orientation of an Eulerian 2k-edge-connected undirected graph is k-arc-connected. A long-standing goal in the area has been to obtain analogous results for vertex-connectivity. Levit, Chandran and Cheriyan recently proved in Levit et al. (2018) that every Eulerian orientation of a hypercube of dimension 2k is k-vertex-connected. Here we provide an elementary proof for this result.We also show other families of 2k-regular graphs for which every Eulerian orientation is k-vertex-connected, namely the even regular complete bipartite graphs, the incidence graphs of projective planes of odd order, the line graphs of regular complete bipartite graphs and the line graphs of complete graphs.Furthermore, we provide a simple graph counterexample for a conjecture of Frank attempting to characterize graphs admitting at least one k-vertex-connected orientation.

Magnetized Kepler manifolds and their quantization

Let k≥1 be an integer and μ be the half of a nonzero integer. It is reported here that the phase space of the (2k+1)-dimensional magnetized Kepler problem with magnetic charge μ can be identified with the elliptic co-adjoint orbit of the real Lie algebra so(2,2k+2) that corresponds to the dominant weight (|μ|,…,|μ|︸k+1,μ).As a corollary, one has the following result: the aforementioned elliptic co-adjoint orbit admits a natural quantization in which the Hilbert space of quantum states is the space of square integrable sections of a Hermitian vector bundle over the punctured Euclidean space in dimension 2k+1; moreover, this Hilbert space provides a geometric realization for the unitary highest weight so(2,2k+2)-module with highest weight (−k−|μ|,|μ|,…,|μ|︸k,μ).

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

A reformulation of the four color theorem is to say that K4 is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension 2k, PC(2k), (that is the Cayley graph (Z22k,{e1,e2,…,e2k,J}, where the ei's are the standard basis and J=e1+e2+⋯+e2k) is a smallest graph of odd-girth 2k+1 to which every planar graph of odd-girth at least 2k+1 admits a homomorphism. This conjecture is related to a conjecture of P. Seymour who claims that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number (more precisely, Seymour conjectured that χ′(G)=⌈χf′(G)⌉ for any planar multigraph G). Note that the restriction of Seymour's conjecture to cubic (planar) graphs is Tait's reformulation of the four color theorem.Both these conjectures are believed to be true for the larger class of K5-minor-free graphs (which includes the class of planar graphs). Motivated by these conjectures and in extension of a recent work of L. Beaudou, F. Foucaud and the second author, which studies homomorphism bounds for the class of K4-minor-free graphs, in this work we first give a necessary and sufficient condition for a graph B of odd-girth 2k+1 to admit a homomorphism from any partial t-tree of odd-girth at least 2k+1. Applying our results to the class of partial 3-trees, which is a rich subclass of K5-minor-free graphs, we prove that PC(2k) is in fact a smallest graph of odd-girth 2k+1 to which every partial 3-tree of odd-girth at least 2k+1 admits a homomorphism. We then apply this result to show that every planar (2k+1)-regular multigraph G whose dual is a partial 3-tree, and whose fractional edge-chromatic number is 2k+1, is (2k+1)-edge-colorable. Both these results are the best known supports for the general cases of the above mentioned conjectures in extension of the four color theorem.

The automorphism group and limit set of a bounded domain I: The finite type case

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to Aut(Bk). Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension 2k−1. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of the Kobayashi metric plays an important role in the paper.

The Independence Number of the Orthogonality Graph in Dimension 2k

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a rank argument due to Frankl who showed the analogous result for $4p^k$-dimensional hypercubes, where $p$ is an odd prime.

Non-Negative Curvature, Elliptic Genus and Unbounded Pontryagin Numbers

AbstractWe discuss the cobordism type of spin manifolds with non-negative sectional curvature. We show that in each dimension 4k⩾ 12, there are infinitely many cobordism types of simply connected and non-negatively curved spin manifolds. Moreover, we raise and analyse a question about possible cobordism obstructions to non-negative curvature.

Existence of regular unimodular triangulations of dilated empty simplices

Given integers k and m with k≥2 and m≥2, let P be an empty simplex of dimension (2k−1) whose δ-polynomial is of the form 1+(m−1)tk. In the present paper, the necessary and sufficient condition for the kth dilation kP of P to have a regular unimodular triangulation will be presented.

Homomorphism bounds and edge-colourings of K4-minor-free graphs

We present a necessary and sufficient condition for a graph of odd-girth 2k+1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k+1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism to the projective hypercube of dimension 2k. This supports a conjecture of the third author which generalizes the four-colour theorem and relates to several outstanding conjectures such as Seymour's conjecture on edge-colourings of planar graphs. Strengthening this result, we show that the Kneser graph K(2k+1,k) satisfies the conditions, thus implying that every K4-minor free graph of odd-girth 2k+1 has fractional chromatic number exactly 2+1k. Knowing that a smallest bound of odd-girth 2k+1 must have at least (k+22) vertices, we build nearly optimal bounds of order 4k2. Furthermore, we conjecture that the suprema of the fractional and circular chromatic numbers for K4-minor-free graphs of odd-girth 2k+1 are achieved by a same bound of odd-girth 2k+1. If true, this improves, in the homomorphism order, earlier tight results on the circular chromatic number of K4-minor-free graphs. We support our conjecture by proving it for the first few cases. Finally, as an application of our work, and after noting that Seymour provided a formula for calculating the edge-chromatic number of K4-minor-free multigraphs, we show that stronger results can be obtained in the case of K4-minor-free regular multigraphs.

Finite part of operatorK–theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds

In this paper, we study lower bounds on the K‐theory of the maximal C ‐algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K‐theory and give a lower bound that is valid for a large class of groups, called the finitely embeddable groups. The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov’s monster groups, virtually torsion-free groups (eg Out.Fn/), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. We apply this result to measure the degree of nonrigidity for any compact oriented manifold M with dimension 4k 1 .k > 1/. In this case, we derive a lower bound on the rank of the structure group S.M/, which is roughly defined to be the abelian group of all pairs .M 0 ;f /, where M 0 is a compact manifold and fW M 0 ! M is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group z S.M/, which measures the size of the collections of compact manifolds homotopic equivalent to but not homeomorphic to M by any homeomorphism at all (not necessary homeomorphism in the homotopy equivalence class). For a compact Riemannian manifold M with dimension greater than or equal to 5 and positive scalar curvature metric, there is an abelian group P.M/ that measures the size of the space of all positive scalar curvature metrics on M . We obtain a lower bound on the rank of the abelian group P.M/ when the compact smooth spin manifold M has dimension 2k 1 .k > 2/ and the fundamental group of M is finitely embeddable. 19K99; 20F99, 58D29

THE SYMMETRY OF spinℂDIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

It is well-known that the spectrum of a <TEX>$spin^{\mathbb{C}}$</TEX> Dirac operator on a closed Riemannian <TEX>$spin^{\mathbb{C}}$</TEX> manifold <TEX>$M^{2k}$</TEX> of dimension 2k for <TEX>$k{\in}{\mathbb{N}}$</TEX> is symmetric. In this article, we prove that over an odd-dimensional Riemannian product <TEX>$M^{2p}_1{\times}M^{2q+1}_2$</TEX> with a product <TEX>$spin^{\mathbb{C}}$</TEX> structure for <TEX>$p{\geq}1$</TEX>, <TEX>$q{\geq}0$</TEX>, the spectrum of a <TEX>$spin^{\mathbb{C}}$</TEX> Dirac operator given by a product connection is symmetric if and only if either the <TEX>$spin^{\mathbb{C}}$</TEX> Dirac spectrum of <TEX>$M^{2q+1}_2$</TEX> is symmetric or <TEX>$(e^{{\frac{1}{2}}c_1(L_1)}{\hat{A}}(M_1))[M_1</TEX><TEX>]</TEX><TEX>=0$</TEX>, where <TEX>$L_1$</TEX> is the associated line bundle for the given <TEX>$spin^{\mathbb{C}}$</TEX> structure of <TEX>$M_1$</TEX>.

On the orbits of magnetized Kepler problems in dimension [formula omitted

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension 2k+1, the non-colliding orbits in the “external configuration space” R2k+1∖{0} are all conics; moreover, a conic orbit is an ellipse, a parabola, or a branch of a hyperbola according as the total energy is negative, zero, or positive. It is also demonstrated that the Lie group SO+(1,2k+1)×R+ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.

Quasiconformal maps, analytic capacity, and non linear potentials

In this paper we prove that if ϕ:C→C is a K-quasiconformal map, with K>1, and E⊂C is a compact set contained in a ball B, then Ċ2K2K+1,2K+1K+1(E)diam(B)2K+1≥c−1(γ(ϕ(E))diam(ϕ(B)))2KK+1, where γ stands for the analytic capacity and Ċ2K2K+1,2K+1K+1 is a capacity associated to a nonlinear Riesz potential. As a consequence, if E is not K-removable (i.e., removable for bounded K-quasiregular maps), it has positive capacity Ċ2K2K+1,2K+1K+1. This improves previous results that assert that E must have non-σ-finite Hausdorff measure of dimension 2K+1. We also show that the indices 2K2K+1, 2K+1K+1 are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are K-removable. So essentially we solve the problem of finding sharp “metric” conditions for K-removability.

HOW LARGE IS THE SHADOW OF A SYMPLECTIC BALL?

Consider the image of the 2n-dimensional unit ball by a symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection onto a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.

Fixed points in absolutely irreducible real representations

It has been an open question whether any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. A positive proposal is also known as the “Ize-conjecture”. Algebraically speaking, this is to ask whether every absolutely irreducible real representation has an odd dimensional fixed point subspace corresponding to some subgroups. Recently, Reiner Lauterbach and Paul Matthews have found counter examples to this conjecture and interestingly, all of the representations are of dimension 4k, for k ∈ N. A natural question arises: what about the case 4k+ 2? In this paper, we give a partial answer to this question and prove that in any 6-dimensional absolutely irreducible real representation of a finite solvable group, there exists an odd dimensional fixed point subspace with respect to subgroups.