2n Dimensions Research Articles

Localization of spinor fields in higher-dimensional braneworlds

In this paper we investigate the localization of spinor fields in braneworld models by reducing a Dirac spinor in 2n + 2-dimensional spacetime to spinors in 2n dimensions. The high-dimensional Dirac can be reduced to low-dimensional spinors including Weyl or Dirac. In conformally flat extra-dimensional spacetime, fermions cannot be localized through minimal coupling with gravity. To achieve the localization of spinor fields, we introduce a tensor coupling term given by Ψ¯ΓMΓNΓP⋯TMNP⋯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\Psi}{\\Gamma}^M{\\Gamma}^N{\\Gamma}^P\\cdots {T}_{MNP\\cdots}\\Psi $$\\end{document}, which ensures SO(n, 1) symmetry. For a tensor TMNP⋯ of odd order, the left and right chiralities of high-dimensional spinors are decoupled. We find that a special form of tensor coupling Ψ¯ΓM∂MFϕRRμνRμν⋯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\Psi}{\\Gamma}^M{\\partial}_MF\\left(\\phi, R,{R}^{\\mu \ u}{R}_{\\mu \ u},\\cdots \\right)\\Psi $$\\end{document} may facilitate the localization of the spinor field when F(ϕ) = ϕn.

Revealing the Boundary Weyl Physics of the Four-Dimensional Hall Effect via Phason Engineering in Metamaterials

Quantum Hall physics has been theoretically predicted in four dimensions and higher. In hypothetical 2n dimensions, the topological characters of both the bulk and the boundary are manifested as quantized nonlinear transport coefficients that respectively connect to the $n\mathrm{th}$ Chern number of the bulk gap projection and to the $n\mathrm{th}$ winding number of the Weyl spectral singularities on the ($2n\ensuremath{-}1$)-dimensional boundaries. Here, we introduce the concept of phason engineering in metamaterials and use it as a vehicle to access and apply the quantum Hall physics in arbitrary dimensions. Using these specialized design principles, we fabricate a reconfigurable two-dimensional aperiodic acoustic crystal with a phason living on a 2-torus, giving us access to the four-dimensional quantum Hall physics. Also, we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as a function of the quasimomenta. We also demonstrate topological wave steering enabled by the Weyl physics of the three-dimensional boundaries.

Yang-Mills sources in biconformal gravity

Biconformal gravity, based on gauging of the conformal group to 2n dimensions, reproduces n-dim scale-covariant general relativity on the co-tangent bundle in any dimension. We generalize this result to include Yang-Mills matter sources formulated as SU(N) gauge theories with a twisted action on the full 2n-dimensional biconformal space. We show that the coupling of the sources to gravity does not stop the reduction of effective dimension 2n→n of the gravity theory, and instead forces the Yang-Mills source to reduce to ordinary n-dimensional Yang-Mills theory on the gravitating cotangent bundle, with the usual Yang-Mills energy tensor as gravitational source. The results apply as well to gravity with sources on Kähler manifolds and in double field theories.

A new kind of quantum field theory of (n – 1)-dimensional defects in 2n dimensions

I describe a project to open a new territory of quantum field theory where the fields live not on a space–time manifold but on certain complete metric spaces of (n − 1)-dimensional objects (defects) in a 2n-dimensional space-time M. These metric spaces are ‘quasi Riemann surfaces’; they are formally analogous to Riemann surfaces. Every construction of a 2d conformal field theory (CFT) is to give an analogous construction of a CFT on the quasi Riemann surfaces, and thereby a cft on M. The global symmetry group of the 2d CFT becomes a local gauge symmetry. Ordinary local quantum fields in space–time are constructed by restricting to small objects. The project is based on writing the free n-form in 2n dimensions as the 2d Gaussian model on the quasi Riemann surfaces.

ADHM construction of (anti-)self-dual instantons in 4n dimensions

The ADHM construction is a very strong scheme to construct the instantons in four dimensions. We study an ADHM construction of instantons in 4n (n ≥ 2) dimensions by generalizing this scheme. The higher-dimensional ADHM construction generates the 4n-dimensional (anti-)self-dual instantons which satisfy the (anti-)self-dual equation in 4n dimensions: F (n) = ± ∗4nF (n). Here F (n) is the nth wedge products of the gauge field strength 2-form F. We also show that our scheme reproduces the known 4n-dimensional one-instantons and there are multi-instanton solutions of the ’t Hooft type in the dilute instanton gas limit. Moreover, we discuss a Harrington-Shepard type caloron in 4n dimen-sions and this monopole limit.

Time machines and AdS solitons with negative mass

We show that in D=2n+1 dimensions, when mass is negative and all angular momenta are non-vanishing, Kerr and Kerr-AdS metrics describe smooth time machines, with no curvature singularity. Turning off the angular momenta appropriately can lead to static AdS solitons with negative mass. Setting zero the cosmological constant yields a class of Ricci-flat K\"ahler metrics in D=2n dimensions. We also show that Euclidean-signatured AdS solitons with negative mass can also arise in odd dimensions. We then construct time machines in D=5 minimal gauged supergravity that carry only magnetic dipole charges. Turning off the cosmological constant, the time machine becomes massless and asymptotically flat. It can be described as a constant time bundle over the Eguchi-Hanson instanton.

Self-duality in higher dimensions

Let ω be a 2-form on a 2n dimensional manifold. In previous work, we called ω “strong self-dual, if the eigenvalues of its matrix with respect to an orthonormal frame are equal in absolute value. In a series of papers, we showed that strong self-duality agrees with previous definitions; in particular if ω is strong self-dual, then, in 2n dimensions, ωn is proportional to its Hodge dual ω and in 4n dimensions, ωn is Hodge self-dual. We also obtained a local expression of the Bonan 4-form on 8 manifolds with Spin7 holonomy, as the sum of the squares of any orthonormal basis of a maximal linear subspace of strong self-dual 2-forms. In the present work we generalize the notion of strong self-duality to odd dimensional manifolds and we express the dual of the Fundamental 3-form 7 manifolds with G2 holonomy, as a sum of the squares of an orthonormal basis of a maximal linear subspace of strong self-dual 2-forms.

Near-symplectic 2n–manifolds

We give a generalization of the concept of near-symplectic structures to 2n dimensions. According to our definition, a closed 2-form \omega on a 2n-manifold M is near-symplectic, if it is symplectic outside a submanifold Z of codimension 3, where \omega^{n-1} vanishes. This extends the concept known in dimension 4. We depict how this notion relates to near-symplectic 4-manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration, or BLF, as a singular map with indefinite folds and Lefschetz-type singularities. We show that given such a map on a 2n-manifold over a symplectic base of codimension 2, then the total space carries such a near-symplectic structure, whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension--3 singular locus Z. We describe a splitting property of the normal bundle N_Z that is also present in dimension four. A tubular neighbourhood for Z is provided, which has as a corollary a Darboux-type theorem for near-symplectic forms.

Analyzing a stochastic time series obeying a second-order differential equation.

The stochastic properties of a Langevin-type Markov process can be extracted from a given time series by a Markov analysis. Also processes that obey a stochastically forced second-order differential equation can be analyzed this way by employing a particular embedding approach: To obtain a Markovian process in 2N dimensions from a non-Markovian signal in N dimensions, the system is described in a phase space that is extended by the temporal derivative of the signal. For a discrete time series, however, this derivative can only be calculated by a differencing scheme, which introduces an error. If the effects of this error are not accounted for, this leads to systematic errors in the estimation of the drift and diffusion functions of the process. In this paper we will analyze these errors and we will propose an approach that correctly accounts for them. This approach allows an accurate parameter estimation and, additionally, is able to cope with weak measurement noise, which may be superimposed to a given time series.

Higher derivative corrections to charged fluids in 2n dimensions

We study anomalous charged fluid in $2n$-dimensions ($n\geq 2$) up to sub-leading derivative order. Only the effect of gauge anomaly is important at this order. Using the Euclidean partition function formalism, we find the constraints on different sub-leading order transport coefficients appearing in parity-even and odd sectors of the fluid. We introduce a new mechanism to count different fluid data at arbitrary derivative order. We show that only the knowledge of independent scalar-data is sufficient to find the constraints. In appendix we further extend this analysis to obtain fluid data at sub-sub-leading order (where both gauge and gravitational anomaly contribute) for parity-odd fluid.

IMPROVING HAMILTONIAN ENERGY EQUATIONS ON THE KÄHLER JET BUNDLES

The aim of this paper is to improve Hamiltonian energy equations for the complex jet bundles using Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated in Hamiltonian energy equations and applications to complex bundle structures. The improved Hamiltonian energy equations have been applied to the presented example in order to test its performance. Moreover, we have presented a new concept of velocity and time dimensions for energy movement equations. Results showed that imaginary Hamiltonian values are opposed to real Hamiltonian values in 2n dimensions. Finally, this study showed a physical application and interpretation of velocity and time dimensions in Hamiltonian energy equations for given examples.

Anomalies and time reversal invariance in relativistic hydrodynamics: The second order and higher dimensional formulations

We present two new results on relativistic hydrodynamics with anomalies and external electromagnetic fields, "Chiral MagnetoHydroDynamics" (CMHD). First, we study CMHD in four dimensions at second order in the derivative expansion assuming the conformal/Weyl invariance. We classify all possible independent conformal second order viscous corrections to the energy-momentum tensor and to the U(1) current in the presence of external electric and/or magnetic fields, and identify eighteen terms that originate from the triangle anomaly. We then propose and motivate the following guiding principle to constrain the CMHD: the anomaly--induced terms that are even under the time reversal invariance should not contribute to the local entropy production rate. This allows us to fix thirteen out of the eighteen transport coefficients that enter the second order formulation of CMHD. We also relate one of our second order transport coefficients to the chiral shear waves. Our second subject is hydrodynamics with (N+1)-gon anomaly in an arbitrary 2N dimensions. The effects from the (N+1)-gon anomaly appear in hydrodynamics at (N-1)'th order in the derivative expansion, and we identify precisely N such corrections to the U(1) current. The time reversal constraint is powerful enough to allow us to find the analytic expressions for all transport coefficients. We confirm the validity of our results (and of the proposed guiding principle) by an explicit fluid/gravity computation within the AdS/CFT correspondence.

Omniscopes: Large area telescope arrays with onlyNlogNcomputational cost

We show that the class of antenna layouts for telescope arrays allowing cheap analysis hardware (with correlator cost scaling as N log N rather than N^2 with the number of antennas N) is encouragingly large, including not only previously discussed rectangular grids but also arbitrary hierarchies of such grids, with arbitrary rotations and shears at each level. We show that all correlations for such a 2D array with an n-level hierarchy can be efficiently computed via a Fast Fourier Transform in not 2 but 2n dimensions. This can allow major correlator cost reductions for science applications requiring exquisite sensitivity at widely separated angular scales, for example 21cm tomography (where short baselines are needed to probe the cosmological signal and long baselines are needed for point source removal), helping enable future 21cm experiments with thousands or millions of cheap dipole-like antennas. Such hierarchical grids combine the angular resolution advantage of traditional array layouts with the cost advantage of a rectangular Fast Fourier Transform Telescope. We also describe an algorithm for how a subclass of hierarchical arrays can efficiently use rotation synthesis to produce global sky maps with minimal noise and a well-characterized synthesized beam.

Partition functions of classical Heisenberg spin chains with arbitrary and different exchange

The classical Heisenberg model has been effective in modelling exchange interactions in molecular magnets. In this model, the partition function is important as it allows the calculation of the magnetization and susceptibility. For an ensemble of N-spin sites, this typically involves integrals in 2N dimensions. Here, for two-, three- and four-spin nearest neighbour open linear Heisenberg chains these integrals are reduced to sums of known functions, using a result due to Gegenbauer. For the case of the three- and four-spin chains, the sums are equivalent in form to the results of Joyce. The general result for an N-spin chain is also obtained.

Low-field susceptibility of classical Heisenberg chains with arbitrary and differentnearest-neighbour exchange

Interest in molecular magnets continues to grow, offering a link between the atomicand nanoscale properties. The classical Heisenberg model has been effective inmodelling exchange interactions in such systems. In this, the magnetizationand susceptibility are calculated through the partition function, where theHamiltonian contains both Zeeman and exchange energy. For an ensemble ofN spins, this requiresintegrals in 2N dimensions. For two, three and four spin nearest-neighbour chains these integralsreduce to sums of known functions. For the case of the three and four spin chains,the sums are equivalent to results of Joyce. Expanding these sums, the effect ofthe exchange on the linear susceptibility appears as Langevin functions withexchange term arguments. These expressions are generalized here to describe anN spin nearest-neighbour chain, where the exchange between each pair of nearest neighboursis different and arbitrary. For a common exchange constant, this reduces to the result ofFisher. The high-temperature expansion of the Langevin functions for the differentexchange constants leads to agreement with the appropriate high-temperature quantumformula of Schmidt et al, when the spin number is large. Simulations are presented foropen linear chains of three, four and five spins with up to four different exchangeconstants, illustrating how the exchange constants can be retrieved successfully.

Counterterms and dual holographic anomalies in CS gravity

The holographic Weyl anomaly associated to Chern-Simons gravity in 2n+1 dimensions is proportional to the Euler term in 2n dimensions, with no contributions from the Weyl tensor. We compute the holographic energy-momentum tensor associated to Chern-Simons gravity directly from the action, in an arbitrary odd-dimensional spacetime. We show, in particular, that the counterterms rendering the action finite contain only terms of the Lovelock type.

Nonplanar graphs and anomalies in chiral noncommutative gaugetheories

The AV (n) one-loop graphs are examined in a 2n-dimensional massless noncommutative gauge model in which both a U(1) axial gauge field A and a U(1) vector gauge field V have adjoint couplings to a Fermion field. A possible anomaly in the divergence of the n + 1 vertices is examined by considering the surface term that can possibly arise when shifting the loop momentum variable of integration. It is shown that despite the fact that the graphs are nonplanar, surface terms do arise in individual graphs, but that in 4n dimensions, a cancellation between the surface term contribution coming from pairs of graphs eliminates all anomalies, while in 4n + 2 dimensions such a cancellation cannot occur and an anomaly necessarily arises.PACS No.: 11.30.Rd

CALCULATION OF ANOMALIES IN GAUGE THEORIES USING A GEOMETRIC APPROACH

Gauge theory and anomalies which arise under quantization of these theories are formulated using differential geometric methods. The Wess–Zumino conditions are developed and some other relevant geometric properties of anomalies are mentioned. An elementary derivation of the non-Abelian anomaly in 2n dimensions is presented using this formalism.

AdS2n spaces from SO(2n − 1, 2) instantons

Conformally invariant sigma models in D = 2n dimensions with target non-compact SO(2n, 1) groups are reviewed. Despite the non-compact nature of the SO(2n, 1), the classical action and Hamiltonian are positive definite. Instanton field configurations obeying a double self-duality condition are found to correspond geometrically to conformal stereographic mappings of R2n into the Euclidean signature AdS2n spaces. The direct link among generalized Yang–Mills systems, double-self-dual generalized Einstein–Cartan backgrounds, AdS2n spaces and p-branes, when d = p + 1 = 2n = 4k is established.

Index of a family of lattice Dirac operators and its relation to the non-Abelian anomaly on the lattice.

In the continuum, a topological obstruction to the vanishing of the non-Abelian anomaly in 2n dimensions is given by the index of a certain Dirac operator in 2n+2 dimensions, or equivalently, the index of a 2-parameter family of Dirac operators in 2n dimensions. In this paper an analogous result is derived for chiral fermions on the lattice in the overlap formulation. This involves deriving an index theorem for a family of lattice Dirac operators satisfying the Ginsparg-Wilson relation. The index density is proportional to Lüscher's topological field in 2n+2 dimensions.