Gauge boson exchange in AdSd+1

We study the amplitude for exchange of massless gauge bosons between pairs of massive scalar fields in anti-de Sitter space. In the AdS/CFT correspondence this amplitude describes the contribution of conserved flavor symmetry currents to 4-point functions of scalar operators in the boundary conformal theory. A concise, covariant, Y2K compatible derivation of the gauge boson propagator in AdS_{d + 1} is given. Techniques are developed to calculate the two bulk integrals over AdS space leading to explicit expressions or convenient, simple integral representations for the amplitude. The amplitude contains leading power and sub-leading logarithmic singularities in the gauge boson channel and leading logarithms in the crossed channel. The new methods of this paper are expected to have other applications in the study of the Maldacena conjecture.     

Towards a quantum field theory of primitive string fields

We denote generating functions of massless even higher-spin fields ??primitive string fields?? (PSF??s). In an introduction we present the necessary definitions and derive propagators and currents of these PDF??s on flat space. Their off-shell cubic interaction can be derived after all off-shell cubic interactions of triplets of higher-spin fields have become known. Then we discuss four-point functions of any quartet of PSF??s. In subsequent sections we exploit the fact that higher-spin field theories in AdS _d+1 are determined by AdS/CFT correspondence from universality classes of critical systems in d-dimensional flat spaces. The O(N) invariant sectors of the O(N) vector models for 1 ?? N ??? play for us the role of ??standard models??, for varying N, they contain, e.g., the Ising model for N = 1 and the spherical model for N = ??. A formula for the masses squared that break gauge symmetry for these O(N) classes is presented for d = 3. For the PSF on AdS space it is shown that it can be derived by lifting the PSF on flat space by a simple kernel which contains the sum over all spins. Finally we use an algorithm to derive all symmetric tensor higher-spin fields. They arise from monomials of scalar fields by derivation and selection of conformal (quasiprimary) fields. Typically one monomial produces a multiplet of spin s conformal higher-spin fields for all s ?? 4, they are distinguished by their anomalous dimensions (in CFT ₃) or by theirmass (in AdS ₄). We sum over these multiplets and the spins to obtain ??string type fields??, one for each such monomial.     

Weyl anomaly of conformal higher spins on six-sphere

This paper is a sequel to arXiv:1309.0785 were we computed the Weyl anomaly a   (Euler density or logarithmic divergence on S^d$S^d$) coefficient for higher-derivative conformal higher spin field in d=4$d=4$ and shown that it matches the expression found in arXiv:1306.5242 by a “holographic” method from a ratio of massless higher spin determinants in AdS₅. Here we repeat the same computation in on 6-sphere and demonstrate that the result matches again the one following from AdS₇. We also discuss explicitly similar matching in the d=2$d=2$ case.     

An xp model on AdS2 spacetime

In this paper we formulate the xp model on the AdS₂ spacetime. We find that the spectrum of the Hamiltonian has positive and negative eigenvalues, whose absolute values are given by a harmonic oscillator spectrum, which in turn coincides with that of a massive Dirac fermion in AdS₂. We extend this result to generic xp models which are shown to be equivalent to a massive Dirac fermion on spacetimes whose metric depend of the xp   Hamiltonian. Finally, we construct the generators of the isometry group SO(2,1)$\mathit{SO}(2,1)$ of the AdS₂ spacetime, and discuss the relation with conformal quantum mechanics.     

Galilean conformal and superconformal symmetries

Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schr?dinger, Galilean conformal, and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c???? from relativistic conformal algebraO(d+1, 2) (d-number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D = 4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complexWeyl supercharges.     

Maximal symmetry and mass generation of Dirac fermions and gravitational gauge field theory in six-dimensional spacetime

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [1] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincar′e symmetry P(1,5)= SO(1,5) P~(1,5) as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.     

AdS branes from partial breaking of superconformal symmetries

It is shown how the static-gauge world-volume superfield actions of diverse superbranes on the AdS_d+1 superbackgrounds can be systematically derived from nonlinear realizations of the appropriate AdS supersymmetries. The latter are treated as superconformal symmetries of flat Minkowski superspaces of the bosonic dimension d. Examples include the N = 1 AdS₄ supermembrane, which is associated with the 1/2 partial breaking of the OSp(1|4) supersymmetry down to the N = 1, d = 3 Poincaré supersymmetry, and the T-duality related L3-brane on AdS₅ and scalar 3-brane on AdS₅ × S¹, which are associated with two different patterns of 1/2 breaking of the SU(2, 2|1) supersymmetry. Another (closely related) topic is the AdS/CFT equivalence transformation. It maps the world-volume actions of the codimension-one AdS_d+1 (super)branes onto the actions of the appropriate Minkowski (super)conformal field theories in the dimension d.     

On the origin of supergravity boundary terms in the AdS /CFT correspondence

The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of AdS_d+1 with a compact Einstein manifold ϵ and fermions on AdS_d+1, are considered. We also discuss conformal transformations of gravity fields on the boundary of AdS and show that they are induced by the isometrics of AdS.     

Higher-order singletons,partially massless fields,and their boundary values in the ambient approach

Using ambient space we develop a fully gauge and o(d,2)$\mathfrak{o}(d,2)$-covariant approach to boundary values of _AdSd+1${\mathit{AdS}}_{d+1}$ gauge fields. It is applied to the study of (partially) massless fields in the bulk and (higher-order) conformal scalars, i.e. singletons, as well as (higher-depth) conformal gauge fields on the boundary. In particular, we identify the corresponding generalized Fradkin–Tseytlin equations as obstructions to the extension of the off-shell boundary value to the bulk, generalizing the usual considerations for the holographic anomalies to the partially massless fields. We also relate the background fields for the higher-order singleton to the boundary values of partially massless fields and prove the appropriate generalization of the Flato–Fronsdal theorem, which is in agreement with the known structure of symmetries for the higher-order wave operator. All these facts support the following generalization of the higher-spin holographic duality: the O(N)$O(N)$ model at a multicritical isotropic Lifshitz point should be dual to the theory of partially massless symmetric tensor fields described by the Vasiliev equations based on the higher-order singleton symmetry algebra.     

Hidden conformal symmetry of self-dual warped AdS<Subscript>3</Subscript> black holes in topological massive gravity

We extend the recently proposal of hidden conformal symmetry to the self-dual warped AdS₃ black holes in topological massive gravity. It is shown that the wave equation of massive scalar field with sufficient small angular momentum can be reproduced by the SL(2, R) Casimir quadratic operator. Due to the periodic identification in the φ direction, it is found that only the left section of hidden conformal symmetry is broken to U(1), while the right section is unbroken, which only gives the left temperature of dual CFT. As a check of the dual CFT conjecture of self-warped AdS₃ black hole, we further compute the Bekenstein–Hawking entropy and absorption cross section and quasinormal modes of scalar field perturbation and show these are just of the forms predicted by the dual CFT.     

Type IIB string theory on AdS5×T

We study Kaluza-Klein spectrum of type IIB string theory compactified on AdS₅×T^nn′ in the context of AdS/CFT correspondence. We examine some of the modes of the complexified 2 form potential as an example and show that for the states at the bottom of the Kaluza-Klein tower the corresponding d=4 boundary field operators have rational conformal dimensions. The masses of some of the fermionic modes in the bottom of each tower as functions of the R charge in the boundary conformal theory are also rational. Furthermore the modes in the bottom of the towers originating from q forms on T¹¹ can be put in correspondence with the BRS cohomology classes of the c=1 non critical string theory with ghost number q.     

Representations of (1,0) and (2,0) Superconformal Algebras in Six Dimensions: Massless and Short Superfields

We construct unitary representations of (1,0) and (2,0) superconformal algebras in six dimensions by using superfields defined on harmonic superspaces with coset manifolds USp(2n)/[U(1)]ⁿ, n=1, 2. In the spirit of the AdS₇/CFT₆ correspondence, massless conformal fields correspond to supersingletons in AdS₇. By tensoring them we produce all short representations corresponding to 1/2 and 1/4 BPS anti-de Sitter bulk states of which massless bulk representations are particular cases.     

Relativistic non-Hamiltonian mechanics

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_μu^μ + c² = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton’s principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton’s principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton’s principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton’s principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.     

Klein Paradox and Disorder-Induced Delocalization of Dirac in Quasiparticles in One-Dimensional Systems

Dirac particle penetration is studied theoretically with Dirac equation in one-dimensional systems. We investigate a one-dimensional system with N barriers where both barrier height and well width are constants randomlydistributed in certain range. The one-parameter scaling theory for nonrelativistic particles is still valid for massive Dirac particles. In the same disorder sample, we find that the localization length of relativistic particles is always larger than that of nonrelativistic particles and the transmission coefficient related to incident particle in both cases fits the form T ∽ exp(-α_L). More interesting, massless relativistic particles are entirely delocalized no matter how big the energy of incident particles is.     

Dynamical and symmetry groups of the SU(2) Kepler problem

It is well known that the MIC–Kepler problem, an extension of the three-dimensional Kepler problems, admits the same dynamical and symmetry groups as the Kepler problem. This paper aims to study dynamical and symmetry groups of the SU(2) Kepler problem, where the SU(2) Kepler problem is defined to be the dynamical system reduced from the eight-dimensional conformal Kepler problem through an SU(2) symmetry and turns out to be an extension of the five-dimensional Kepler problem. It is shown that the SU(2) Kepler problem admits a dynamical group SO^*(8) and that the phase space of the SU(2) Kepler problem is symplectomorphic with a co-adjoint orbit of SO^*(8), on which the Kirillov–Kostant–Souriau form is defined. It is further shown that the subgroups, SU(4), SU^*(4), and Sp(2)×_SR⁵, of SO^*(8) provide the symmetry groups, SU(4)/Z₂SO(6), SU^*(4)/Z₂SO₀(1,5), and (Sp(2)×_SR⁵)/Z₂SO(5)×_SR⁵, of the SU(2) Kepler problem with negative, positive, and zero energies, respectively, where ×_S denotes a semi-direct product. Furthermore, constants of motion for the SU(2) Kepler problem are found together with their Poisson brackets. The symmetry Lie algebra formed by constants of motion is shown to be isomorphic with so(6)su(4), so(1,5)su^*(4), or so(5)_SR⁵sp(2)_SR⁵, depending on whether the energy is negative, positive, or zero, where _S denotes a semi-direct sum. These Lie algebras are subalgebras of so^*(8)so(2,6).     

Protecting the Conformal Symmetry via Bulk Renormalization on Anti deSitter Space

The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field j{\varphi} on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field f{\phi} on d + 1-dimensional Anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the one-loop “fish diagram” on AdS₄ by differential renormalization, and calculate the anomalous dimension of the composite boundary field j²{\varphi^2} with bulk interaction kf⁴{\kappa \phi^4}.     

Electric/magnetic deformations of S3 and AdS3, and geometric cosets

We analyze asymmetric marginal deformations of SU(2)_k and SL(2,ℝ)_k WZW models. These appear in heterotic string backgrounds with non‐vanishing Neveu–Schwarz three‐forms plus electric or magnetic fields, depending on whether the deformation is elliptic, hyperbolic or parabolic. Asymmetric deformations create new families of exact string vacua. The geometries which are generated in this way, deformed S³ or AdS₃, include in particular geometric cosets such as S², AdS₂ or H₂. Hence, the latter are consistent, exact conformal sigma models, with electric or magnetic backgrounds. We discuss various geometric and symmetry properties of the deformations at hand as well as their spectra and partition functions, with special attention to the supersymmetric AdS₂ × S² background. We also comment on potential holographic applications.     

Non-commutative Geometry in Massless and Massive Particles

In this paper, we study the symmetries of massless and massive particles action. By considering the non-commutative space-time, we find appropriate non-commutative relation for relativistic particles which leaves invariant the non-commutative Minkowski space-time. We show that non-commutativity break the scale and conformal invariance in massless and massive action. So, in non-commutative space-time the massless and massive particles have same symmetry.