Arbitrary Spacetime Dimensions Research Articles

Scattering of massless particles in arbitrary dimensions.

We present a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimensions. The new formula for the scattering of n particles is given by an integral over the positions of n points on a sphere restricted to satisfy a dimension-independent set of equations. The integrand is constructed using the reduced Pfaffian of a 2n×2n matrix, Ψ, that depends on momenta and polarization vectors. In its simplest form, the gravity integrand is a reduced determinant which is the square of the Pfaffian in the Yang-Mills integrand. Gauge invariance is completely manifest as it follows from a simple property of the Pfaffian.

Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles

We generalize the scattering equations to include both massless and massive particles. We construct an expression for the tree-level n-point amplitude with n-2 gluons or gravitons and a pair of massive scalars in arbitrary spacetime dimension as a sum over the (n-3)! solutions of the scattering equations, a la Cachazo, He, and Yuan. We derive the BCJ relations obeyed by these massive amplitudes.

The polynomial form of the scattering equations

The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reformulated in polynomial form. The scattering equations for $N$ particles are shown to be equivalent to a Moebius invariant system of $N-3$ equations, $\tilde h_m=0$, $2 \leq m \leq N-2$, in $N$ variables, where $\tilde h_m$ is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Moebius invariance appropriately, yields polynomial equations $h_m=0$, $1 \leq m \leq N-3$, in $N-3$ variables, where $h_m$ has degree $m$. The linearity of the equations in the individual variables facilitates computation, e.g the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $\tilde h_m$ and $h_m$. The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.

Scattering of massless particles: scalars, gluons and gravitons

In a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U(N) color structures while the second is a Pfaffian. The S-matrix of a U(N)xU(N') cubic scalar theory is obtained by simply replacing the Pfaffian with a U(N') version of the previous U(N) factor. Given that gravity amplitudes are obtained by replacing the U(N) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. We find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials.

Interpreting canonical tensor model in minisuperspace

Canonical tensor model is a theory of dynamical fuzzy spaces in arbitrary space–time dimensions. Examining its simplest case, we find a connection to a special case of minisuperspace model of general relativity in arbitrary dimensions. This is a first step in interpreting variables in canonical tensor model based on the known language of general relativity.

Unified description for $$\kappa $$ κ -deformations of orthogonal groups

In this paper we provide universal formulas describing Drinfeld-type quantization of inhomogeneous orthogonal groups determined by a metric tensor of an arbitrary signature living in a spacetime of arbitrary dimension. The metric tensor does not need to be in diagonal form and \(\kappa \)-deformed coproducts are presented in terms of classical generators. It opens the possibility for future applications in deformed general relativity. The formulas depend on the choice of an additional vector field which parametrizes classical \(r\)-matrices. Non-equivalent deformations are then labeled by the corresponding type of stability subgroups. For the Lorentzian signature it covers three (non-equivalent) Hopf-algebraic deformations: time-like, space-like (a.k.a. tachyonic) and light-like (a.k.a. light-cone) quantizations of the Poincaré algebra. Finally the existence of the so-called Majid–Ruegg (non-classical) basis is reconsidered.

Forced fluid dynamics from gravity in arbitrary dimensions

We consider long wavelength solutions to the Einstein-dilaton system with negative cosmological constant which are dual, under the AdS/CFT correspondence, to solutions of the conformal relativistic Navier-Stokes equations with a dilaton-dependent forcing term. Certain forced fluid flows are known to exhibit turbulence; holographic duals of forced fluid dynamics are therefore of particular interest as they may aid efforts towards an explicit model of holographic steady state turbulence. In recent work, Bhattacharyya et al. have constructed long wavelength asymptotically locally AdS5 bulk space-times with a slowly varying boundary dilaton field which are dual to forced fluid flows on the 4–dimensional boundary. In this paper, we generalise their work to arbitrary space-time dimensions; we explicitly compute the dual bulk metric, the fluid dynamical stress tensor and Lagrangian to second order in a boundary derivative expansion.

Renormalization group flow and symmetry restoration in de Sitter space

We compute the renormalization group flow of O(N) scalar field theories in de Sitter space using nonperturbative renormalization group techniques in the local potential approximation. We obtain the flow of the effective potential on superhorizon scales for arbitrary space–time dimension D=d+1. We show that, due to strong infrared fluctuations, the latter is qualitatively similar to the corresponding one in Euclidean space RD with D=0. It follows that spontaneously broken symmetries are radiatively restored in any space–time dimension and for any value of N.

Note on black hole no hair theorems for massive forms and spin-1/2fields

We give a proof of the non-perturbative no hair theorems for a massive 2-form field with 3-form field strength for general stationary axisymmetric and static (anti)-de Sitter or asymptotically flat black hole spacetimes with some suitable geometrical properties. The generalization of this result for higher form fields is discussed. Next, we discuss the perturbative no hair theorems for massive spin-1/2 fields for general static backgrounds with electric or magnetic charge. Some generalization of this result for stationary axisymmetric spacetimes are also discussed. All calculations are done in arbitrary spacetime dimensions.

Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynman’s master integrals

We investigate the structure of the constraints on three-point correlation functions emerging when conformal invariance is imposed in momentum space and in arbitrary space-time dimensions, presenting a derivation of their solutions for arbitrary scalar operators. We show that the differential equations generated by the requirement of symmetry under special conformal transformations coincide with those satisfied by generalized hypergeometric functions (Appell's functions). Combined with the position space expression of this correlator, whose Fourier transform is given by a family of generalized Feynman (master) integrals, the method allows to derive the expression of such integrals in a completely independent way, bypassing the use of Mellin-Barnes techniques, which have been used in the past. The application of the special conformal constraints generates a new recursion relation for this family of integrals.

Off-shell Green functions at one-loop level in Maxwell-Chern-Simons quantum electrodynamics

We derive the off-shell photon propagator and fermion-photon vertex at one-loop level in Maxwell-Chern-Simons quantum electrodynamics in arbitrary covariant gauge, using four-component spinors with parity-even and parity-odd mass terms for both fermions and photons. We present our results using a basis of two, three and four point integrals, some of them not known previously in the literature. These integrals are evaluated in arbitrary space-time dimensions so that we reproduce results derived earlier under certain limits.

On unitary subsectors of polycritical gravities

We study higher-derivative gravity theories in arbitrary space-time dimension d with a cosmological constant at their maximally critical points where the masses of all linearized perturbations vanish. These theories have been conjectured to be dual to logarithmic conformal field theories in the (d-1)-dimensional boundary of an AdS solution. We determine the structure of the linearized perturbations and their boundary fall-off behaviour. The linearized modes exhibit the expected Jordan block structure and their inner products are shown to be those of a non-unitary theory. We demonstrate the existence of consistent unitary truncations of the polycritical gravity theory at the linearized level for odd rank.

Solving the 3D Ising model with the conformal bootstrap

We study the constraints of crossing symmetry and unitarity in general 3D conformal field theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and product-expansion coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Interacting spin-2 fields

We construct consistent theories of multiple interacting spin-2 fields in arbitrary spacetime dimensions using a vielbein formulation. We show that these theories have the additional primary constraints needed to eliminate potential ghosts, to all orders in the fields, and to all orders beyond any decoupling limit. We postulate that the number of spin-2 fields interacting at a single vertex is limited by the number of spacetime dimensions. We then show that, for the case of two spin-2 fields, the vielbein theory is equivalent to the recently proposed theories of ghost-free massive gravity and bi-metric gravity. The vielbein formulation greatly simplifies the proof that these theories have an extra primary constraint which eliminates the Boulware-Deser ghost.

Thermodynamic Geometry and Topological Einstein–Yang–Mills Black Holes

From the perspective of the statistical fluctuation theory, we explore the role of the thermodynamic geometries and vacuum (in)stability properties for the topological Einstein–Yang–Mills black holes. In this paper, from the perspective of the state-space surface and chemical Weinhold surface of higher dimensional gravity, we provide the criteria for the local and global statistical stability of an ensemble of topological Einstein–Yang–Mills black holes in arbitrary spacetime dimensions D ≥ 5. Finally, as per the formulations of the thermodynamic geometry, we offer a parametric account of the statistical consequences in both the local and global fluctuation regimes of the topological extremal Einstein–Yang–Mills black holes.

RG FLOW OF TRANSPORT QUANTITIES

The RG flow equation of various transport quantities are studied in arbitrary space–time dimensions, in the fixed as well as fluctuating background geometry both for the Maxwellian and DBI type of actions. The regularity condition on the flow equation of the conductivity at the horizon for the DBI action reproduces naturally the leading order result of Hartnoll et al. [J. High Energy Phys. 04, 120 (2010)]. Motivated by the result of van der Marel et al. [Science 425, 271 (2003], we studied, analytically, the conductivity versus frequency plane by dividing it into three distinct parts: ω < T, ω > T and ω ≫ T. In order to compare, we choose (3+1)-dimensional bulk space–time for the computation of the conductivity. In the ω < T range, the conductivity does not show up the Drude like form in any space–time dimensions. In the ω > T range and staying away from the horizon, for the DBI action with unit dynamical exponent, nonzero magnetic field and charge density, the conductivity goes as ω-2/3, whereas the phase of the conductivity, goes as, arctan ( Im σxx/ Re σxx) = π/6 and arctan ( Im σxy/ Re σxy) = -π/3. There exists a universal quantity at the horizon that is the phase angle of conductivity, which either vanishes or an integral multiple of π. Furthermore, we calculate the temperature dependence to the thermoelectric and the thermal conductivity at the horizon. The charge diffusion constant for the DBI action is studied.

Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes

We investigate whether a symmetric, second order Wick polynomial T of a free scalar field, including derivatives, is essentially self-adjoint on the natural (Wightman) domain in a quasi-free (i.e., Fock space) Hadamard representation. Our results apply to arbitrary spacetime dimensions d ≥ 2, but we do restrict our attention to the case where T is smeared with a test-function from a particular class \documentclass[12pt]{minimal}\begin{document}${\cal S}$\end{document}S, namely the class of sums of squares of test-functions. (This class of smearing functions is smaller than the class of all non-negative test-functions – a fact which follows Hilbert's theorem.) Combining techniques from microlocal and functional analysis we prove that T is essentially self-adjoint if it is a Wick square (without derivatives). In the presence of derivatives we prove the weaker result that T is essentially self-adjoint if its compression to the one-particle Hilbert space is essentially self-adjoint. For the latter result, we use Wüst's theorem and an application of Konrady's trick in Fock space. In the presence of derivatives, we also prove that one has some control over the spectral projections of T, by describing it as the strong graph limit of a sequence of essentially self-adjoint operators.

Warped de Sitter compactifications in the scalar–tensor theory

We present new solutions of warped compactifications in the higher-dimensional gravity coupled to the scalar and the form field strengths. These solutions are constructed in the D-dimensional spacetime with matter fields, with the internal space that has a finite volume. Our solutions give explicit examples where the cosmological constant or 0-form field strength leads to a de Sitter spacetime in arbitrary dimensions.

Black holes and boson stars with one Killing field in arbitrary odd dimensions

We extend the recent $D=5$ results of Dias, Horowitz, and Santos by finding asymptotically anti--de Sitter rotating black hole and boson star solutions with scalar hair in arbitrary odd spacetime dimension. Both the black holes and the boson stars are invariant under a single Killing vector field which corotates with the scalar field and, in the black hole case, is tangent to the generator of the horizon. Furthermore, we explicitly construct boson star and small black hole (${r}_{+}\ensuremath{\ll}\ensuremath{\ell}$) solutions perturbatively assuming a small amplitude for the scalar field, resulting in solutions valid for low energies and angular momenta. We find that just as in $D=5$, the angular momentum is primarily carried by the scalar field in $D>5$, whereas unlike $D=5$ the energy is also primarily carried by the scalar field in $D>5$; the thermodynamics in $D=5$ are governed by both the black hole and scalar field whereas in $D>5$ they are governed primarily by the scalar field alone. We focus on cataloging these solutions for the spacetime dimensions of interest in string theory, namely $D=5,7,9,11$.

All the fundamental massless fields in bosonic string theory

AbstractA systematic analysis of the massless fields in the mass spectra of bosonic strings is carried in arbitrary spacetime dimension D > 2. The emphasis is put on the derivations of their propagators, their polarization aspects and the underlying constraints involved. The treatment is given, in the presence of external sources, in the celebrated Coulomb gauge for the second rank tensors and vector fields, which ensures positivity – a result which is also established in the process. No constraints are imposed on the external sources so that their components may be varied independently generating complete expressions for the propagators. This latter condition is an important one in the generation of dynamical theories with constraints involving such modifications as Faddeev‐Popov factors.