Klein Reduction Research Articles

Einstein–Bumblebee-dilaton black hole solution

We obtain new black hole solutions in a Einstein–Bumblebee-scalar theory. By starting with a Einstein–Bumblebee theory in (D+d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D+d$$\\end{document}) dimensions, the scalar dilaton field and its interaction with the gravitational and Bumblebee fields are obtained by Kaluza–Klein (KK) reduction over the extra dimensions. Considering the effects of both the Bumblebee vacuum expectation value (VEV) and the fluctuations over the VEV, we obtained new charged solutions in (3+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3+1)$$\\end{document} dimensions. For a vanishing dilaton, the black hole turned out to be a charged de Sitter-Reissner–Nordstrom solution, where the transverse mode is the Maxwell field and the longitudinal mode is the cosmological constant. The stability of these new solutions is investigated by means of the analysis of the black hole thermodynamics. The temperature, entropy, and heat capacity show that these modified black holes are thermodynamic stable.

A braneworld model in a massive gravity

A Randall–Sundrum-like braneworld scenario is constructed in a 5D extension of the Lorentz-violating massive gravity. The gauge hierarchy problem is solved in current model. The linear perturbations are calculated, and it is found that the tensor and vector perturbations are robust and free from the ghost and tachyonic instabilities, however, the scalar perturbation is a ghost filed. After Kaluza–Klein reduction, all the tensor, vector and scalar modes are massive and possess the mass splitting of order of TeV in their respective mass spectra. The massive ground states of tensor and scalar modes propagate only along the brane, however, the vector ground state is absent in the mass spectrum. By introducing the Goldberger-Wise mechanism to stabilize the extra dimension, the 4D effective theory on the brane includes a nearly massless graviton plus three towers of very massive spin-2, spin-1 and ghost spin-0 particles.

Topology in thermodynamics of regular black strings with Kaluza–Klein reduction

We study the topological defects in the thermodynamics of regular black strings (from a four-dimensional perspective) that is symmetric under the double Wick rotation and constructed in the high-dimensional spacetime with an extra dimension compactified on a circle. We observe that the thermodynamic phases of regular black strings can be topologically classified by the positive and negative winding numbers (at the defects) which correspond to the thermodynamically stable and unstable branches. This topological classification implies a phase transition due to the decay of a thermodynamically unstable regular black string to another which is thermodynamically stable. We confirm these topological properties of the thermodynamics of regular black strings by investigating their free energy, heat capacity, and Ruppeiner scalar curvature of the state space. The Ruppeiner scalar curvature of regular black strings is found to be always negative, implying that the interactions among the microstructures of regular black strings are only attractive.

Quasinormal Modes of a Charged Black Hole with Scalar Hair

Based on the five-dimensional Einstein–Maxwell theory, Bah et al. constructed a singularity-free topology star/black hole [Phys. Rev. Lett. 126, 151101 (2021)]. After performing the Kaluza–Klein reduction, i.e., integrating the extra space dimension, it can obtain an effective four-dimensional spherically static charged black hole with scalar hair. In this paper, we study the quasinormal modes (QNMs) of the scalar, electromagnetic, and gravitational fields in the background of this effective four-dimensional charged black hole. The radial parts of the perturbed fields all satisfy a Schrödinger-like equation. Using the asymptotic iteration method, we obtain the QNM frequencies semianalytically. For low-overtone QNMs, the results obtained using both the asymptotic iteration method and the Wentzel–Kramers–Brillouin approximation method agree well. In the null coordinates, the evolution of a Gaussian package is also studied. The QNM frequencies obtained by fitting the evolution data also agree well with the results obtained using the asymptotic iteration method.

Momentum gauge fields from curved momentum space through Kaluza–Klein reduction

In this work we investigate the relation between curved momentum space and momentum-dependent gauge fields. While the former is a classic idea that has been shown to be tied to minimal-length models, the latter constitutes a relatively recent development in quantum gravity phenomenology. In particular, the gauge principle in momentum space amounts to a modification of the position operator of the form akin to a gauge-covariant derivative in momentum space according to the minimal coupling prescription. Here, we derive both effects from a Kaluza–Klein reduction of a higher-dimensional geometry exhibiting curvature in momentum space. The interplay of the emerging gauge fields as well as the remaining curved momentum space lead to modifications of the Heisenberg algebra. While the gauge fields imply Moyal-type noncommutativity dependent on the analogue field strength tensor, the dimensionally reduced curved momentum space geometry translates to a Snyder-type noncommutative geometry.

Noncommutative $$SO(2,3)_{\star }$$ gauge theory of gravity

Topological gravity (in the sense that it is metric-independent) in a 2n-dimensional spacetime can be formulated as a gauge field theory for the AdS gauge group $$SO(2,2n-1)$$ by adding a multiplet of scalar fields. These scalars can break the gauge invariance of the topological gravity action, thus making a connection with Einstein’s gravity. This review is about a noncommutative (NC) star-product deformation of the four-dimensional AdS gauge theory of gravity, including Dirac spinors and the Yang–Mills field. In general, NC actions can be expanded in powers of the canonical noncommutativity parameter $$\theta$$ using the Seiberg–Witten map. The leading-order term of the expansion is the classical action, while the higher-order $$\theta$$ -dependent terms are interpreted as new types of coupling between classical fields due to spacetime noncommutativity. We study how these perturbative NC corrections affect the field equations of motion and derive some phenomenological consequences, such as NC-deformed Landau levels of an electron. Finally, we discuss how topological gravity in four dimensions (both classical and noncommutative) appears as a low-energy sector of five-dimensional Chern–Simons gauge theory in the sense of Kaluza–Klein reduction.

Shadow of a charged black hole with scalar hair

Seeking singularity free solutions are important for further understanding black holes in quantum level. Recently, a five-dimensional singularity free black hole/topological star was constructed (Bah and Heidmann in Phys Rev Lett 126:151101, 2021). Through the Kaluza–Klein reduction, an effective four-dimensional static spherically symmetric charged black hole with scalar hair can be obtained. In this paper, we study shadow of this charged black hole with scalar hair in terms of four kinds of observers, i.e., static observers, surrounding observers, freely falling observers, and escaping observers in four-dimensional spacetime. For a spherically symmetric black hole, the shadow is circular for any observer, but the shadow size depends on the motion status of the observer. On the other hand, the effect of plasma is also investigated by a simple model. The radius of the photon sphere depends on the plasma model. Most importantly, we find that the shadow sizes do not monotonically decrease with r in some cases.

Monstrous M-Theory

In 26+1 space–time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to 25+1 dimensions. This theory generalizes M-theory in many respects, and we name it Monstrous M-theory, or M2-theory. Upon Kaluza–Klein reduction to 25+1 dimensions, the M2-theory spectrum irreducibly splits as 1 ⊕ 196,883, where 1 is identified with the dilaton, and 196,883 is the dimension of the smallest non-trivial representation of the Monster. This provides a field theory explanation of the lowest instance of the Monstrous Moonshine, and it clarifies the definition of the Monster as the automorphism group of the Griess algebra by showing that such an algebra is not merely a sum of unrelated spaces, but descends from massless states for M2-theory, which includes Horowitz and Susskind’s bosonic M-theory as a subsector. Further evidence is provided by the decomposition of the coefficients of the partition function of Witten’s extremal Monster SCFT in terms of representations of SO24, the massless little group in 25+1; the purely bosonic nature of the involved SO24-representations may be traced back to the unique feature of 24 dimensions, which allow for a non-trivial generalization of the triality holding in 8 dimensions. Last but not least, a certain subsector of M2-theory, when coupled to a Rarita–Schwinger massless field in 26+1, exhibits the same number of bosonic and fermionic degrees of freedom; we cannot help but conjecture the existence of a would-be N=1 supergravity theory in 26+1 space–time dimensions.

Kaluza–Klein reductions of maximally supersymmetric five-dimensional Lorentzian spacetimes

A recent study of filtered deformations of (graded subalgebras of) the minimal five-dimensional Poincaré superalgebra resulted in two classes of maximally supersymmetric spacetimes. One class are the well-known maximally supersymmetric backgrounds of minimal five-dimensional supergravity, whereas the other class does not seem to be related to supergravity. This paper is a study of the Kaluza–Klein (KK) reductions to four dimensions of this latter class of maximally supersymmetric spacetimes. We classify the Lorentzian and Riemannian KK reductions of these backgrounds, determine the fraction of the supersymmetry preserved under the reduction and in most cases determine explicitly the geometry of the four-dimensional quotient. Among the many supersymmetric quotients found, we highlight a number of novel non-homogeneous four-dimensional Lorentzian spacetimes admitting N = 1 supersymmetry, whose supersymmetry algebra is not a filtered deformation of any graded subalgebra of the four-dimensional N = 1 Poincaré superalgebra. Any of these four-dimensional Lorentzian spacetimes may serve as the arena for the construction of new rigidly supersymmetric field theories.

Noncommutative D=5 Chern–Simons gravity: Kaluza–Klein reduction and chiral gravitational anomaly

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter theta using the Seiberg–Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical (theta =0) action while higher-order terms give us theta -dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern–Simons (CS) gauge and gravity theories, showing that non-trivial theta -dependence exists only for spacetime dimensions Dge 5, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in theta . Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza–Klein reduction procedure on the NC D=5 CS theory for the conformal gauge group SO(4, 2), to obtain an effective, theta -dependent four-dimensional theory of gravity that has Einstein–Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing theta -dependent Pontryagin density.

11D supergravity on AdS 4 × S 7 versus AdS 7× S 4

The maximally supersymmetric Freund–Rubin vacua for 11 dimensional supergravity, namely AdS 4 × S 7 and AdS 7 × S 4, admit an analytic continuation to S 4 × S 7. From the full harmonic expansions on S 4 × S 7, it is shown that by analytical continuation to either AdS 4, or to AdS 7, the detailed structure of the Kaluza–Klein spectrum can be obtained for both vacua in a unified manner. The results are shown to be related by a simple rule which interchanges the spacetime and internal space representations. We also obtain the linearized field equations for the singletons and doubletons but they can be gauged away by fixing certain Stuckelberg shift symmetries inherited from the Kaluza–Klein reduction.

Hitchhiker’s guide to Courant algebroid relations

Courant algebroids provide a useful mathematical tool (not only) in string theory. It is thus important to define and examine their morphisms. To some extent, this was done before using an analogue of canonical relations known from symplectic geometry. However, it turns out that applications in physics require a more general notion. We aim to provide a self-contained and detailed treatment of Courant algebroid relations and morphisms. A particular emphasis is placed on providing enough motivating examples. In particular, we show how Poisson–Lie T-duality and Kaluza–Klein reduction of supergravity can be interpreted as Courant algebroid relations compatible with generalized metrics (generalized isometries).

Consistent KK truncations for M5-branes wrapped on Riemann surfaces

We construct a consistent Kaluza–Klein reduction of D = 11 supergravity on , where or H2, or a quotient thereof, at the level of the bosonic fields. The result is a gauged N = 4, D = 5 supergravity theory coupled to three vector multiplets, with the gauging lying in an subgroup of the global symmetry group of the ungauged theory. For , the D = 5 theory has a maximally supersymmetric AdS5 vacuum which uplifts to the known solution of D = 11 supergravity corresponding to M5-branes wrapping a Riemann surface with genus greater than one and dual to an N = 2 SCFT in d = 4. For , we find two AdS5 solutions, one of which is new, and both of which are unstable. There is an additional subtruncation to an N = 2 gauged supergravity coupled to two vector multiplets, with very special real manifold , and a single hypermultiplet, with quaternionic Kähler manifold and gauging associated with an subgroup.

Kaluza–Klein reduction on a maximally non-Riemannian space is moduli-free

We propose a novel Kaluza–Klein scheme which assumes the internal space to be maximally non-Riemannian, meaning that no Riemannian metric can be defined for any subspace. Its description is only possible through Double Field Theory but not within supergravity. We spell out the corresponding Scherk–Schwarz twistable Kaluza–Klein ansatz, and point out that the internal space prevents rigidly any graviscalar moduli. Plugging the same ansatz into higher-dimensional pure Double Field Theory and also to a known doubled-yet-gauged string action, we recover heterotic supergravity as well as heterotic worldsheet action. In this way, we show that 1) supergravity and Yang–Mills theory can be unified into higher-dimensional pure Double Field Theory, free of moduli, and 2) heterotic string theory may have a higher-dimensional non-Riemannian origin.

A Kaluza–Klein reduction of super-integrable systems

Given a super-integrable system in $n$ degrees of freedom, possessing an integral which is linear in momenta, we use the "Kaluza-Klein construction" in reverse to reduce to a lower dimensional super-integrable system. We give two examples of a reduction from 3 to 2 dimensions. The constant curvature metric (associated with the kinetic energy) is the same in both cases, but with two different super-integrable extensions. For these, we use different elements of the algebra of isometries of the kinetic energy to reduce to $2-$dimensions. Remarkably, the isometries of the reduced space can be derived from those of the $3-$dimensional space, even though it requires the use of {\em quadratic} expressions in momenta.

Boosted Kaluza–Klein magnetic monopole

We consider a Kaluza–Klein vacuum solution which is closely related to the Gross–Perry–Sorkin (GPS) magnetic monopole. The solution can be obtained from the Euclidean Taub–NUT solution with an extra compact fifth spatial dimension within the formalism of Kaluza–Klein reduction. We study its physical properties as appearing in (3+1) spacetime dimensions, which turns out to be a static magnetic monopole. We then boost the GPS magnetic monopole along the extra dimension, and perform the Kaluza–Klein reduction. The resulting four-dimensional spacetime is a rotating stationary system, with both electric and magnetic fields. In fact, after the boost the magnetic monopole turns into a string connected to a dyon.

Teleparallel conformal invariant models induced by Kaluza–Klein reduction

We study the extensions of teleparallism in the Kaluza–Klein (KK) scenario by writing the analogous form to the torsion scalar in terms of the corresponding antisymmetric tensors, given by , in the four-dimensional new general relativity (NGR) with arbitrary coefficients a, b and c. After the KK dimensional reduction, the Lagrangian in the Einstein-frame can be realized by taking 2a + b + c = 0 with the ghost-free condition for the one-parameter family of teleparallelism. We demonstrate that the pure conformal invariant gravity models can be constructed by the requirements of 2a + b = 0 and c = 0. In particular, the torsion vector can be identified as the conformal gauge field, while the conformal gauge theory can be obtained by 2a + b + 4c = 0 or 2a + b = 0, which is described on the Weyl–Cartan geometry Y4 with the ghost-free conditions 2a + b + c > 0 and . We also consider the weak field approximation and discuss the non-minimal coupled term of the scalar current and torsion vector. For the conformal invariant models with 2a + b = 0, we find that only the anti-symmetric tensor field is allowed rather than the symmetric one.

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in {mathcal {N}} = 6 U_k(N)times mathrm{U}_{-k}(N) mass-deformed ABJM theory and the LLM geometries with {mathbb {Z}}_k orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension Delta = 1, which is given by langle {mathcal {O}}^{(Delta =1)}rangle = N^{frac{3}{2}} , f_{(Delta =1)}, for large N and k=1. Here the factor f_{(Delta )} is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of kne 1 for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.

Black holes and stars in Horndeski theory

We review black hole and star solutions for Horndeski theory. For non-shift symmetric theories, black holes involve a Kaluza–Klein reduction of higher dimensional Lovelock solutions. On the other hand, for shift symmetric theories of Horndeski and beyond Horndeski, black holes involve two classes of solutions: those that include, at the level of the action, a linear coupling to the Gauss–Bonnet term and those that involve time dependence in the galileon field. We analyze the latter class in detail for a specific subclass of Horndeski theory, discussing the general solution of a static and spherically symmetric spacetime. We then discuss stability issues, slowly rotating solutions as well as black holes coupled to matter. The latter case involves a conformally coupled scalar field as well as an electromagnetic field and the (primary) hair black holes thus obtained. We review and discuss the recent results on neutron stars in Horndeski theories.

Newton–Cartan, Galileo–Maxwell and Kaluza–Klein

We study Kaluza–Klein reduction in Newton–Cartan gravity. In particular we show that dimensional reduction and the nonrelativistic limit commute. The resulting theory contains Galilean electromagnetism and a nonrelativistic scalar. It provides the first example of back-reacted couplings of scalar and vector matter to Newton–Cartan gravity. This back-reaction is interesting as it sources the spatial Ricci curvature, providing an example where nonrelativistic gravity is more than just a Newtonian potential.