We consider scalar perturbations in the time-dependent Hourava-Witten Model in order to probe its stability. We show that during the non-singular epoque the model evolves without instabilities until it encounters the curvature singularity where a big crunch is supposed to occur. We compute the frequencies of the scalar field oscillation during the stable period and show how the oscillations can be used to prove the presence of such a singularity.
Horava-Witten spacetimes necessarily include two branes of opposite tension. If these branes are BPS we are led to a puzzle: a negative tension brane should be unstable as it can loose energy by expanding, whereas a BPS brane should be stable as it resides at a minimum of the energy. We provide a detailed analysis of the energy of such braneworld spacetimes in 5 dimensions. This allows us to show by a non-perturbative positive energy theorem that Horava-Witten spacetimes are stable, essentially because the dynamics of the branes is entirely accounted for by the behaviour of the bulk fields. We…
As a supersymmetric extension of the Randall-Sundrum model, we consider a 5-dimensional Horava-Witten type theory, and derive its low energy effective action. The model we consider is a two-brane system with a bulk scalar field satisfying the BPS condition. We solve the bulk equations of motion using a gradient expansion method, and substitute the solution into the original action to get the 4-dimensional effective action. The resultant effective theory can be casted into the form of Einstein gravity coupled with two scalar fields, one arising from the radion, the degree of freedom of the inte…
We discuss various superstring effective actions and, in particular, their common sector which leads to the so-called pre-big-bang cosmology (cosmology in a weak coupling limit of heterotic superstring). Then, we review the main ideas of the Horava-Witten theory which is a strong coupling limit of heterotic superstring theory. Using the conformal relationship between these two theories we present Kasner asymptotic solutions of Bianchi type IX geometries within these theories and make predictions about possible emergence of chaos. Finally, we present a possible method of generating Horava-Witte…
Bianchi type I and type IX ('Mixmaster') geometries are investigated within the framework of Hořava-Witten cosmology. We consider the models for which the fifth coordinate is a $S^1/Z_2$ orbifold while the four coordinates are such that the 3-space is homogeneous and has geometry of Bianchi type I or IX while the rest six dimensions have already been compactified on a Calabi-Yau space. In particular, we study Kasner-type solutions of the Bianchi I field equations and discuss Kasner asymptotics of Bianchi IX field equations. We are able to recover the isotropic 3-space solutions found by Lukas …
Brane world scenarios offer a way of setting the cosmological constant to zero after supersymmetry breaking provided there is a sufficient number of adjustable integration constants/parameters. In the case of the Horava-Witten theory compactified on a Calabi-Yau threefold, we argue that it is difficult to find enough freedom to get a zero (or small) cosmological constant after supersymmetry breaking.
We study the behavior of gravitational waves and their backreaction on the background in cosmological solutions of the five-dimensional Hořava-Witten theory. As a dynamical background, we consider two cosmological solutions with spatially flat expanding FRW branes, called $(\uparrow)$- and $(\downarrow)$-solutions, in which the orbifold size increases and decreases in time, respectively. % For these background solutions, the wave equation for the tensor perturbation can be solved by the method of separation of variables, and the mode functions are classified by a separation constant which can …
The cosmological solutions of Horava-Witten theory discovered by Lukas, Ovrut and Waldram are generalized to allow non vanishing spatial curvature. The solution with closed spatial sections has initial and final curvature singularities. We find two solutions with open spatial sections, both of which evolve from an initial curvature singularity to the supersymmetric domain wall solution at late times. We also present a solution with open spatial sections and a non-zero Ramond-Ramond scalar. The behaviour of the solutions in eleven dimensions is discussed.
The general structure of the matter Kahler metric in the $κ^{2/3}$ expansion of Horava-Witten M-theory with nonstandard embeddings is examined. It is shown that phenomenological models based on this structure can lead to Yukawa and V$_{\rm CKM}$ hierarchies (consistent with all data) without introducing ad hoc small parameters if the 5-branes lie near the distant orbifold plane and the instanton charges of the physical plane vanish. M-theory thus offers an alternate way of describing these hierarchies, different from the conventional models of Yukawa textures.
We construct exact time-dependent solutions of the supergravity equations of motion in which two initially non-singular branes, one with positive and the other with negative tension, move together and annihilate each other in an all-enveloping spacetime singularity. Among our solutions are the Horava-Witten solution of heterotic M-theory and a Randall-Sundrum I type solution, both of which are supersymmetric, i.e. BPS, in the time-independent case. In the absence of branes our solutions are of Kasner type, and the source of instability may ascribed to a failure to stabilise some of the modulus…
We study classical BPS five-brane solutions in the Horava-Witten supergravity. The presence of the eleventh dimension add a new feature, namely the dependence of the solution on this new coordinate. For gauge five-branes with an instanton size less than the eleventh radius and in the neighborhood of the center of the neutral five-brane, important corrections to the ten-dimensional solution appear for all values of the string coupling constant. We compute the mass and magnetic charge of the five-brane solitons and the result is shown to agree with the membrane and five-brane quantization condit…
We study the radion stability and radion mass in the framework of the Horava-Witten (HW) heterotic M-Theory on $S^{1}/Z_{2}$, and find that the radion is stable and its mass can be of the order of GeV. The gravity is localized on the visible brane, and the spectrum of the gravitational Kaluza-Klein (KK) modes is discrete and can have a mass gap of TeV. The corrections to the 4D Newtonian potential from the higher order gravitational KK modes are exponentially suppressed. Applying such a setup to cosmology, we find the generalized Friedmann-like equations on each of the two orbifold branes.
Orbifold branes are studied in the framework of the 11-dimensional Horava-Witten heterotic M-Theory. It is found that the effective cosmological constant can be easily lowered to its current observational value by the mechanism of large extra dimensions. The domination of this constant over the evolution of the universe is only temporarily. Due to the interaction of the bulk and the branes, the universe will be in its decelerating expansion phase again in the future, whereby all problems connected with a far future de Sitter universe are resolved.
We review the Seiberg-Witten instability of topological black holes in Anti-de Sitter space due to nucleation of brane-anti-brane pairs. We start with black holes in general relativity, and then proceed to discuss the peculiar property of topological black holes in Hořava-Lifshitz gravity -- they have instabilities that occur at only finite range of distance away from the horizon. This behavior is not unique to black holes in Hořava-Lifshitz theory, as it is also found in the relatively simple systems of charged black hole with dilaton hair that arise in low energy limit of string theory.
The cosmology of the Horava-Witten M-Theory reduced to five dimensions retaining the volume modulus is considered. Brane matter is considered as a perturbation on the vacuum solution, and the question of under what circumstances does the theory give rise to the standard RWF cosmology is examined. It is found that for static solutions, one obtains a consistent solution of the bulk field equations and the brane boundary conditions only for pure radiation on the branes. (A similar result holds if additional 5-branes are added in the bulk.) If one stabilizes the fifth dimension in an ad hoc manner…
There is good support for the embedding of the Standard Model fermions in the chiral 16 representation of SO(10). Such an embedding is provided by the realistic free fermionic heterotic-string models. In this talk we demonstrate the existence of solutions with 3 generations and SO(10) observable gauge group, in the case of compactification on a torus-fibred Calabi-Yau space over a Hirzebruch base surface. The SO(10)symmetry is broken to SU(5)xU(1) by a Wilson line. The overlap with the realistic free fermionic heterotic-string models is discussed.
We discuss simple cosmological solutions of Horava-Witten theory describing the strongly coupled heterotic string. At energies below the grand-unified scale, the effective theory is five- not four-dimensional, where the additional coordinate parameterizes a S^1/Z_2 orbifold. Furthermore, it admits no homogeneous solutions. Rather, the vacuum state, appropriate for a reduction to four-dimensional supersymmetric models, is a BPS domain wall. Relevant cosmological solutions are those associated with this BPS state. In particular, such solutions must be inhomogeneous, depending on the orbifold coo…
The cosmology in the Hubble expansion era of the Horava-Witten M-theory compactified on a Calabi-Yau threefold is studied in the reduction to five-dimensions where the effects of the Calabi-Yau manifold are summarized by the volume modulus, and all perturbative potentials are included. Matter on the branes are treated as first order perturbations of the static vacuum solution, and all equations in the bulk and all boundary conditions on both end branes are imposed. It is found that for a static volume modulus and a static fifth dimension, y, one can recover the four dimensional Robertson Fried…
Recent advances in 11 dimensional Horava-Witten M-theory based on non-standard embeddings with torus fibered Calabi-Yau manifolds have allowed the construction of three generation models with Wilson line breaking to the Standard Model gauge symmetry. Central to these constructions is the existence of a set of 5-branes in the bulk. We examine within this framework the general structure of the matter Yukawa couplings and show that M-theory offers an alternate possible way of achieving the CKM and quark mass hierarchies without introducing undue fine tuning or (as in conventional analysis) small …
The combined Einstein equations and scalar equation of motion in the Horava-Witten scenario of the strongly coupled heterotic string compactified on a Calabi-Yau manifold are solved in the presence of additional matter densities on the branes. We take into account the universal Calabi-Yau modulus phi with potentials in the 5-d bulk and on the 3-branes, and allow for an arbitrary coupling of the additional matter to phi and an arbitrary equation of state. No ad hoc stabilization of the five dimensional radius is assumed. The matter densities are assumed to be small compared to the potential for…
Space- vs Time-dependence in taming the infrared instability of projectable Hořava Gravity
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo（东京大学理学研究院早期宇宙研究中心）；Department of Physics and Astronomy, McMaster University（麦克马斯特大学物理与天文学系）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）
We consider the cosmology of the reduced 5D Horava-Witten M-Theory (HW) with volume modulus and treating matter on the orbifold planes to first order. It is seen that one can recover the FRW cosmology in the Hubble expansion era with relativistic matter, but if a solution exist with non-relativistic (massive) matter it must be non-static with a Hubble constant that depends on the fifth dimension. (The same result holds when 5-branes are present.) This difficulty is traced to the fact that in HW, the volume modulus couples to the bulk and brane cosmological constants (so that the net 4D constan…
We consider the Horava-Witten based model with 5-branes situated near the distant orbifold plane and with vanishing instanton numbers on the physical plane. This model has a toric fibered Calabi-Yau with del Pezzo base dP_7 which allows three generations with Standard Model gauge group at the GUT scale. Previous analysis showed that the quark hierarchy at the electroweak scale could be achieved qualitatively without undue fine tuning due to the effects of the 5-branes on the Kahler potential. We extend here this analysis to include the leptons. A new mechanism is introduced to obtain neutrino …
In this paper, $E_6$ and especially $E_7$ GUT are considered in the F-theory setting in view of the free fermionic construction of the $4D$ heterotic string. In particular, the NAHE-Based LRS model of \cite{Cleaver:2000ds, Cleaver:2002ps} is revisited as an illustration where the starting point was taken to be the $N=4$, $E_7 \times E_7 \times SO(16)$ which through the use of boundary condition basis vectors is reduced to obtain the flipped $SO(10)$ GUT symmetry. We also seek to extend the results of \cite{Faraggi:2002ah} in the case of the flipped $SU(5)$ to home in on the flipped $SO(10)$ va…
The field equations and boundary conditions of Horava-Witten theory, compactified on a smooth compact spin quotient of CH^3, where CH^3 denotes the hyperbolic cousin of CP^3, are studied in the presence of Casimir energy density terms. If the Casimir energy densities near one boundary result in a certain constant of integration taking a value greater than around 10^5 in units of the d = 11 gravitational length, a form of thick pipe geometry is found that realizes TeV-scale gravity by the ADD mechanism, with that boundary becoming the inner surface of the thick pipe, where we live. Three altern…
In this paper we show that both of the Green-Schwarz anomaly factorization formula for the gauge group $E_8\times E_8$ and the Hořava-Witten anomaly factorization formula for the gauge group $E_8$ can be derived through modular forms of weight 14. This answers a question of J. H. Schwarz. We also establish generalizations of these factorization formulas and obtain a new Hořava-Witten type factorization formula.
We study stability of singularity-free cosmological solutions with positive cosmological constant based on projectable Hořava-Lifshitz (HL) theory. In HL theory, the isotropic and homogeneous cosmological solutions with bounce can be realized if spacial curvature is non-zero. By performing perturbation analysis around non-flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime, we derive a quadratic action and discuss the stability, i.e, ghost and tachyon-free conditions. Although the squared effective mass of scalar perturbation must be negative in infrared regime, we can avoid tachyon inst…
We find stable singularity-free cosmological solutions in non-flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime in the context of Hořava-Lifshitz (HL) theory. Although we encounter the negative squared effective masses of the scalar perturbations in the original HL theory, the behaviors can be remedied by relaxing the projectability condition. In our analysis, the effects from the background dynamics are taken into account as well as the sign of the coefficients in the quadratic action for perturbations. More specifically, we give further classification of the gradient stability/instab…
By computing the Lyapunov exponent, which is the inverse of the instability time scale associated with this geodesic motion we show that there is two region of space which in both of them the equatorial timelike geodesics are stable via Lyapunov measure of stability.
A proposal for a power-counting renormalizable theory of quantum gravity at a Lifshitz point was recently put forth by Horava (arXiv:0901.3775), and has been since dubbed as Horava-Lifshitz gravity. The theory explicitly breaks Lorentz invariance, which introduces an apparent extra scalar degree of freedom. In this note, we show that the low energy limit of (non-projectible) Horava-Lifshitz gravity is uniquely given by the quadratic cuscuton model: a covariant scalar field theory with an infinite speed of sound and a quadratic potential, which is minimally coupled to Einstein gravity. This imp…
Inclusion of $f(R)$ term in the action of Horava-Lifshitz quantum gravity with projectability but without detailed balance condition is investigated, where $R$ denotes the 3-spatial dimensional Ricci scalar. Conditions for the spin-0 graviton to be free of ghosts and instability are studied. The requirement that the theory reduce to general relativity in the IR makes the scalar mode unstable in the Minkowski background but stable in the de Sitter. It is remarkable that the dark sector, dark matter and dark energy, of the universe has a naturally geometric origin in such a setup. Bouncing unive…
As shown by Hořava and Witten, there are gravitational anomalies at the boundaries of $M^{10}\times S^1/Z_2$ of 11 dimensional supergravity. They showed that only 10 dimensional vector multiplets belonging to $E_8$ gauge group can be consistently coupled to this theory. Thus, the dimensional reduction of this theory should be the low energy limit of the $E_8\times E_8$ heterotic string. Here we assume that M-theory is a theory of supermembranes which includes twisted supermembranes. We show that for a target space $M^{10}\times S^1/Z_2$, in the limit in which $S^1/Z_2$ is small, the effective …
This article reviews basic construction and cosmological implications of a power-counting renormalizable theory of gravitation recently proposed by Horava. We explain that (i) at low energy this theory does not exactly recover general relativity but instead mimic general relativity plus dark matter; that (ii) higher spatial curvature terms allow bouncing and cyclic universes as regular solutions; and that (iii) the anisotropic scaling with the dynamical critical exponent z=3 solves the horizon problem and leads to scale-invariant cosmological perturbations even without inflation. We also comme…
We consider an extended theory of Horava-Lifshitz gravity with the detailed balance condition softly breaking, but without the projectability condition. With the former, the number of independent coupling constants is significantly reduced. With the latter and by extending the original foliation-preserving diffeomorphism symmetry $ {Diff}(M, {\cal{F}})$ to include a local U(1) symmetry, the spin-0 gravitons are eliminated. Thus, all the problems related to them disappear, including the instability, strong coupling, and different speeds in the gravitational sector. When the theory couples to a …
Motivated by the ``universe as a brane'' idea, we investigate the motion of a $(D-2)$-brane (or domain wall) that couples to bulk matter. Usually one would expect the spacetime outside such a wall to be time dependent however we show that in certain cases it can be static, with consistency of the Israel equations yielding relationships between the bulk metric and matter that can be used as ansätze to solve the Einstein equations. As a concrete model we study a domain wall coupled to a bulk dilaton with Liouville potentials for the dilaton both in the bulk and on the wall. The bulk solutions we…
We investigate, from a spacetime perspective, some aspects of Horowitz's recent conjecture that black strings may catalyze the decay of Kaluza-Klein spacetimes into a bubble of nothing. We identify classical configurations that interpolate between flat space and the bubble, and discuss the energetics of the transition. We investigate the effects of winding tachyons on the size and shape of the barrier and find no evidence at large compactification radius that tachyons enhance the tunneling rate. For the interesting radii, of order the string scale, the question is difficult to answer due to th…
We demonstrate the existence of a class of N=1 supersymmetric nonperturbative vacua of Horava-Witten M-theory compactified on a torus fibered Calabi-Yau 3-fold Z with first homotopy group π_{1}(Z)= Z2, having the following properties: 1) SO(10) grand unification group, 2) net number of three generations of chiral fermions in the observable sector, and 3) potentially viable matter Yukawa couplings. These vacua correspond to semistable holomorphic vector bundles V_{Z} over Z having structure group SU(4)_C, and generically contain M5-branes in the bulk space. The nontrivial first homotopy group a…
A recent anomaly computation of Horava and Witten is proved and generalized in the form of two index theorems in odd dimensions. Theorem A is a fixed point formula for orientation-reversing involutions. Theorem B is an index theorem for manifolds with boundary using local boundary conditions. Both hold for families of operators.
We show the complete cancellation of gauge and gravitational anomalies in the M-theory of Horava and Witten using their boundary contribution, and a term coming from the existence of two and five-branes. A factor of three discrepancy noted in an earlier work is resolved. We end with a comment on flux quantization.
Using the Witten-Sakai-Sugimoto model in the D0-D4 background, we holographically compute the vacuum decay rate of the Schwinger effect in this model. Our calculation contains the influence of the D0-brane density which could be identified as the $θ$ angle or chiral potential in QCD. Under the strong electromagnetic fields, the instability appears due to the creation of quark-antiquark pairs and the associated decay rate can be obtained by evaluating the imaginary part of the effective Euler-Heisenberg action which is identified as the action of the probe brane with a constant electromagnetic …
We compactify the M-theory proposed by Horava and Witten on a Calabi-Yau manifold with boundary $S_1/Z_2$. A no-scale-like Kähler potential, the superpotential, and the gauge kinetic function are obtained in this 4-dimensional $E_6\times E_8$ model. We also study the general phenomenological consequences of the resulting M-theory-inspired model, which may include very light gravitinos, axions and axinos.
Perturbative spectra and related factorization properties of one-loop open string amplitudes in the presence of a constant external background B are analysed in detail. While the pattern of the closed string spectrum, obtained after a careful study of the properly symmetrized amplitudes, turns out to be unaffected by the presence of B, a series of double open-string poles, which would be absent when B is turned off, can couple owing to a partial symmetry loss. These features are studied first in a bosonic setting and then generalized in the more satisfactory superstring context. When the backg…
We provide the first example of a cosmological solution of the Horava-Witten supergravity. This solution is obtained by exchanging the role of time with the radial coordinate of the transverse space to the five-brane soliton. On the boundary this corresponds to rotating an instanton solution into a tunneling process in a space with Lorentzian signature, leading to an expanding universe. Due to the freedom to choose different non-trivial Yang-Mills backgrounds on the boundaries, the two walls of the universe ( visible and hidden worlds) expand differently. However at late times the anisotropy i…
We compactify M-theory in the Horava-Witten formulation on S^1/Z_2 \times K3 \times T^2. Focusing on the moduli-space of vector multiplets of the resulting four-dimensional N=2 theory, we determine the prepotential as an expansion in two dimensionless parameters which both scale as κ^{2/3}. We determine the prepotential completely to relative order κ^{4/3} and compare the expression with the results obtained for the perturbative string theories. We find complete agreement to relative order κ^{4/3} between the strong and weak coupling regimes. The sources of higher order perturbative and non-pe…
We study the warped geometry of heterotic M-Theory in five dimensions where five-branes are included in the bulk. Five-branes wrapping holomorphic curves lead to BPS configurations where the junction conditions are automatically satisfied. We consider five-branes wrapped around non-supersymmetric cycles and show that the configuration is unstable. We describe explicitly the resulting time-dependent geometry where the bulk five-branes move towards the Horova-Witten boundary walls. The five-branes collide with the boundary walls in a finite time resulting in the restoration of supersymmetry.
Linde has recently argued that compact flat or negatively curved spatial sections should, in many circumstances, be considered typical in Inflationary cosmologies. We suggest that the "large brane instability" of Seiberg and Witten eliminates the negative candidates in the context of string theory. That leaves the flat, compact, three-dimensional manifolds -- Conway's *platycosms*. We show that deep theorems of Schoen, Yau, Gromov and Lawson imply that, even in this case, Seiberg-Witten instability can be avoided only with difficulty. Using a specific cosmological model of the Maldacena-Maoz t…
Witten long ago pointed out that the simplest Kaluza-Klein theory, without supersymmetry, is subject to a catastrophic instability. There are a variety of string theories which are potentially subject to these instabilities. Here we explore a number of questions: how generic are these instabilities? what happens when a potential is generated on the moduli space? in the presence of supersymmetry breaking, is there still a distinction between supersymmetric and non-supersymmetric states?
