Hyperbolic Kac–Moody algebras and chaos in Kaluza–Klein models

Damour, Thibault; Henneaux, Marc; Juliá, B.; Nicolai, Hermann

doi:10.1016/s0370-2693(01)00498-1

Cited by 156 publications

(301 citation statements)

References 29 publications

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“…This chaotic behavior is a new effect not present in any classical treatment. It is generally well known that chaos plays an important role in general relativity [11], quantum field theories [12,13,14], and string theories [15]. The main result of our consideration is that the chaotic field theories considered naturally generate a small cosmological constant and have the scope to offer simultaneous solutions to the cosmological coincidence and uniqueness problem.…”

Section: Introductionmentioning

confidence: 89%

Chaotic scalar fields as models for dark energy

Beck

2004

Phys. Rev. D

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We consider stochastically quantized self-interacting scalar fields as suitable models to generate dark energy in the universe. Second quantization effects lead to new and unexpected phenomena if the self interaction strength is strong. The stochastically quantized dynamics can degenerate to a chaotic dynamics conjugated to a Bernoulli shift in fictitious time, and the right amount of vacuum energy density can be generated without fine tuning. It is numerically observed that the scalar field dynamics distinguishes fundamental parameters such as the electroweak and strong coupling constants as corresponding to local minima in the dark energy landscape. Chaotic fields can offer possible solutions to the cosmological coincidence problem, as well as to the problem of uniqueness of vacua.

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Section: Introductionmentioning

confidence: 89%

Chaotic scalar fields as models for dark energy

Beck

2004

Phys. Rev. D

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“…) and 10 The dominant linear forms can be identified in many physically relevant cases with the simple roots of an hyperbolic Kac-Moody algebra [32,33,34].…”

Section: Hamiltonian Approach In Iwasawa Variablesmentioning

confidence: 99%

“…the various space derivatives appearing in the r.h.s. of (33) can 'bring down', when operating on e −2w A (p•(x))τ , one (∂ x ) or two (∂ 2 x ) powers of τ .…”

Section: Construction Of a Fuchsian System For The 'Differenced Variamentioning

confidence: 99%

Describing general cosmological singularities in Iwasawa variables

2008

Self Cite

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Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that the description of the asymptotic behavior of a generic solution of Einstein equations near a spacelike singularity could be drastically simplified by considering that the time derivatives of the metric asymptotically dominate (except at a sequence of instants, in the 'chaotic case') over the spatial derivatives. We present a precise formulation of the BKL conjecture (in the chaotic case) that consists of basically three elements: (i) we parametrize the spatial metric g ij by means of Iwasawa variables (β a , N a i ); (ii) we define, at each spatial point, a (chaotic) asymptotic evolution system made of ordinary differential equations for the Iwasawa variables; and (iii) we characterize the exact Einstein solutions β, N whose asymptotic behavior is described by a solution β [0] , N [0] of the previous evolution system by means of a 'generalized Fuchsian system' for the differenced variablesβ = β − β [0] ,N = N − N [0] , and by requiring thatβ andN tend to zero on the singularity. We also show that, in spite of the apparently chaotic infinite succession of 'Kasner epochs' near the singularity, there exists a well-defined asymptotic geometrical structure on the singularity : it is described by a partially framed flag. Our treatment encompasses Einstein-matter systems (comprising scalar and p-forms), and also shows how the use of Iwasawa variables can simplify the usual ('asymptotically velocity term dominated') description of non-chaotic systems.

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“…This root is the highest weight of the fundamental 56-representation of E 7 (7) . As a consequence of this the affine extension of E 8(8) has the same Dynkin diagram as it would have E 9 (9) formally continuing the E r(r) series to r > 8.…”

Section: Systematics Of the Affine Extensionmentioning

confidence: 80%

The general pattern of Kač–Moody extensions in supergravity and the issue of cosmic billiards

Fré¹,

Gargiulo²,

Rulik³

et al. 2006

Nuclear Physics B

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In this paper we study the systematics of the affine extension of supergravity duality algebras when one steps down from D = 4 to D = 2 which is instrumental for the study of cosmic billiards. For all D = 4 supergravities (with N ≥ 3) there is a universal field theoretical mechanism promoting the extension, which relies on the coexistence of two non locally related lagrangian descriptions of the corresponding D = 2 degrees of freedom: the Ehlers lagrangian and the Matzner-Misner one. This and the existence of a generalized Kramer-Neugebauer non local transformation relating the two models, provide a Chevalley-Serre presentation of the affine Kač-Moody algebra which follows a universal pattern for all supergravities. This is an extension of the mechanism considered by Nicolai for pure N=1 supergravity, but has general distinctive features in extended theories (N≥3) related to the presence of vector fields and to their symplectic description. Moreover the novelty is that in the general case the Matzner-Misner lagrangian is structurally different from the Ehlers one, since half of the scalars are replaced by gauge 0-forms subject to SO(2n, 2n) electric-magnetic duality rotations representing in D = 2 the Sp(2n, R) rotations of D = 4. The role played by the symplectic bundle of vectors in this context, suggests that the mechanism of the affine extension can be studied also for N = 2 supergravity, where one deals with geometries rather than with algebras, the scalar manifold being not necessarily a homogeneous manifold U/H. We also show that the mechanism of the affine extension commutes with the Tits Satake projection of the relevant duality algebras. This is very important for the issue of cosmic billiards, as we show in a separate paper. Finally we also comment on the general field theoretical mechanism of the further hyperbolic extension obtained in D = 1, which we plan to analyze in detail in a forthcoming paper. The possible uses of our results and their relation to outstanding problems are pointed out.

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Hyperbolic Kac–Moody algebras and chaos in Kaluza–Klein models

Cited by 156 publications

References 29 publications

Chaotic scalar fields as models for dark energy

Chaotic scalar fields as models for dark energy

Describing general cosmological singularities in Iwasawa variables

The general pattern of Kač–Moody extensions in supergravity and the issue of cosmic billiards

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