Bernard Julia

Supergravity in theory in 11 dimensions

We present the action and transformation laws of supergravity in 11 dimensions which is expected to be closely related to the O(8) theory in 4 dimensions after dimensional reduction.

The SO(8) supergravity

We present the derivation of the SO(8) supergravity theory by dimensional reduction of the supergravity theory in 11 dimensions to 4 dimensions. It has been found that the equations of motion are invariant under the global non-compact group E7(+7). They can be derived from a family of Lagrangians invariant under a local compact group SU(8). The general procedure to deal with non-compact global internal symmetry without introducing ghosts is discussed in connection with the appearance of an associated compact local symmetry and the use of a non-linear realization of the non-compact group. The supersymmetry transformation rules have been partially derived by dimensional reduction; their complete form follows from the assumption of covariance with respect to E7 and SU(8). We also present briefly the O(N) supergravities N = 7, 6, 5 and explain the symmetry SU(4)×SU(1, 1) found for the O(4) supergravity.

The N = 8 supergravity theory. I. The lagrangian

Abstract The SO(8) supergravity action is constructed in closed form. A local SU(8) group as well as the exceptional group E7 are invariances of the equations of motion and of a new first order lagrangian.

Spontaneous Symmetry Breaking and Higgs Effect in Supergravity Without Cosmological Constant

The super Higgs effect is studied in the (2,32) + (12, 0+, 0−) model. The most general action is obtained using the recently developed tensor calculus: it contains an arbitrary function of two variables G(A,B), A and B being the 0+ scalar and 0− pseudoscalar fields of the matter system. The conditions are given which G must satisfy in order that both the gravitino ψμ becomes massive and no cosmological term is induced. Explicit examples are given, a class of them leading to the mass formula mA2 + mB2 = 4mψ2.

Super-higgs effect in supergravity with general scalar interactions

Using the recently established tensor calculus for supergravity, we construct the most general action for the scalar multiplet coupling. We discuss under which conditions supersymmetry is broken spontaneously and show explicitly that the gravitino acquires a mass by absorbing the Goldstone fermion. Parity violation as well as a cosmological constant can be avoided.

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

Some time ago, it was found that the never-ending oscillatory chaotic behaviour discovered by Belinskii, Khalatnikov and Lifshitz (BKL) for the generic solution of the vacuum Einstein equations in the vicinity of a spacelike (“cosmological”) singularity disappears in spacetime dimensions D ≡ d + 1 > 10. Recently, a study of the generalization of the BKL chaotic behaviour to the superstring effective Lagrangians has revealed that this chaos is rooted in the structure of the fundamental Weyl chamber of some underlying hyperbolic Kac–Moody algebra. In this Letter we show that the same connection applies to pure gravity in any spacetime dimension 4, where the relevant algebras are AEd . In this way the disappearance of chaos in pure gravity models in D 11 dimensions becomes linked to the fact that the Kac–Moody algebras AEd are no longer hyperbolic for d 10.  2001 Elsevier Science B.V. All rights reserved.

Null-Killing vector dimensional reduction and Galilean geometrodynamics *

The solutions of Einsteins equations admitting one non-null Killing vector field are best studied with the projection formalism of Geroch. When the Killing vector is lightlike, the projection onto the orbit space still exists and one expects a covariant theory with degenerate contravariant metric to appear, its geometry is presented here. Despite the complications of indecomposable representations of the local Euclidean subgroup, one obtains an absolute time and a canonical, Galilean and so-called Newtonian, torsionless connection. The quasi-Maxwell field (Kaluza Klein one-form) that appears in the dimensional reduction is a non-separable part of this affine connection, in contrast to the reduction with a non-null Killing vector. One may define the Kaluza Klein scalar (dilaton) together with the absolute time coordinate after having imposed one of the equations of motion in order to prevent the emergence of torsion. We present a detailed analysis of the dimensional reduction using moving frames, we derive the complete equations of motion and propose an action whose variation gives rise to all but one of them. Hidden symmetries are shown to act on the space of solutions.

Currents and superpotentials in classical gauge-invariant theories: I. Local results with applications to perfect fluids and general relativity

Noethers general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance. Bulk charges are replaced by fluxes at a suitable singularity (in general, at infinity) of so-called superpotentials, namely local functions of the gauge fields (or more generally of the gauge forms). Some gauge-invariant bulk charges and current densities may subsist when distinguished one-dimensional subgroups are present. We shall study mostly local consequences of gauge invariance. Quite generally there exist local superpotentials analogous to those of Freud or Bergmann for general relativity. They are parametrized by infinitesimal gauge transformations, but are afflicted by topological ambiguities which one must handle on a case-by-case basis. The choice of variational principle: variables, surface terms and boundary conditions is crucial. As a first illustration we propose a new affine action that reduces to general relativity upon gauge fixing the dilatation (Weyl-1918-like) part of the connection and elimination of auxiliary fields. We can also reduce it by similar considerations either to the Palatini action or to the Cartan-Weyl moving frame action and compare the associated superpotentials. This illustrates the concept of Noether identities. We formulate a vanishing theorem for the superpotential and the current when there is a (Killing) global isometry or its generalization. We distinguish between asymptotic symmetries and symmetries defined in the bulk. A second and independent application is a geometrical reinterpretation of the convection of vorticity in barotropic non-viscous fluids first established by Helmholtz-Kelvin, Eckart and Ertel. In the homentropic case it can be seen to follow by a general theorem from the vanishing of the superpotential corresponding to the time-independent relabelling symmetry. The special diffeomorphism symmetry is, in the absence of dynamical gauge field and spin, associated with a vanishing internal transverse momentum flux density. We also consider the non-homentropic case. We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an impulsive forcing for creating or destroying selectively helicity (respectively enstrophies) in odd (respectively even) dimensions. This is an example of a new and general forcing rule.

Duality and moduli spaces for time-like reductions

Abstract We consider the dimensional reduction/compactification of supergravity, string and M-theories on tori with one time-like circle. We find the coset spaces in which the massless scalars take their values, and identify the discrete duality groups.

Hyperbolic billiards of pure D = 4 supergravities

We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz (BKL) limit for all pure supergravities in D = 4 spacetime dimensions, as well as for D = 4, N = 4 supergravities coupled to k (N = 4) Maxwell supermultiplets. We find that just as for the cases N = 0 and N = 8 investigated previously, these billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. Hence, the dynamics is chaotic in the BKL limit. A new feature arises, however, which is that the relevant Kac-Moody algebra can be the lorentzian extension of a twisted affine Kac-Moody algebra, while the N = 0 and N = 8 cases are untwisted. This occurs for N = 5, where one gets A4(2), and for N = 3 and 2, for which one gets A2(2). An understanding of this property is provided by showing that the data relevant for determining the billiards are the restricted root system and the maximal split subalgebra of the finite-dimensional real symmetry algebra characterizing the toroidal reduction to D = 3 spacetime dimensions. To summarise: split symmetry controls chaos.