The parallelizing S7 torsion in gauged N=8 supergravity

Using the parallelizing S⁷ torsion as an ansatz we investigate two solutions of gauged N=8 supergravity with SO(7) invariance. Supersymmetry is uniformly broken. We calculate the masses for these solutions which are both unstable. Certain apparent discrepancies with the results obtained by spontaneous compactification of d=11 supergravity are discussed. We establish that the compactification on the parallelized S⁷ has an SO(7) invariance and clarify the issue of supersymmetry breaking. The lack of stability in d=4 indicates that this d=11 solution is unstable.     

N = 2, D = 10 non-chiral supergravity and its spontaneous compactification

The non-chiral N = 2, D = 10 supergravity theory is constructed using dimensional reduction from N = 1, D = 11 supergravity. It is shown that this theory may spontaneously compactify, yielding S⁴ × S², CP² × S² and S² × S² × S² spaces for the extra dimensions.     

An attractor universe in six-dimensional N = 2 supergravity Kaluza-Klein theory

Cosmological solutions are investigated in six-dimensional, N = 2 supergravity Kaluza-Klein theory. It is shown that the solution of (the four-dimensional Friedmann universe)×(a constant S²) is the attractor, i.e. all the cosmological solutions starting from arbitrary initial conditions (apart from the time reversal ones) approach the above space-time asymptotically without any fine-tuning. The Friedmann solution is asymptotically “unique” in the later stage of the universe in six-dimensional N = 2 supergravity.     

Compactification of d = 11 supergravity on S4 (or 11 = 7 + 4, too)

Eleven-dimensional supergravity can compactify spontaneously on either S⁷ or S⁴. We study the latter case, with attention to its connection with a possible gauged N = 4 supergravity in d = 7. We derive the linearized field equations and supersymmetry transformation rules for the effective d = 7 supergravity multiplet. There are five third rank antisymmetric tensors in this multiplet which are in the 5 representation of the gauge group USp(4) and they are propagated with a self-duality condition in 7 dimensions. There is also a 14 of scalar fields and they are found to propagate with a non-conformal wave operator.     

Superunification from eleven dimensions

We show how N = 8 supersymmetry can break spontaneously to N = 1 at the Planck scale via a Kaluza-Klein compactification of d = 11 supergravity on the squashed seven-sphere. Features unique to Kaluza-Klein supergravity are (i) the massless gravitino of the N = 1 phase comes from a massiveN = 8 supermultiplet, (ii) the scalars developing nonzero VEVs also belong to massive N = 8 supermultiplets, (iii) parity remains unbroken when N = 8 breaks to N = 1.Next we ask whether the resulting N = 1 theory can provide a realistic SU(3) × SU(2) × U(1) unification and speculate that it might if some of the gauge bosons and fermions are composite as in the EGMZ model. In contrast to their model, however, we avoid unwanted helicities and problems with their non-compact E₇. Moreover, we suggest a scheme in which the electroweak SU(2) × U(1) is a subgroup of the d = 11 general coordinate group but that the strong SU(3) is a subgroup of the d = 11 local Lorentz group and are not, therefore, to be combined into a GUT. The special properties of the seven-sphere also suggest a possible solution of the cosmological constant problem involving fermion condensates.     

A nontrivial vacuum state in D = 10, N = 1 supergravity

We find a ground state of D = 10, N = 1 supergravity of the form (AdS(3) × R¹) × S³ × T³ which preserves all supersymmetries and should provide a gauged D = 4, N = 4 supergravity coupled to supermatter after dimensional reduction.     

Quantum Kaluza-Klein cosmologies (III)

The “ground state” proposal for the quantum state of the universe is generalized to the case of a noncompact spacelike three-hyperboloid as the configuration space. The most probable evolution of the universe must come from a gravitational instanton by quantum tunneling. We show that under some minisuperspace ansatz, there exists only S₄ × S₇ gravitational instanton in d = 11 supergravity. From the point of view of quantum cosmology this fact must be related to the fact that our observed spacetime is four-dimensional.     

N = 4 supergravity coupled to N = 4 matter and hidden symmetries

We present the N = 1 supergravity in 10 dimensions obtained by truncating the reduced N = 1 supergravity from 11 dimensions. This is further reduced to 4 dimensions to give SU(4) supergravity coupled to six SO(4) vector multiplets. As the reduction is from 10 dimensions, the theory is expected to have the symmetry SL(6R)_global×SO(6)_local, but we give a theoretical argument that this can be extended to SO(6,6)×SU(1,1)_global and SO(6)×SO(6)×U(1)_local.     

N = 8 supergravity

The complete structure of N = 8 supergravity is presented with an optional local SO(8) invariance. The SO(8) gauge interactions break E₇ invariance, but leave the local SU(8) unaffected. Exploiting E₇ × SU(8) invariance and using explicit lowest order results, we first derive the complete action and transformation laws. Subsequently, we introduce local SO(8) invariance and prove the consistency of the theory. Possible implications of our results are discussed.     

Geometry of coset spaces and massless modes of the squashed seven-sphere in supergravity

We derive some general results for Killing vectors on arbitrary coset manifolds and explicitly exhibit the squashed seven-sphere as the coset space Sp₄×Sp₂/Sp₂×Sp₂. Using these results, we then analyze the zero-mass sector of supergravity of the squashed S⁷ and argue that it is not interpretable as a spontaneously broken version of N=8 supergravity. We also point out the existence of a new solution which combines squashing and torsion.     

Formulation of 11-dimensional supergravity in superspace

We formulate on-shell 11-dimensional supergravity in superspace and express its equations of motion in terms of purely geometrical quantities. All torsion and curvature components are solved in terms of a single superfield W_rstu, totally antisymmetric in its (flat vector) indices. The dimensional reduction of this formulation is expected to be related to the superspace formulation of N=8 extended supergravity and might explain the origin of the hidden (local) SU(8) and (global) E₇ symmetries present in this theory.     

Antigravitating black hole solitons with scalar hair in N=4 supergravity

We present some new solutions of the equations of the N=4 supergravity theory which represent black holes with scalar, electric and magnetic charges. The solutions are parameterized by the mass and 6 electric and 6 magnetic charges which can be assembled into a complex 6-vector, Z^N. One can act on the solutions with SO(6)×U(1) to obtain new solutions with the same mass M but charges Z^N related by SO(6)×U(1) transformations, the U(1) factor corresponding to the duality subgroup of the hidden SU(1, 1) ssymetry of the N=4 model. In a certain limiting case the black holes have zero temperature and behave like solitons. In this case multisoliton solutions are exhibited which antigravitate, i.e. are in static equilibrium. We also present some solutions of the Kaluza-Klein theory which were anticipated by Scherk which also antigravitate. However, these latter solutions contain naked singularities. A discussion is also given of the relation of these solutions to dimensional reduction which has relevance for the black holes in the N=8 supergravity theory.     

Extended supergravity with local SO(5) invariance

We present a version of N = 5 supergravity with local SO(5) invariance and a lowest order calculation for N = 8 supergravity with local SO(8). The implications of these results and related aspects are discussed.     

An SU(4) invariant compactification of d = 11 supergravity on a stretched seven-sphere

We construct on S⁷ an SU(4) invariant solution of d = 11 supergravity in which the metric on the seven-sphere, regarded as a U(1) bundle overCP³, is distorted by stretching the U(1) fibers, and the four-index field strength F_MNPQ is non-zero in the S⁷ directions. This solution presumably corresponds to the SU(4) invariant extremum of the de Wit-Nicolai potential.     

Discrete symmetries in Kaluza-Klein theories

We investigate discrete symmetries in theories of higher-dimensional (d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d ? 4 dimensional internal space. If the action of pure d-dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d-dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor.A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong-CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.     

Anisotropic supergravity cosmologies

Anisotropic cosmologies of Bianchi types I, II, VI₀, VII₀, VIII and IX are studied in the bosonic sector of d = 11 supergravity. Exact solutions are given for all types in s = 4 dimensions coupled to the “round” (d ? s = 7) seven-sphere solutions due to Freund, Rubin, Duff and Pope, and Englert. The generalized Kasner, the Taub-NUT-de Sitter and some “mixmaster” solutions with a cosmological constant are derived. In addition, we wish to point out that the isotropic solution given recently by Myung and Kim does not exist. Another inconsistent solution was also given by Gleiser et al.     

N = 3 and N = 1 supersymmetry in a new class of solutions for d = 11 supergravity

We present a new class of compactifying solutions for d = 11 supergravity. The internal 7-spaces are described by coset manifolds N^pqr of the form SU(3) × U(1)/U(1) × U(1). The three integers p, q, r characterize the embedding of the stability subgroup U(1) × U(1) in SU(3) × U(1).Their supersymmetry content is quite remarkable. For a particular choice of p, q, r the isometry of N^pqr is SU(3) × SU(2): in this case we find that N = 3 supersymmetry survives. For all the other values of p, q, r, supersymmetry is broken to N = 1, and the isometry group is SU(3) × U(1).We also find a class of solutions with internal photon curl F_αβγδ ≠ 0, breaking all supersymmetries.     

Counterterms in extended supergravities

Geometrical invariants respecting all necessary symmetries of the theory are shown to exist, starting from the 8th (4th) loop approximation in N = 8 (N = 4) on-shell supergravity. 3-loop counterterms are presented on a linearized level for N = 4 and N = 8 theories. The corresponding 3-loop non-linear invariants are discussed.     

Zilch currents,supersymmetry and Kaluza-Klein consistency

We give a simple explanation for the fact that one can always perform a consistent truncation of a normal second-order Kaluza-Klein supergravity theory to the irreducible massless graviton supermulitplet. The basic reason for consistency is the highly restrictive requirement that massive fields of spins s⩾1 couple at lowest order to conserved currents constructed from field strengths of the massless fields. Inparticular, one understands in this way the consistency of the truncation of D=11 supergravity compactified on the seven-sphere to d=4, N=8 gauged supergravity.     

Compactification and supersymmetry in d = 11 supergravity

We examine the conditions under which the ground state of d = 11 supergravity can be supersymmetric and be of the form M⁴ ? B⁷ with M⁴ Minkowski spacetime and B⁷ a compact seven-dimensional manifold. Since we have in mind a background that renders the effective action stationary we make no use of the classical field equations. We find that the requirement that the four-space be flat is very restrictive. It requires all components of the background four-index field to vanish and the compact manifold to be Ricci-flat and hence to have at most the abelian symmetries associated with tori.