Cartan Action Research Articles

Cartan actions of higher rank abelian groups and their classification

We study R k × Z ℓ \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.

The dark side of the torsion: dark energy from propagating torsion

An extension to the Einstein–Cartan (EC) action is discussed in terms of cosmological solutions. The torsion incorporated in the EC Lagrangian is assumed to be totally anti-symmetric, represented by a time-like axial vector S^mu . The dynamics of torsion is invoked by a novel kinetic term. Here we show that this kinetic term gives rise to dark energy, while the quadratic torsion term, emanating from the EC part, represents a stiff fluid that leads to a bouncing cosmology solution. A constraint on the bouncing solution is calculated using cosmological data from different epochs.

Mimetic-metric-torsion with induced axial mode and phantom barrier crossing

We extend the basic formalism of mimetic-metric-torsion gravity theory, in a way that the mimetic scalar field can manifest itself geometrically as the source of not just the trace mode of torsion, but its axial (or, pseudo-trace) mode as well. Specifically, we consider the mimetic field to be (i) coupled explicitly to the well-known Holst extension of the Riemann–Cartan action, and (ii) identified with the square of the associated Barbero–Immirzi field, presumably a pseudo-scalar. The conformal symmetry originally prevaling in the theory would still hold, as the associated Cartan transformations do not affect the torsion pseudo-trace, and hence the Holst term. Demanding the theory to preserve the spatial parity symmetry as well, we show a geometric unification of the cosmological dark sector, and the feasibility of a super-accelerating regime in the course of evolution of the universe. From the observational perspective, assuming the cosmological evolution profile to be very close to that for varLambda CDM, we further illustrate a smooth crossing of the so-called phantom barrier at a low red-shift, albeit with a restricted parametric domain. Subsequently, we determine the extent of the super-acceleration by examining the evolution of the relevant torsion parameters.

Deviations of ergodic sums for toral translations II. Boxes

We study the Kronecker sequence $\{n\alpha \}_{n\leq N}$ on the torus ${\mathbf {T}}^{d}$ when $\alpha $ is uniformly distributed on ${\mathbf {T}}^{d}$ . We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln ^{d} N$ , converges as $N\to \infty $ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.

The minimal Fried average entropy for higher-rank Cartan actions

We find the minimal value of the Fried average entropy by proving new lower bounds for regulators of totally real number fields.

First-order formulation of teleparallel gravity and dual loop gravity

There are at least two ways to encode gravity into geometry: Einstein’s general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein–Cartan action, which were studied recently in the literature. The first discretization leads to the loop gravity kinematical phase space where the zero torsion condition is enforced first and the other is the dual loop gravity kinematical phase space where curvature is imposed to vanish first. Our argument is based on the observation that the GR first-order Einstein–Cartan action can also be seen as a first-order action for teleparallel gravity up to a boundary term. The results of our paper suggest that the dual loop gravity framework is a natural discretization of teleparallel gravity, whereas loop gravity is naturally related to the standard GR metric description.

Equivalence between the Lovelock\u2013Cartan action and a constrained gauge theory

We show that the four-dimensional Lovelock–Cartan action can be derived from a massless gauge theory for the SO(1, 3) group with an additional BRST trivial part. The model is originally composed of a topological sector and a BRST exact piece and has no explicit dependence on the metric, the vierbein or a mass parameter. The vierbein is introduced together with a mass parameter through some BRST trivial constraints. The effect of the constraints is to identify the vierbein with some of the additional fields, transforming the original action into the Lovelock–Cartan one. In this scenario, the mass parameter is identified with Newton’s constant, while the gauge field is identified with the spin connection. The symmetries of the model are also explored. Moreover, the extension of the model to a quantum version is qualitatively discussed.

Spin-2 fields from torsion: Dark energy and bouncing cosmology

We study a modified theory of gravitation built on a restricted Riemann–Cartan space-time, characterized by reduction in the number of degrees of freedom of torsion. The dynamics is defined by the Einstein–Cartan action of the Einstein–Cartan–Sciama–Kibble (ECSK) theory, while the kinematical structure is defined in such a way that the general-relativistic kinematics is recovered in the associated Riemannian space-time. In this case, the torsion is automatically decomposed into two spin-2 fields with energies of opposite sign. Investigated is the special case where these spin-2 fields are “sterile” (i.e., only interact with other matter fields through gravitation) and have their number of degrees of freedom further reduced to only one. When these fields are furnished with a conservative self-interaction potential, the spatially homogeneous and isotropic cosmological solution obtained via numerical integration of the field equations exhibits accelerated expansion compatible with the current observational data, while the equation of state of both spin-2 fields have values confined in the interval between −1 and −1/3 during the whole cosmic history. Moreover, the present model exhibits nonsingular (bouncing) cosmological solutions.

Simultaneous dense and nondense orbits for commuting maps

We show that, for two commuting automorphisms of the torus and for two elements of the Cartan action on compact higher rank homogeneous spaces, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, we show that the set of points that have dense orbit under one map and nondense orbit under the second has full Hausdorff dimension.

CLIFFORD SPACE GRAVITATIONAL FIELD EQUATIONS AND DARK ENERGY

We continue with the study of Clifford-space (C-space) gravity and analyze further the C-space generalized gravitational field equations which are obtained from a variational principle based on the generalization of the Einstein–Hilbert–Cartan action. One of the main features is that the C-space connection requires torsion in order to have consistency with the Clifford algebraic structure associated with the curved C-space basis generators. Hence no spin matter is required to induce torsion since it already exists in the vacuum. The field equations in C-spaces associated to a Clifford algebra in D dimensions are not equivalent to the ordinary gravitational equations with torsion in higher 2D dimensions. The most physically relevant conclusion, besides the presence of torsion in the vacuum, is the contribution of the higher grade metric components gμ1μ2ν1ν2, gμ1μ2μ3ν1ν2ν3,… of the C-space metric to dark energy/dark matter.

Algebraic properties of the Einstein–Cartan action

We study algebraic properties of matrix operators which act on the torsion in the Einstein–Cartan action. Multiplication rules are written and the corresponding equations of motion are explored in different variables. Bibliography: 6 titles.

LANCZOS–LOVELOCK–CARTAN GRAVITY FROM CLIFFORD-SPACE GEOMETRY

A rigorous construction of Clifford-space (C-space) gravity is presented which is compatible with the Clifford algebraic structure and permits the derivation of the expressions for the connections with torsion in C-spaces. The C-space generalized gravitational field equations are derived from a variational principle based on the extension of the Einstein–Hilbert–Cartan action. We continue by arguing how Lanczos–Lovelock–Cartan (LLC) higher curvature gravity with torsion can be embedded into gravity in C-spaces and suggest how this might also occur for extended gravitational theories based on f(R), f(Rμν), … actions, for polynomial-valued functions. In essence, the LLC curvature tensors appear as Ricci-like traces of certain components of the C-space curvatures. Torsional gravity is related to higher-order corrections of the bosonic string-effective action. In the torsionless case, black-strings and black-brane metric solutions in higher dimensions D > 4 play an important role in finding specific examples of solutions to LL gravity.

Topological self-joinings of Cartan actions by toral automorphisms

We show that if r≥3 and α is a faithful Zr-Cartan action on a torus Td by automorphisms, then any closed subset of (Td)2 which is invariant and topologically transitive under the diagonal Zr-action by α is homogeneous, in the sense that it is either the full torus (Td)2, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.

GAUGED ADS–MAXWELL ALGEBRA AND GRAVITY

We deform the anti-de Sitter algebra by adding additional generators [Formula: see text], forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with the help of a constrained BF theory. It turns out that the resulting theory is described by the Einstein–Cartan action with Holst term, and the gauge fields associated with the Maxwell generators [Formula: see text] appear only in topological terms that do not influence dynamical field equations. We briefly comment on the extension of this construction, which would lead to a nontrivial Maxwell fields dynamics.

Rigidity of commutative non-hyperbolic actions by toral automorphisms

AbstractBerend [Multi-invariant sets on tori. Trans. Amer. Math. Soc.280(2) (1983), 509–532] obtained necessary and sufficient conditions on a ℤr-action α on a torus 𝕋d by toral automorphisms in order for every orbit to be either finite or dense. One of these conditions is that for every common eigendirection of the ℤr-action there is an element n∈ℤr such that αn expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit {αn⋅x:n∈Ω} where Ω is a set of elements which acts in an approximately isometric way on a given set of eigendirections. This analysis is used in an essential way in the work of the author with E. Lindenstrauss [Topological self-joinings of Cartan actions by toral automorphisms. Preprint, 2010] classifying topological self-joinings of maximal ℤr-actions on tori for r≥3 .

On the generalized Einstein–Cartan action with fermions

From the freedom exhibited by the generalized Einstein action proposed in Dubois-Violette and Lagraa (2010 Lett. Math. Phys. 91 83–91), we show that we can construct the standard effective Einstein–Cartan action coupled to the fermionic matter without the usual current–current interaction and therefore an effective action which does not depend either on the Immirzi parameter or on the torsion. This establishes the equivalence between the Einstein–Cartan theory and the theory of general relativity minimally coupled to the fermionic matter.

Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data"

Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data"

Quantum Regge Calculus of Einstein–Cartan theory

We study the Quantum Regge Calculus of Einstein–Cartan theory to describe quantum dynamics of Euclidean space–time discretized as a 4-simplices complex. Tetrad field eμ(x) and spin-connection field ωμ(x) are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and local gauge-invariant Einstein–Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)-group valued ωμ(x) fields and Dirac-matrix valued eμ(x) fields over 4-simplices complex.

THE EXCEPTIONAL E8 GEOMETRY OF CLIFFORD (16) SUPERSPACE AND CONFORMAL GRAVITY YANG–MILLS GRAND UNIFICATION

We continue to study the Chern–Simons E8 Gauge theory of Gravity developed by the author which is a unified field theory (at the Planck scale) of a Lanczos–Lovelock Gravitational theory with a E8 Generalized Yang–Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The Exceptional E8 Geometry of the 256-dim slice of the 256 × 256-dimensional flat Clifford (16) space is explicitly constructed based on a spin connection [Formula: see text], that gauges the generalized Lorentz transformations in the tangent space of the 256-dim curved slice, and the 256 × 256 components of the vielbein field [Formula: see text], that gauge the nonabelian translations. Thus, in one-scoop, the vielbein [Formula: see text] encodes all of the 248 (nonabelian) E8 generators and 8 additional (abelian) translations associated with the vectorial parts of the generators of the diagonal subalgebra [Cl(8) ⊗ Cl(8)] diag ⊂ Cl(16). The generalized curvature, Ricci tensor, Ricci scalar, torsion, torsion vector and the Einstein–Hilbert–Cartan action is constructed. A preliminary analysis of how to construct a Clifford Superspace (that is far richer than ordinary superspace) based on orthogonal and symplectic Clifford algebras is presented. Finally, it is shown how an E8 ordinary Yang–Mills in 8D, after a sequence of symmetry breaking processes E8 → E7 → E6 → SO(8, 2), and performing a Kaluza–Klein–Batakis compactification on CP2, involving a nontrivial torsion, leads to a (Conformal) Gravity and Yang–Mills theory based on the Standard Model in 4D. The conclusion is devoted to explaining how Conformal (super) Gravity and (super) Yang–Mills theory in any dimension can be embedded into a (super) Clifford-algebra-valued gauge field theory.

On the Classification of Cartan Actions

We study higher-rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by \(\mathbb{R}^{k}\) with k ≥ 3 whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic \(\mathbb{Z}^{k}\)-actions for k ≥ 3.