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In this article we try to give a simple Quantum Gravity Lagrangean that behaves quite well. Feynman calculus for unpolarized crosssections, and the diagrams involved, behave good. The action is renormalizable by dimension counting. It implies standard Einstein gravity for a massless graviton. Further investigations have to be done. 1. A More or Less Well Behaved Quantum Gravity Lagrangean in Dimension 4? Note the identity 1 D/ 2 = +R/ = +Rab [Γ,Γ] 4 using the Dirac or Clifford Algebra representation Γ = θ + θ∗a [θ] a ON basis on the tangent space Tp(X) over a point p and [θ ∗a] a raised ON basis on the cotangent space T ∗ p (X) over the same point p, just like in the notation of E. Cartan, who wrote the metric g in terms of the veilbeins eμ = θ a μ as gμν = e a μδabe b ν , δab the Kronecker delta, and We sometimes suppress a minus-sign or a imaginary unit i in the following. 58 E.B. Torbrand Dhrif D/ = Γ(∂μ + ωμ + Aμ) = Γ ∇a Γ = eμΓ . We thus also conclude θ∗D/ θ = θ∗( +R/ )θ = θ∗ θ +R with R the Ricci scalar. Here θ is the graviton or vierbein. Here we have suppressed a term, including a coupling constant, 16πG. Notice that this is the Hilbert-Einstein action SGravity,Einstein = ∫

We adopt a vierbein formalism to study pseudo-Finsler spaces modeled on a pseudo-Minkowski space. We show that it is possible to obtain closed expressions for most of the geometric objects of the theory, including Berwald's curvature, Landsberg's tensor, Douglas' curvature, non-linear connection and Ricci scalar. These expressions are particularly convenient in computations since they factor the dependence on the base and the fiber. As an illustration, we study Lorentz-Finsler spaces modeled on the Bogoslovsky Lorentz-Minkowski space, and give sufficient conditions which guarantee the Berwald property. We then specialize to a recently proposed Finslerian pp-wave metric. Finally, the paper points out that non-trivial Berwald spaces have necessarily indicatrices which admit some non-trivial linear group of symmetries., 15 pages, no figures. Accepted by "Reports on Mathematical Physics"

A Hamiltonian approach to the equations of general relativity is proposed using the powerful mathematical language of multivector-valued differential forms. In the approach, the gravitational coordinates are the 12 spatial components of the line interval (the vierbein) including their antisymmetric parts, and their 12 conjugate momenta. A feature of the proposed formalism is that it allows Lorentz gauge freedoms to be imposed on the Lorentz connections rather than on the vierbein, which may facilitate numerical integration in some challenging problems. The 40 Hamilton's equations comprise 12 + 12 = 24 equations of motion, 10 constraint equations (first class constraints, which must be arranged on the initial hypersurface of constant time, but which are guaranteed thereafter by conservation laws), and 6 identities (second class constraints). The 6 identities define a trace-free spatial tensor that is the gravitational analog of the magnetic field of electromagnetism. If the gravitational magnetic field is promoted to an independent field satisfying its own equation of motion, then the system becomes the WEBB system, which is known to be strongly hyperbolic. Some other approaches, including ADM, BSSN, WEBB, and Loop Quantum Gravity, are translated into the language of multivector-valued forms, bringing out their underlying mathematical structure., v1: 14 pages. v2: 15 pages. v3: 21 pages, extensive revision, added discussion of WEBB formalism

We point out that the Cartan geometry known as the second-order conformal structure provides a natural differential geometric framework underlying gauge theories of conformal gravity. We are concerned by two theories: the first one will be the associated Yang-Mills-like Lagrangian, while the second, inspired by J.T. Wheeler in Phys. Rev. D90 (2014), will be a slightly more general one which will relax the conformal Cartan geometry. The corresponding gauge symmetry is treated within the BRST language. We show that the Weyl gauge potential is a spurious degree of freedom, analogous to a Stueckelberg field, that can be eliminated through the dressing field method. We derive sets of field equations for both the studied Lagrangians. For the second one, they constrain the gauge field to be the `normal conformal Cartan connection'. Finally, we provide in a Lagrangian framework a justification of the identification, in dimension $4$, of the Bach tensor with the Yang-Mills current of the normal conformal Cartan connection, as proved in Class. Quant. Grav. 20 (2003).

We derive a null tetrad from axially-symmetric vierbein field. The \(f(T)\)-Maxwell field equations with cosmological constant, where T is the scalar torsion, are applied to the null tetrad. An exact non-vacuum solution having three constants of integration is derived which is a solution to the f (T) -Maxwell field equations provided that \(f(T)=T_{0}\) and \( f_{T}=\frac{d f(T)}{d T}=1\), where \( T_{0}\) is a constant. The scalar torsion related to this solution is constant, i.e., \( T=T_{0}\), and differs from the classical general relativity when \( f(T)\neq T_{0}\). We study the singularities of this solution using curvature and torsion invariants. We consider a slow rotation and show that the derived solution behaves asymptotically as de Sitter spacetime and display the existence of Nariai spacetime as a background solution. We assume a perturbation of Nariai spacetime till the first order and investigate the behavior of the black hole horizon. Finally, we explain that the anti-evaporation occurs on the classical level in the f (T) gravitational theories.

In the description of general covariance, the vierbein and the Lorentz connection can be treated as independent fundamental fields. With the usual gauge Lagrangian, the Lorentz connection is characterized by an asymptotically free running coupling. When running from high energy, the coupling gets large at a scale which can be called the Planck mass. If the Lorentz connection is confined at that scale, the low energy theory can have the Einstein Lagrangian induced at low energy through dimensional transmutation. However, in general there will be new divergences in such a theory and the Lagrangian basis should be expanded. I construct a conformally invariant model with a larger basis size which potentially may have the same property., Comment: 15 pages - corrected typos, updated references, small changes in discussion

In the final chapter of this volume we investigate how our approach to Cartan geometries can be modified to encompass the geometry of second-order ordinary differential equations.

The spin connection enters the theory of gravity as a nonabelian gauge field associated with local Lorentz transformations. Normally it is eliminated from making an extra assumption - that of the metricity of the vierbein field. However, treated by itself with the usual gauge action, it has a negative beta function, implying that it is asymptotically free. I suggest that the spin connection could be confined (or perhaps partially confined) in the same way as other nonabelian gauge fields. This would remove the need to make the extra assumption of metricity, as the spin connection would not be present in the low energy theory, leaving the symmetry to be realized only using metric variables., Comment: 6 pages - some new references, Euclidean discussion enhanced and partial-Euclidean discussion removed

It is shown that there exists a 2-dimensional matrix representation of complex quaternions over real quaternions, which allows to define Pauli matrix in 4 dimensions over the quaternionic field and leads to the quaternionic spinor group previously proposed. It is also attempted to apply complex quaternions to general relativity at the level of the variational formalism. Linear gravitational Lagrangian in Riemann-Cartan space-time U4 is derived using quaternion caluculus; namely scalar curvature in U4 is put into a quaternionic form. Consequently, Einstein-Hilbert Lagrangian in Riemann space R4 is also defined over quaternions, as first shown by Sachs. The matter fields coupled to gravity are assumed to be the scalar and the Dirac fields. The quaternionic variational formalism corresponds to the firstorder formalism but with a limited pattern of allowed fields such that the quaternionic fields carry only coordinate tensor indices but no local Lorentz indices which are contracted with that possessed by the basis of complex quaternions. In particular, both the quaternionic vierbein field and Lorentz gauge field (corresponding to the spin connection) are regarded as coordinate vectors which are independently varied, obtaining Einstein and Cartan equations, respectively. It is incidentally shown that the consistent condition of Einstein equation in U4 is proved via the variational formalism and the anti-symmetric part of Einstein equation together with Cartan equation in U4 leads to an identity which expresses the anti-symmetric part of the enegy-momentum tensor by means of the covariant divergence of the spin angular momentum tensor, both of Dirac field. We also present pedagogical proofs of Bianchi and Bach-Lanczos identities in U4 using the quaternionic formalism.

Lattice spinor gravity is a proposal for regularized quantum gravity based on fermionic degrees of freedom. In our lattice model the local Lorentz symmetry is generalized to complex transformation parameters. The difference between space and time is not put in a priori, and the euclidean and Minkowski quantum field theory are unified in one functional integral. The metric and its signature arise as a result of the dynamics, corresponding to a given ground state or cosmological solution. Geometrical objects as the vierbein, spin connection or the metric are expectation values of collective fields built from an even number of fermions. The quantum effective action for the metric is invariant under general coordinate transformations in the continuum limit. The action of our model is found to be also invariant under gauge transformations. We observe a "geometrical entanglement" of gauge- and Lorentz-transformations due to geometrical objects transforming non-trivially under both types of symmetry transformations., Comment: 38 pages

We express the first jet bundle of curves in Euclidean space as homogeneous spaces associated to a Galilean-type group. Certain Cartan connections on a manifold with values in the Lie algebra of the Galilean group are characterized as geometries associated to systems of second order ordinary differential equations. We show these Cartan connections admit a form of normal coordinates, and that in these normal coordinates the geodesic equations of the connection are second order ordinary differential equations. We then classify such connections by some of their torsions, extending a classical theorem of Chern involving the geometry associated to a system of second order differential equations., 6 pages, Latex2e

International audience; We study effectively the Cartan geometry of Levi-nondegenerate C^6-smooth hypersurfaces M^3 in C^2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M^3 in C^2 to the Heisenberg sphere H^3 , such M's being necessarily real analytic.

Some conceptual issues concerning f ( T ) theories – a family of modified gravity theories based on absolute parallelism – are analyzed. Due to the lack of local Lorentz invariance, the autoparallel frames satisfying the field equations are evasive to an a priori physical understanding. We exemplify this point by working out the vierbein (tetrad) fields for closed and open Friedmann–Robertson–Walker cosmologies.

The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text] being a parallel π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: The Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

We give the notion of bi-null Cartan curves in semi-Euclidean spaces of index 2, together with the unique Frenet frame and the Cartan curvatures. We also discuss some properties of bi-null Cartan curves in terms of the Cartan curvatures.

We consider a particular set of ghost-free interactions for three spin-2 fields in which we freeze out the dynamics of the metric tensor that couples to the matter sector. Integrating out the non-dynamical degrees of freedom in vacuum results in a ghost-free bimetric theory. In the presence of the matter source, which we treat as a small perturbation, the equations for the non-dynamical field can be solved perturbatively for its vierbein. This results in ghost-free bimetric theory in vierbein formulation with a modified matter coupling. To lowest order in matter perturbations, we precisely obtain an effective matter coupling that has been suggested earlier in the literature. This coupling contains a linear combination of the two vierbeine whose corresponding metric fluctuation coincides with the massless spin-2 mode. In the past, bimetric theory with this symmetric coupling has been treated as an effective theory valid at low energies without a ghost-free completion. Our results demonstrate that the effective matter coupling can be rendered entirely ghost-free by including the higher-order corrections obtained from the trimetric setup., 15 pages, fixed typo in equation (4.6)

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

Linear transformations of the eight-dimensional state space that leave the phase of plane waves invariant form the SO(4, 4) group. The transformations that do not mix the x-space and the separated k-space form the subgroup GL(4, R), and this transformation can be viewed as vierbeins in Einstein’s gravity theory. The defining representation and the spinor representation of the phase invariance group are concretely constructed. Plane waves are considered as a Gaussian solution of eight-dimensional simple harmonic oscillations with negative potential.

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. On this manifold, we construct a Cartan connection and find its torsion tensor. We also find the geodesics of the Cartan connection.

We revise the notion of a Cartan extension and consider in detail simple examples of known Cartan extensions.

The main goal of the present work is to analyze the cosmological scenario of the induced gravity theory developed in previous works. Such a theory consists on a Yang-Mills theory in a four-dimensional Euclidian spacetime with $SO(m,n)$ such that $m+n=5$ and $m\in\{0,1,2\}$ as its gauge group. This theory undergoes a dynamical gauge symmetry breaking via an In\"on\"u-Wigner contraction in its infrared sector. As a consequence, the $SO(m,n)$ algebra is deformed into a Lorentz algebra with the emergency of the local Lorentz symmetries and the gauge fields being identified with a vierbein and a spin connection. As a result, gravity is described as an effective Einstein-Cartan-like theory with ultraviolet correction terms and a propagating torsion field. We show that the cosmological model associated with this effective theory has three different regimes. In particular, the high curvature regime presents a de Sitter phase which tends towards a $\Lambda$CDM model. We argue that $SO(m,n)$ induced gravities are promising effective theories to describe the early phase of the universe., Comment: 22 pages, 3 figures

We construct a null tetrad from a non-diagonal spherically symmetric vierbein field, involving two unknown functions of radial coordinate r. This null tetrad is applied to the equations of motion of non-minimal torsion-matter coupling extension of f(T) gravity. Exact vacuum solution with two constants of integration is obtained. The scalar torsion related to this special solution is constant. The energy of this solution is calculated using the Euclidean continuation method and the known value is derived. We also calculate the thermodynamical quantities of charged Schwarzschild spacetime and investigate the validity of the first law of thermodynamics and quantum statistical relation.

Inspired by the $f(R)$ non-linear massive gravity, we propose a new kind of modified gravity model, namely $f(T)$ non-linear massive gravity, by adding the dRGT mass term reformulated in the vierbein formalism, to the $f(T)$ theory. We then investigate the cosmological evolution of $f(T)$ massive gravity, and constrain it by using the latest observational data. We find that it slightly favors a crossing of the phantom divide line from the quintessence-like phase ($w_{de} > -1$) to the phantom-like one ($w_{de} < -1$) as redshift decreases., 12 pages, 4 figures, revtex4, Commun. Theor. Phys. in press; v2: published version

We derive the most general effective low-energy potential to order O(1/m) for slow Dirac fermions with mass m, coupled to gravitational, chameleon and torsion fields in the Einstein-Cartan gravity. The obtained results can be applied to the experimental analysis of gravitational, chameleon and torsion interactions in terrestrial laboratories. We discuss the use of rotating coordinate systems, caused by rotations of devices, for measurements of the torsion vector and tensor components, caused by minimal torsion--fermion couplings (Ivanov and Wellenzohn, Phys. Rev. D92, 065006 (2015)). Using the most general form of a metric tensor of curved spacetimes in rotating coordinate systems, proposed by Obukhov, Silenko, and Teryaev (Phys. Rev. D84, 024025 (2011)), we extend this metric by the inclusion of the chameleon field and calculate the set of vierbein fields, in terms of which Dirac fermions couple to torsion vector and tensor components through minimal torsion-fermion couplings. For such a set of vierbein fields we discuss a part of the effective low-energy potential for slow Dirac fermions, coupled to gravitational, chameleon and torsion fields to order O(1) in the large fermion mass expansion., Comment: 16 pages, no figures, the paper is accepted for publication in Physical Review D

We propose an effective Lorentz violating electrodynamics model via static de Sitter metric which is deviated from Minkowski metric by a minuscule amount depending on the cosmological constant. We obtain the electromagnetic field equations via the vierbein decomposition of the tensors. In addition, as an application of the electromagnetic field equations obtained, we get the solutions of electrostatic field and magnetostatic field due to a point charge and a circle current respectively and discussed the implication of the effect of Lorentz violation in our electromagnetic theory., 13 pages

Given a constrained variational problem on the 1-jet extension J1Y of a fibre bundle p:Y→X, under certain conditions on the constraint submanifold S⊂J1Y, we characterize the space of admissible infinitesimal variations of an admissible section s:X→Y as the image by a certain first order differential operator, Ps, of the space of sections Γ(X,s∗VY). In this way we obtain a constrained first variation formula for the Lagrangian density Lω on J1Y, which allows us to characterize critical sections of the problem as admissible sections s such that Ps+ELω(s)=0, where Ps+ is the adjoint operator of Ps and ELω(s) is the Euler–Lagrange operator of the Lagrangian density Lω as an unconstrained variational problem. We introduce a Cartan form on J2Y which we use to generalize the Cartan formalism and Noether theory of infinitesimal symmetries to the constrained variational problems under consideration. We study the relation of this theory with the Lagrange multiplier rule as well as the question of regularity in this framework. The theory is illustrated with several examples of geometrical and physical interest.

Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein Gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin connection variables and auxiliary two-form fields. In the systematic study of all possible embeddings of Einstein gravity into that formulation with auxiliary fields, the introduction of a ``bi-complex'' algebra possesses crucial technical advantages. Certain components of the new two-form fields directly provide canonical momenta for spatial components of all Cartan variables, whereas the remaining ones act as Lagrange multipliers for a large number of constraints, some of which have been proposed already in different, less radical approaches. The time-like components of the Cartan variables play that role for the Lorentz constraints and others associated to the vierbein fields. Although also some ternary ones appear, we show that relations exist between these constraints, and how the Lagrange multipliers are to be determined to take care of second class ones. We believe that our formulation of standard Einstein gravity as a gauge theory with consistent local Poincare algebra is superior to earlier similar attempts., more corrected typos, added references

As a simple example of the "blurring" influence of noncommutative geometry, a noncommutative version of a high-frequency perturbation of a background metric is calculated. The relation between the principal null vectors of the wave and those of the induced symplectic structure is in this case particularly clear.

It has been proposed to abandon the requirement that parallel transporters in gauge theories are unitary (or pseudoorthogonal). This leads to a geometric interpretation of Vierbein fields as parts of gauge fields, and nonunitary parallel transport in extra directions yields Higgs fields. In such theories, the holonomy group H is larger than the gauge group G. Here we study a 1-dimensional model with fermions which retains only the extra dimension, and which is soluble in the sense that its renormalization group flow may be exactly computed, with G=SU(2) and noncompact subgroup H of GL(2,C), or G=U(2), H=GL(2,C). In all cases the asymptotic behavior of the Higgs potential is computed, and with one possible exception for G=SU(2), H=GL(2,C), there is a flow of the action from an UV-fix point which describes a SU(2)-gauge theory with unitary parallel transporters, to a IR-fixpoint. We explain how a splitting of the masses of fermions of different flavor can arise through spontaneous symmetry breaking., 34 pages, 1 figure

In a previous paper, we addressed the method of Abelian decomposition to the case of SU (N) Yang–Mills theory. Here, we extend the decomposition method further to the general case. With the Cartan–Weyl basis we decompose semisimple group connection and discuss the SO(3,1) group in particular. In terms of the vierbein projection, we propose two two-forms as the U(1) gauge fields in the SO(3,1) gauge theory and show that these two-forms are just the different cosmic string tensors. Meanwhile, these two-forms indicate that the cosmic strings appear naturally in the Lorentz spacetime, i.e. the torsion in the Riemann–Cartan spacetime is not necessary for the cosmic strings.

The canonical covariant formalism (CCF) of the topological five-dimensional Chern–Simons gravity is constructed. Because this gravity model naturally contains a Gauss–Bonnet term, the extended CCF valid for higher curvature gravity must be used. In this framework, the primary constraint and the total Hamiltonian are found. By using the equations of the CCF, it is shown that the bosonic five-form which defines the total Hamiltonian is a first-class dynamical quantity strongly conserved. In this context the equations of motion are also analyzed. To determine the effective interactions of the model, the toroidal dimensional reduction of the five-dimensional Chern–Simons gravity is carried out. Finally the first-order CCF and the usual canonical vierbein formalism (CVF) are related and the Hamiltonian as generator of time evolution is constructed in terms of the first-class constraints of the coupled system.

Following the ideas of Elie Cartan (1928), we use Cartan’s equivalence method and the notion of Cartan’s affine generalized space and development to geometrize non-holonomic mechanics.

We study the cosmological perturbations in teleparallel dark energy models in which there is a dynamical scalar field with a non-minimal coupling to gravity [1]. We use vierbein perturbation scenario in this work [2], and we find that the propagating degrees of freedom are the same as in quintessence cosmology despite that variables of the perturbed vierbein field are greater than those in metric theories. We numerically show some evident discrepancy from general relativity in the evolutions of the perturbations on all scales of the universe as seen in figure 1. We also demonstrate that the gravitational interactions are enhanced during the unique tracker evolutions in these models [3,4].

We present a first order theory of gravity (vierbein formulation) on noncommutative spacetime. The first order formalism allows to couple the theory to fermions. This NC action is then reinterpreted (using the Seiberg-Witten map) as a gravity theory on commutative spacetime that contains terms with higher derivatives and higher powers of the curvature and depend on the noncommutativity parameter \(\theta \). When the noncommutativity is switched off we recover the usual gravity action coupled to fermions. The first nontrival corrections to the usual gravity action coupled to fermions are explicitly calculated.

This paper deals with various interrelations between strings and surfaces in three-dimensional ambient space, two-dimensional integrable models, and two-dimensional and four-dimensional decomposed SU(2) Yang-Mills theories. Initially, a spinor version of the Frenet equation is introduced in order to describe the differential geometry of static three-dimensional stringlike structures. Then its relation to the structure of the su(2) Lie algebra valued Maurer-Cartan one-form is presented, while by introducing time evolution of the string a Lax pair is obtained, as an integrability condition. In addition, it is shown how the Lax pair of the integrable nonlinear Schrodinger equation becomes embedded into the Lax pair of the time extended spinor Frenet equation, and it is described how a spinor-based projection operator formalism can be used to construct the conserved quantities, in the case of the nonlinear Schrodinger equation. Then the Lax pair structure of the time extended spinor Frenet equation is related to properties of flat connections in a two-dimensional decomposed SU(2) Yang-Mills theory. In addition, the connection between the decomposed Yang-Mills and the Gauss-Codazzi equation that describes surfaces in three-dimensional ambient space is presented. In that context the relation between isothermic surfaces and integrable models is discussed. Finally, the utility of the Cartan approach to differential geometry is considered. In particular, the similarities between the Cartan formalism and the structure of both two-dimensional and four-dimensional decomposed SU(2) Yang-Mills theories are discussed, while the description of two-dimensional integrable models as embedded structures in the four-dimensional decomposed SU(2) Yang-Mills theory are presented.

We investigate a possibility that our four-dimensional world is a brane-like object embedded in a higher dimensional spacetime. In such a situation, the transverse coordinates of the brane become the Nambu-Goldstone bosons which appear as a result of spontaneous breaking of the translation symmetry. We determine the form of the effective action of the system, finding the explicit form of the vierbein induced on the brane in terms of the Nambu-Goldstone boson variables and the bulk vielbein. As was pointed out in the previous paper, the Kaluza-Klein mode couplings are suppressed by the effect of the brane fluctuation and the suppression is stronger if the brane tension is smaller. However, we here show that the brane tension cannot be arbitrarily small since the inverse of the brane tension gives the coupling constant of the Nambu-Goldstone bosons. A rather stringent bound is obtained for the brane tension and the fundamental (`string') scale from the consideration of the cooling process of the supernova., Comment: 35 pages, 5 eps figures

In the light of the local Lorentz transformations and the general Noether theorem, a new formulate of the general covariant energy-momentum conservation law in f(R) gravity is obtained, which does not depend on the coordinative choice.

We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous d\'emontrons que les seules vari\'et\'es projectives de Calabi--Yau qui poss\`edent des g\'eom\'etrie holomorphes de Cartan sont les vari\'et\'es ab\'eliennes.), Comment: 4 pages

Instantons in regular and singular gauges and explicit multi-instanton solutions are generalized to curvilinear coordinates. Expressions can be simplified by fitting gauge orientation of the solutions to the vierbein. The gauge transformation replaces η -symbols used in Cartesian formalism by constant ξ -symbols and generates a compensating addition to the gauge potential of pseudoparticles. Typical examples (4-spherical, (2+2) - and (3+1) -cylindrical coordinates) are studied explicitly. Effects of singularities of the compensating field on gauge dependent quantities are discussed. PACS: 12.38.Aw, 12.38.Lg, 11.15.-q, 11.15.Kc, Keywords: Instantons, Multi-instanton solutions, Curvilinear coordinates, Confinement models

The Poincare-Cartan (PC) form of a Lagrangian on the bundle J2 = J2(N,M) is, as a general rule, defined on J3 thus leading to a non-equivalence between Euler-Lagrange and Hamilton-Cartan equations. This naturally leads to the problem of determining what Lagrangians have a PC form projectable onto J2, as they will then admit a second-order Hamiltonian formalism. There are specific examples of this phenomenon in field theory. This paper provides an explicit classification of such Lagrangians.

We show how some basic dynamical ideas can be brought to bear on the study of Cartan geometries. In our main results, we give conditions for certain types of Cartan geometries to have constant curvature. We also consider ergodic actions of `higher-rank' semisimple groups on bundles supporting (not necessarily invariant) Cartan connections and show that these are `standard locally homogeneous' actions provided that some noncompact 1-parameter subgroup `preserves' a Cartan geometry.

We give a simple, general approach for constructing a twistorspace Z with a CR structure for a manifold Mwith a geometric structure. The starting point is a principalG-bundle π: P → M with a Cartanconnection. The twistor space is an appropriate subgroup ofG. The integrability condition for the CR structure isdescribed in terms of the curvature of the Cartan connection.

JBW*-triples can be described (modulo W*-algebras, compare [13]) by those of type I. Among these the (complex) Cartan factors are the building blocks. We determine for every complex Cartan factorU all conjugations of the underlying complex Banach space and hence all real forms (in the sense of [15]) ofU, calledreal Cartan factors. We also give a concrete list of all isomorphy classes of real Cartan factors which generalizes the classification of LOOS [23] to infinite dimensions. Furthermore, we give an explicit description of the full automorphism group as well as the group of all surjective ℝ-linear isometries for every non-exceptional real Cartan factor and decide which of the real or complex Cartan factors are isometrically equivalent to each other as real Banach spaces.

Massive gravity in three dimensions accepts several different formulations. Recently, the three-dimensional bigravity de Rham--Gabadadze--Tolley (dRGT) model in first-order form, Zwei-Dreibein gravity, was considered by Bergshoeff et al., and it was argued that the Boulware-Deser mode is killed by extra constraints. We revisit this assertion and conclude that there are sectors on the space of initial conditions, or subsets of the most general such model, where this mode is absent. But, generically, the theory does carry 3 degrees of freedom, and thus the Boulware-Deser mode is still active. Our results also sheds light on the equivalence between metric and vierbein formulations of the dRGT model.

We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter \theta. We thus express the action at order \theta^n+2 in terms of noncommutative fields of order at most \theta^n+1 and, iterating the procedure, in terms of noncommutative fields of order at most \theta^n. This in particular provides the explicit expression of the action at order \theta^2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations., Comment: 14 pages

A coupling between the spacetime geometry and a scalar field involving the Euler four-form can have important consequences in General Relativity. The coupling is a four-dimensional version of the Jackiw-Teitelboim action, in which a scalar couples to the Euler two-form in two dimensions. In this case the first order formalism, in which the vierbein (or the metric) and the spin connection (or the afine connection) are varied independently, is not equivalent to the second order one, in which the geometry is completely determined by the metric. This is because the torsion postulate (T=0) is not valid and one cannot algebraically solve the spin connection from its own field equation. The direct consequence of this obstruction is that the torsion becomes a new source for the metric curvature, and even if the scalar field is very slowly varying over cosmic scales as to have no observable astronomical efects at the galactic scale, it has important dynamical efects that can give rise to a cosmological evolution radically diferent from the standard FRWL model.

We study the post-Newtonian limit in the teleparallel equivalent of General Relativity with a scalar field which non-minimally couples to gravity. The metric perturbation is obtained from the vierbein field expansion with respect to the Minkowski background. Due to the structure of the teleparallel gravity Lagrangian, the potential of the scalar field shows no effect to the parametrized post-Newtonian parameters, and compatible results with Solar System observations are found., Comment: 18 pages, no figure, revised version accepted for publication in PRD

We present the complete form of the decoupling limit of ghost-free massive gravity with a Minkowski reference metric, including the full interactions of the helicity-1 and helicity-0 modes of the massive spin-2 field. While in the metric language the square root structure of the mass terms makes it difficult to find a simple way to write down the interactions, we show that using the vierbein formulation of massive gravity, including Stueckelberg fields for both diffeomorphism and local Lorentz symmetries, we can find an explicitly resummed expression for the helicity-1 field interactions. We clarify the equations of motion for the Lorentz Stueckelberg fields and how these generate the symmetric vierbein condition which guarantees equivalence between the vierbein and metric formulations of massive gravity., Comment: 20 pages, typos corrected, references added