Holst Term Research Articles

Dark torsion oscillations and bounces induced by Barbero-Immirzi and quantum correction on Einstein-Cartan gravity with very light inflatons

In this paper we present several examples of how to constrain loop quantum gravity Barbero-Immirzi (BI) parameter, either by using quantum corrections in Nieh–Yan (NY) modified Einstein-Cartan (EC) gravity with the Holst term. The reason why we introduce some extra terms in the actions, is that even when one adds both NY and Holst terms, being topological terms, they do not contribute to the dynamics of the problem and to the equations of motion (EOM). Examples run from quantum corrections and torsion mass in the dark universe with very light inflaton approximation to bouncing cosmologies in NY gravity. This follows similar model from Tukhashvili and Steinhardt (2023) of a bouncing cosmology induced by torsion condensates in EC gravity. The Lagrangian in this paper, is shown to be expressed in terms of NY polynomial form. The quantum one-loop correction is taken in NY Lagrangian as the inverse of BI parameter. This has been recently appeared in He et al. (2024) quantum corrections in Einstein-Cartan-Nieh–Yan one loop quantum gravity in Starobinsky inflation. We show that to have a cosmological NY dark universe one needs a real BI parameter. Inflation Hubble scale of the order of O(100eV) and very light inflaton of mass ∼10−12TeV are used due to its relation with dark matter. The very light mass character of inflatons and much heavier torsion are responsible to induced cosmological bounce in dark universe. When one introduces the quantum correction parameter b as second-order in front of the NY topological term in squared and introduce the NY term back into the ECH gravity, the EC gravity limit where BI parameter β→∞, yields a chiral fermion density ρ5. This result lies very close to the upper bound in the range of 0≤β≤1.185, previously obtained by Panza et al. Analysis of fermionic currents shows that Holst term can be eliminated by choosing β=0.288. Massive torsion oscillates in Einstein-Cartan gravity in the BI limit of β→∞.

Matter-antimatter asymmetry induced by Barbero-Immirzi parameter

We review the baryon asymmetry generated in the early Universe in the framework of the metric-spinor gravity with the Holst term, which accounts for the Barbero-Immirzi parameter. For the generation of the matter-antimatter asymmetry, we have considered the model in which the time derivative of the Ricci scalar couples with matter currents (this model is known in the literature as gravitational baryogenesis). By using the current bounds on the parameter of the asymmetry, the ratio of baryon number to the entropy density, we infer a bound on the parameters characterizing the model under consideration (that is, the cosmological model in the presence of the torsion and Barbero-Immirzi parameter).

Null boundary gravitational charges from local Lorentz symmetries

In this paper, we revisit the null boundary gravitational charge in the Newman-Penrose formalism with special emphasis on the charges from local Lorentz transformations. We find that there is one more charge derived from the local Lorentz transformation and the new charge is purely from the Holst term. This reveals a remarkable fact that trivial terms which do not change classical equations of motion can not only affect the boundary degrees of freedom through their contributions to the boundary charges but also have their own rights to create new boundary degrees of freedom.

A comparison between the Jordan and Einstein frames in Brans-Dicke theories with torsion

In recent years, gravitational models motivated by quantum corrections to gravity which introduce higher order terms like \(R^{2}\) or terms in which the Riemann tensor is not symmetric have been studied by several authors in the form of a general Brans-Dicke type model containing the Ricci scalar, the Holst term and the Nieh-Yan invariant. In this paper we focus on the less explored Jordan frame of such theories and in the comparison between both this frame and the Einstein one. Furthermore, we discuss the role of the transformation of the torsion under conformal transformations and show that the transformation proposed in this paper (extended conformal transformation) contains a special case of the projective transformation of the connection used in some of the papers that motivated this work. We discuss the role and advantages of the extended conformal transformation and show that this new approach can have interesting consequences by working with different variables such as the metric and torsion. Moreover, we study the stability of the system via a dynamical analysis in the Jordan frame, this in order to analyze whether or not we have the fixed points that can be later identified as the inflationary attractor and the unstable fixed point where inflation could take place. Finally we study the scale invariant case of the general model in the Jordan frame. We find out that both the scalar spectral index and the tensor-to-scalar ratio are in agreement with the latest Planck results.

Einstein–Cartan–Holst–Proca dynamos massive photons as dark matter and GWs

In this paper, we extend Shaposhnikov et al.'s parity violating the Einstein–Cartan–Holst action by adding to Holst term H, a Proca photon–torsion coupling χpt = 10−24 as computed by De Sabbata, Garcia de Andrade, and Sivaram. The strength of Holst term is obtained, in the case of early universe torsion as, T ∼ 1 MeV. The gravity part is the Planck mass product with the Holst term. The photon mass in this case yields m2 γ H ∼ 10−69 GeV4. Therefore, in this case, the photon interacts very weakly with spin-torsion matter in Einstein–Cartan (EC) gravity and the strength of the Maxwell–Proca–Holst interaction is much weaker than in the pure gravity sector. For dark photons, this situation changes drastically since a dark photon mass reaches 1.5 GeV, and from this mass and T = 10−3 GeV, one obtains a Maxwell–Proca–Holst strength of 10−6 GeV4, which is now comparable with the gravity sector. Dynamo mechanism competes with the chirality and dissipation by the handness of the magnetic field with respect to the torsion trace vector to regenerate the magnetic field decay. Bombacigno and Mantovani recently found a sign of the Immirzi parameter in the Holst–Nieh–Yan extension of EC gravity. Here, we also find torsion waves for the Immirzi inverse parameter in the sense that this divergence of torsion trace is porportional to the Immirzi field.

Imprints from a Riemann–Cartan space-time on the energy levels of Dirac spinors

In this work, we investigate the effects of the torsion–fermionic interaction on the energy levels of fermions within a Riemann–Cartan geometry using a model-independent approach. We consider the case of fermions minimally coupled to the background torsion as well as non-minimal extensions via additional couplings with the vector and axial fermionic currents which include parity-breaking interactions. In the limit of zero-curvature, and for the cases of constant and spherically symmetric torsion, we find a Zeeman-like effect on the energy levels of fermions and anti-fermions depending on whether they are aligned/anti-aligned with respect to the axial vector part of the torsion (or to specific combination of torsion quantities), and determine the corresponding fine-structure energy transitions. We also discuss non-minimal couplings between fermionic fields and torsion within the Einstein–Cartan theory and its extension to include the (parity-breaking) Holst term. Finally we elaborate on the detection of torsion effects related to the splitting of energy levels in astrophysics, cosmology and solid state physics using current capabilities.

Gravitational SL(2, \u211d) algebra on the light cone

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini-Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the SL(2, ℝ) symmetries of the boundary fields are manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental SL(2, ℝ) variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.

Comments on the symmetry breaking condition in MacDowell–Mansouri action

In this work, we study the symmetry breaking conditions, given by a (anti)de Sitter-valued vector field, of a full (anti)de Sitter-invariant MacDowell–Mansouri inspired action. We show that under these conditions, the action breaks down to General Relativity with a cosmological constant, the four-dimensional topological invariants, as well as the Holst term. We obtain the equations of motion of this action, and analyze the symmetry breaking conditions.

Parity violating metric-affine gravity theories

We study a metric-affine gravitational theory given by the Einstein–Hilbert (EH) action plus a parity violating contribution (which we will refer to as the Hojman term, also known as Holst term) in vacuum. We find out that for a certain value of the Barbero–Immirzi (BI) parameter the total action possesses a remarkable invariance under particular transformations of the affine connection. We prove that in all cases, with appropriate gauge choices, the connection reduces to the Levi-Civita one and that the theory turns out to be equivalent to general relativity (GR) in vacuum. Subsequently, we generalize our discussion and analyze the case of metric-affine f(R) gravity plus the Hojman term. In particular, we show that for f′(R) ≠ constant the theory results to be on-shell equivalent to a metric-compatible torsionless scalar–tensor model with a propagating pseudo-scalar. Matter coupling of the aforementioned models is also discussed, together with explicit examples and applications.

Higgs inflation with the Holst and the Nieh–Yan term

The action of loop quantum gravity includes the Holst term and/or the Nieh-Yan term in addition to the Ricci scalar. These terms are expected to couple non-minimally to the Higgs. Thus the Holst and Nieh-Yan terms contribute to the classical equations of motion, and they can have a significant impact on inflation. We derive inflationary predictions in the parameter space of the non-minimal couplings, including non-minimally coupled terms up to dimension 4. Successful inflation is possible even with zero or negative coupling of the Ricci scalar. Notably, inflation supported by the non-minimally coupled Holst term alone gives almost the same observables as the original metric formulation plateau Higgs inflation. A non-minimally coupled Nieh-Yan term alone cannot give successful inflation. When all three terms are considered, the predictions for the spectral index and tensor-to-scalar ratio span almost the whole range probed by upcoming experiments. This is not true for the running of the spectral index, and many cases are highly tuned.

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.

A note on dual gravitational charges

Dual gravitational charges have been recently computed from the Holst term in tetrad variables using covariant phase space methods. We highlight that they originate from an exact 3-form in the tetrad symplectic potential that has no analogue in metric variables. Hence there exists a choice of the tetrad symplectic potential that sets the dual charges to zero. This observation relies on the ambiguity of the covariant phase space methods. To shed more light on the dual contributions, we use the Kosmann variation to compute (quasi-local) Hamiltonian charges for arbitrary diffeomorphisms. We obtain a formula that illustrates comprehensively why the dual contribution to the Hamiltonian charges: (i) vanishes for exact isometries and asymptotic symmetries at spatial infinity; (ii) persists for asymptotic symmetries at future null infinity, in addition to the usual BMS contribution. Finally, we point out that dual gravitational charges can be equally derived using the Barnich-Brandt prescription based on cohomological methods, and that the same considerations on asymptotic symmetries apply.

Hamiltonian derivation of dual gravitational charges

We provide a Hamiltonian derivation of recently discovered dual BMS charges. In order to do so, we work in the first order formalism and add to the usual Palatini action, the Holst term, which does not contribute to the equations of motion. We give a method for finding the leading order integrable dual charges à la Wald-Zoupas and construct the corresponding charge algebra. We argue that in the presence of fermions, the relevant term that leads to dual charges is the topological Nieh-Yan term.

Implications of the Holst term in a f(R) theory with torsion

We analyze a modified $f(R)$ theory of gravity in the Palatini formulation, when an Holst term endowed with a dynamical Immirzi field is included. We study the basic features of the model, especially in view of liminating the torsion field via the Immirzi field and the scalar-tensor degrees of freedom of the $f(R)$ model. The main task of this study is the investigation of the morphology of the gravitational wave polarization when their coupling to a circle of test particles is considered. We first observe that the dynamics of the scalar mode of the $f(R)$ Lagrangian is frozen out, since its first order term identically vanishes. This allows a detailed characterization of the linearized theory, which outlines the emergence of a modified Newtonian potential in the static limit, and when time independence is relaxed a standard gravitational wave plus the scalar wave associated to the Immirzi field. Investigating the effect of the coupling of this scalar-tensor wave on a circle of test particles, we arrive to define two effective gravitational polarizations, corresponding to an equivalent phenomenological wave, whose morphology is anomalous with respect the standard case of General Relativity. In fact, the particle circle suffers modifications as it was subjected to modified plus and cross modes, whose specific features depend on the model free parameters and are, in principle, detectable via a data analysis procedure.

Noncommutative gravity with self-dual variables

We build a noncommutative extension of Palatini–Holst theory on a twist-deformed spacetime, generalizing a model that has been previously proposed by Aschieri and Castellani. The twist deformation entails an enlargement of the gauge group, and leads to the introduction of new gravitational degrees of freedom. In particular, the tetrad degrees of freedom must be doubled, thus leading to a bitetrad theory of gravity. The model is shown to exhibit new duality symmetries. The introduction of the Holst term leads to a dramatic simplification of the dynamics, which is achieved when the Barbero–Immirzi parameter takes the value , corresponding to a self-dual action. We study in detail the commutative limit of the model, focusing in particular on the role of torsion and non-metricity. The effects of spacetime noncommutativity are taken into account perturbatively, and are computed explicitly in a simple example. Connections with bimetric theories and the role of local conformal invariance in the commutative limit are also explored.

Comments on MacDowell–Mansouri gravity and torsion

Starting with the MacDowell–Mansouri formulation of gravity with a [Formula: see text] gauge group, we introduce new parameters into the action to include the nondynamical Holst term, and the topological Nieh–Yan and Pontryagin classes. Then, we consider the new parameters as fields and analyze the solutions coming from their equations of motion. The new fields introduce torsional contributions to the theory that modify Einstein’s equations.

Discussion on possible effects of the Barbero-Immirzi parameter at the TeV-scale particle physics

In this paper, we analyze a curvature- and torsion-square quantum gravity action with an additional Holst term minimally coupled to a massive Dirac field in four dimensions. The main purpose here is to try to estimate and compare the value of the Barbero-Immirzi parameter with its currently known results. To do that, we work out the physical mass of the fermion as a function of this parameter in a perturbative one-loop calculation, assuming the scenario of a physics at the TeV scale.

Quantum Einstein–Cartan theory with the Holst term

The Holst term represents an interesting addition to the Einstein–Cartan theory of gravity with torsion. When this term is present the contact interactions between vector and axial vector fermion currents gain an extra parity-violating component. We re-derive this interaction using a simple representation for the Holst term. The same representation serves as a useful basis for the calculation of one-loop divergences in the theory with external fermionic currents and cosmological constant. Furthermore, we explore the possibilities of the on-shell version of the renormalization group and construct equations for the running of dimensionless parameters related to currents and for the effective Barbero–Immirzi parameter.

Note on Kerr/CFT correspondence in a first order formalism

In symmetry-based approaches to black hole entropy, we calculate the central charge of the Virasoro algebra in the first order formulation of gravity for both Palatini and Holst actions. In these calculations, we make use of the near-horizon extremal Kerr metric and the Kerr/conformal field theory correspondence. For the Palatini action the results obtained in the second order formulation are reproduced. We also argue that the Holst term does not contribute to the charge algebra no matter what geometry/boundary conditions one is considering.

Fermion Interactions, Cosmological Constant and Space-Time Dimensionality in a Unified Approach Based on Affine Geometry

One of the main features of unified models, based on affine geometries, is that all possible interactions and fields naturally arise under the same standard. Here, we consider, from the effective Lagrangian of the theory, the torsion induced 4-fermion interaction. In particular, how this interaction affects the cosmological term, supposing that a condensation occurs for quark fields during the quark-gluon/hadron phase transition in the early universe. We explicitly show that there is no parity-violating pseudo-scalar density, dual to the curvature tensor (Holst term) and the spinor-bilinear scalar density has no mixed couplings of A-V form. On the other hand, the space-time dimensionality cannot be constrained from multidimensional phenomenological models admitting torsion.