Three-form Field Research Articles

The dynamics of three-forms in thick branes

In this work, we investigate thick brane models with a single three-form field. We find novel solutions for thick braneworlds where only three-forms exist and interact gravitationally in the bulk, both with and without matter fields. We use an additional scalar field as proxy for the matter fields. As an initial study, we consider the results here in contrast to the single scalar field thick braneworld case. The properties of the specific three-form parameterisation limits the freedom we have to choose the form of the warp factor, leading to a closed system of equations with nontrivial yet unstable solutions. The stability of the gravitational sector for thick brane three-forms is investigated and the models are shown to be unstable against small perturbations of the metric, further indicating that three-forms cannot exist stably in thick braneworld settings.

Rotating traversable wormhole geometries in the presence of three-form fields

In this work, we study the rotating wormhole geometries supported by a three-form field. We demonstrate for particular choices of parameters that it is possible for the matter fields threading the wormhole to satisfy the null and weak energy conditions throughout the spacetime, when the three-form field is present. In this case, the form field is interpreted as supporting the wormhole and the energy condition violation is restricted only to the 3-form fields. Thus, the three-form curvature terms, which may be interpreted as a gravitational fluid, sustain these wormhole geometries. Additionally, we also address the ergoregion of the solutions.

Contributions to the muon g − 2 from a three-form field

We examine contributions to the muon dipole moment g − 2 from a 3-form field Ω, which naturally arises from many fundamental theories, such as the string theory and the hyperunified field theory. In particular, by calculating the one-loop Feynman diagram, we have obtained the leading-order Ω-induced contribution to the muon g − 2, which is found to be finite. Then we investigate the theoretical constraints from perturbativity and unitarity. Especially, the unitarity bounds are yielded by computing the tree-level μ+μ− scattering amplitudes of various initial and final helicity configurations. As a result, despite the strong unitarity bounds imposed on this model of Ω, we have still found a substantial parameter space which can accommodates the muon g − 2 data.

Fractional quantum Hall effect for extended objects: from skyrmionic membranes to dyonic strings

It is well known that in two spatial dimensions the fractional quantum Hall effect (FQHE) deals with point-like anyons that carry fractional electric charge and statistics. Moreover, in presence of a SO(3) order parameter, point-like skyrmions emerge and play a central role in the corresponding quantum Hall ferromagnetic phase. In this work, we show that in six spatial dimensions, the FQHE for extended objects shares very similar features with its two-dimensional counterpart. In the higher-dimensional case, the electromagnetic and hydrodynamical one-form gauge fields are replaced by three-form gauge fields and the usual point-like anyons are replaced by membranes, namely two-dimensional extended objects that can carry fractional charge and statistics. We focus on skyrmionic membranes, which are associated to a SO(5) order parameter and give rise to an higher-dimensional generalizaton of the quantum Hall ferromagnetism. We show that skyrmionic membranes naturally couple to the curved background through a generalized Wen-Zee term and can give us some insights about the chiral conformal field theory on the boundary. We then present a generalization of the Witten effect in six spatial dimensions by showing that one-dimensional extended monopoles (magnetic strings) in the bulk of the FQH states can acquire electric charge through an axion field by becoming dyonic strings.

The E10 Wheeler-DeWitt operator at low levels

We consider the Wheeler-DeWitt operator associated with the bosonic part of the Hamiltonian of D = 11 supergravity in a formulation with only the spatial components of the three-form and six-form fields, and compare it with the E10 Casimir operator at low levels, to show that these two operators precisely match modulo spatial gradients up to and including mathfrak{gl} 10 level ℓ = 2. The uniqueness of the E10 Casimir operator eliminates all ordering ambiguities in the quantum Hamiltonian, at least up to the level considered. Beyond ℓ ≥ 3 the two operators are expected to start to differ from each other, as they do so for the classical expressions. We then consider truncations of the E10 Wheeler-DeWitt operator for various finite-dimensional subgroups of E10 in order to exhibit the automorphic properties of the associated wave functions and to show that physically sensible wave functions generically vanish at the cosmological singularity, thus providing new and more sophisticated examples of DeWitt’s proposed mechanism for singularity resolution in quantum gravity. Our construction provides novel perspectives on several unresolved conceptual issues with the Wheeler-DeWitt equation, such as the question of observables in quantum gravity, or the issue of emergent space and time in a purely algebraic framework. We also highlight remaining open questions of the E10 framework.

Spin(11, 3), particles, and octonions

The fermionic fields of one generation of the Standard Model (SM), including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S+ of the group Spin(11, 3). We describe an octonionic model for Spin(11, 3) in which the semi-spinor representation gets identified with S+=O⊗Õ, where O,Õ are the usual and split octonions, respectively. It is then well known that choosing a unit imaginary octonion u∈Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion ũ∈Im(Õ) equips Õ with a complex structure J̃, except that there are now two inequivalent complex structures, one parameterized by a choice of a timelike and the other of a spacelike unit ũ. In either case, the identification S+=O⊗Õ implies that there are two natural commuting complex structures J,J̃ on S+. Our main new observation is that the subgroup of Spin(11, 3) that commutes with both J,J̃ on S+ is the direct product Spin(6) × Spin(4) × Spin(1, 3) of the Pati–Salam and Lorentz groups, when ũ is chosen to be timelike. The splitting of S+ into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S+ into eigenspaces of J̃ corresponds to splitting of Lorentz Dirac spinors into two different chiralities. This provides an efficient bookkeeping in which particles are identified with components of such an elegant structure as O⊗Õ. We also study the simplest possible symmetry breaking scenario with the “Higgs” field taking values in the representation that corresponds to three-forms in R11,3. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM and thus breaks Spin(11, 3) down to the product of the SM, Lorentz, and U(1)B−L groups, with the last one remaining unbroken. This three-form Higgs field also produces the Dirac mass terms for all the particles.

Small-instanton transitions in F-theory

We study the phase transition between G-instantons and D3-branes quantitatively. A G-instanton is a classical solution to the self-dual equation of the M/F-theory three-form tensor field C in the complex fourfold. This phase transition is dual to that between ‘vertical’ small instantons and 5-branes in the heterotic string. Using G as a background gauge flux, we may dynamically control the gauge symmetry breaking and connect between different vacua of F-theory. We may understand the amount of flux undergoing the phase transition and the resulting number of D3-branes in terms of group-theoretical quantities. We also discuss the resulting chirality change and preservation of anomaly freedom.

Relieve the H0 tension with a new coupled generalized three-form dark energy model

In this work we propose a new coupled generalized three-form dark energy model, in which dark energy are represented by a three-form field and dark matter are represented by point particles that can be regarded as dust. We first perform a dynamical analysis on the new model and obtain four fixed points, including a saddle point representing a radiation dominated Universe, a saddle point representing a matter dominated Universe, and two attractors representing two dark energy dominated Universes. We then use the observational data, including cosmic microwave background (CMB) data, baryon acoustic oscillations (BAO) data, and Type Ia supernovae (SN Ia) data to constrain the model parameters of the coupled generalized three-form dark energy model. For comparison, we also consider the coupled three-form dark energy model, generalized three-form dark energy model, and Λ CDM model, we find that parameters α and β in the coupled generalized three-form dark energy model are essential for such model to reduce the H 0 tension to a more acceptable level, with H 0 = 70 . 1 − 1 . 5 + 1 . 4 km/s/Mpc, which is consistent with R19 at 2.0 σ confidence level. However, analysis with Akaike information criterion (AIC) show that the coupled generalized three-form dark energy model is least supported by CMB, BAO, and SN Ia datasets since the model have more free parameters, analysis also show that the Λ CDM model is most supported by CMB, BAO, and SN Ia datasets although it leads to a strongest H 0 tension. We also investigate the best-fit dynamical behavior of the coupled generalized three-form dark energy model, and show that our model is equivalent to a quintom dark energy model, in which dark energy, at early epoch, behaves like some form of early dark energy with a small positive equation of state.

Dynamical system analysis of self-interacting three-form field cosmological model: stability and bifurcation

The present work deals with Cosmological model of a three-form field, minimally coupled to gravity and interacting with cold dark matter in the background of flat FLRW space-time. By suitable choice of the dimensionless variables, the evolution equations are converted to an autonomous system and cosmological study is done by dynamical system analysis. The critical points are determined and the stability of the (non-hyperbolic) equilibrium points are examined by center manifold Theory. Possible bifurcation scenarios have been examined by the Poincaré index theory to identify possible cosmological phase transition. Also stabilities of the critical points have been analyzed globally using geometric features.

Static spherically symmetric three-form stars

We consider interior static and spherically symmetric solutions in a gravity theory that extends the standard Hilbert–Einstein action with a Lagrangian constructed from a three-form field A_{alpha beta gamma }, which generates, via the field strength and a potential term, a new component in the total energy-momentum tensor of the gravitational system. We formulate the field equations in Schwarzschild coordinates and investigate their solutions numerically for different equations of state of neutron and quark matter, by assuming that the three-field potential is either a constant or possesses a Higgs-like form. Moreover, stellar models, described by the stiff-fluid, radiation-like, bag model and the Bose–Einstein condensate equations of state are explicitly obtained in both general relativity and three-form gravity, thus allowing an in-depth comparison between the astrophysical predictions of these two gravitational theories. As a general result we find that, for all the considered equations of state, three-form field stars are more massive than their general relativistic counterparts. As a possible astrophysical application, we suggest that the 2.5 M_{odot } mass compact object, associated with the GW190814 gravitational wave event, could be in fact a neutron or a quark star described by the three-form gravity theory.

Regular black hole interior spacetime supported by three-form field

In this paper, we show that a minimally coupled 3-form endowed with a proper potential can support a regular black hole interior. By choosing an appropriate form for the metric function representing the radius of the 2-sphere, we solve for the 3-form field and its potential. Using the obtained solution, we construct an interior black hole spacetime which is everywhere regular. The singularity is replaced with a Nariai-type spacetime, whose topology is text {dS}_2 times text {S}^2, in which the radius of the 2-sphere is constant. So long as the interior continues to expand indefinitely, the geometry becomes essentially compactified. The 2-dimensional de Sitter geometry appears despite the negative potential of the 3-form field. Such a dynamical compactification could shed some light on the origin of de Sitter geometry of our Universe, exacerbated by the Swampland conjecture. In addition, we show that the spacetime is geodesically complete. The geometry is singularity-free due to the violation of the null energy condition.

Dynamical system analysis of three-form field dark energy model with baryonic matter

A cosmological model having matter field as (non) interacting dark energy (DE) and baryonic matter and minimally coupled to gravity is considered in the present work with flat FLRW space time. The DE is chosen in the form of a three-form field while radiation and dust (i.e; cold dark matter) are the baryonic part. The cosmic evolution is studied through dynamical system analysis of the autonomous system so formed from the evolution equations by suitable choice of the dimensionless variables. The stability of the non-hyperbolic critical points are examined by Center manifold theory and possible bifurcation scenarios have been examined.

Local self-tuning mechanism for the cosmological constant

Recently the global variation of the Planck mass in the General Relativistic Einstein-Hilbert action was proposed as a self-tuning mechanism of the cosmological constant preventing vacuum energy from freely gravitating. We show that this global mechanism emerges for generic local scalar-tensor theories with additional coupling of the scalar field to the field strength of a three-form gauge field that turns the scalar field constant on the domain of the action. Evaluation of the resulting integral constraint equation over the observable Universe yields a self-consistent framework with General Relativistic field equations and arbitrary radiatively stable residual cosmological constant. We argue that the expectation value for this residual is in good agreement with the magnitude of the observed cosmic acceleration.

Black hole and naked singularity geometries supported by three-form fields

We investigate static and spherically symmetric solutions in a gravity theory that extends the standard Hilbert–Einstein action with a Lagrangian constructed from a three-form field A_{alpha beta gamma }, which is related to the field strength and a potential term. The field equations are obtained explicitly for a static and spherically symmetric geometry in vacuum. For a vanishing three-form field potential the gravitational field equations can be solved exactly. For arbitrary potentials numerical approaches are adopted in studying the behavior of the metric functions and of the three-form field. To this effect, the field equations are reformulated in a dimensionless form and are solved numerically by introducing a suitable independent radial coordinate. We detect the formation of a black hole from the presence of a Killing horizon for the timelike Killing vector in the metric tensor components. Several models, corresponding to different functional forms of the three-field potential, namely, the Higgs and exponential type, are considered. In particular, naked singularity solutions are also obtained for the exponential potential case. Finally, the thermodynamic properties of these black hole solutions, such as the horizon temperature, specific heat, entropy and evaporation time due to the Hawking luminosity, are studied in detail.

Numerical study of a cosmological relaxation model of the Higgs boson mass

In light of no new physics being discovered at the LHC, ideas which tackle the hierarchy problem without novelties around the TeV scale must be taken seriously. Such is a cosmological relaxation model of the Higgs boson mass, proposed in the pre-LHC era, which does not rely on new physics below the Planck scale. This scenario introduces a different notion of naturalness according to which the vacuum with a small expectation value of the Higgs field corresponds to an infinitely enhanced entropy point of the vacuum landscape that becomes an attractor of cosmological inflationary evolution. In this framework we study numerically the evolution of the Higgs vacuum expectation value (VEV). We model the inflationary vacuum-to-vacuum transitions that are triggered by nucleation of branes charged under three-form fields as a random walk. In particular, we investigate the impact of the number of coupled three-forms on the convergence rate of the Higgs VEV. We discover an enhanced rate when increasing the number of brane charges. Moreover, we show that for late times the inclusion of more charges is equivalent to additional brane nucleations.

Einstein manifolds with torsion and nonmetricity

Manifolds endowed with torsion and nonmetricity are interesting both from the physical and the mathematical points of view. In this paper, we generalize some results presented in the literature. We study Einstein manifolds (i.e., manifolds whose symmetrized Ricci tensor is proportional to the metric) in d dimensions with nonvanishing torsion that has both a trace and a traceless part, and analyze invariance under extended conformal transformations of the corresponding field equations. Then, we compare our results to the case of Einstein manifolds with zero torsion and nonvanishing nonmetricity, where the latter is given in terms of the Weyl vector (Einstein-Weyl spaces). We find that the trace part of the torsion can alternatively be interpreted as the trace part of the nonmetricity. The analysis is subsequently extended to Einstein spaces with both torsion and nonmetricity, where we also discuss the general setting in which the nonmetricity tensor has both a trace and a traceless part. Moreover, we consider and investigate actions involving scalar curvatures obtained from torsionful or nonmetric connections, analyzing their relations with other gravitational theories that appeared previously in the literature. In particular, we show that the Einstein-Cartan action and the scale invariant gravity (also known as conformal gravity) action describe the same dynamics. Then, we consider the Einstein-Hilbert action coupled to a three-form field strength and show that its equations of motion imply that the manifold is Einstein with totally antisymmetric torsion.

A coupled generalized three-form dark energy model

A coupled dark energy model is considered, in which dark energy is represented by a generalized three-form field and dark matter by dust. By assuming the functions $N$ and $I$ in the model's Lagrangian as two power-law functions of the three-form field, we obtain two fixed points of the autonomous system of evolution equations, consisting of a attractor and a tracking saddle point which can be used to alleviate the coincidence problem. After marginalizing the present three-form field $\kappa X_{0}$ which is unable to be strictly restricted, we confront the model with the latest Type Ia Supernova (SN \uppercase\expandafter{\romannumeral1}a), Baryon Acoustic Oscillations (BAO) and Cosmic Microwave Backround (CMB) radiation observations with the fitting results $\Omega_{m0}= 0.280_{-0.048}^{+0.048}$ and $\lambda=0.011_{-0.032}^{+0.032}$ in the $2\sigma$ confidence level, we also find that the best fitting effective dark energy equation of state (EOS) crosses $ -1$ at redshift around 0.2.

Embedding the SU(3) sector of SO(8) supergravity in D=11

The SU(3)--invariant sector of maximal supergravity in four dimensions with an SO(8) gauging is uplifted to $D=11$ supergravity. In order to do this, the SU(3)--neutral sector of the tensor and duality hierarchies of the $D=4$ ${\cal N}=8$ supergravity is first worked out. The consistent $D=11$ embedding of the full, dynamical SU(3) sector is then expressed at the level of the $D=11$ metric and three-form gauge field in terms of these $D=4$ tensors. The redundancies introduced by this approach are eliminated at the level of the $D=11$ four-form field strength by making use of the $D=4$ duality hierarchy. Our results encompass previously known truncations of $D=11$ supergravity down to sectors of SO(8) supergravity with symmetry larger than SU(3), and include new ones. In particular, we obtain a new consistent truncation of $D=11$ supergravity to minimal $D=4$ ${\cal N}=2$ gauged supergravity.

Supersymmetric solutions of matter-coupled 7D N=2 gauged supergravity

We study supersymmetric solutions within seven-dimensional $N=2$ gauged supergravity coupled to three vector multiplets in seven dimensions. The gauged supergravity contains six vector fields that gauge the $SO(4)\sim SO(3)\times SO(3)$ symmetry and admits two $N=2$ supersymmetric $AdS_7$ vacua with $SO(4)$ and $SO(3)_{\textrm{diag}}\subset SO(3)\times SO(3)$ symmetries. We consider solutions interpolating between two asymptotically locally $AdS_7$ geometries in the presence of a three-form field. For a particular value of the two $SO(3)$ gauge coupling constants, the $SO(3)_{\textrm{diag}}$ supersymmetric $AdS_7$ vacuum does not exist, but the solutions can be uplifted to eleven dimensions by a known reduction ansatz. We also study solutions of this type and their embedding in M-theory. We further extend these solutions to include the $SO(3)_{\textrm{diag}}$ gauge fields and argue that, in general, this generalization does not lead to supersymmetric solutions.

Quantum behavior of the “Little Sibling” of the Big Rip induced by a three-form field

A canonical quantization à la Wheeler-DeWitt is performed for a model of three-form fields in a homogeneous and isotropic universe. We start by carrying out the Hamiltonian formalism of this cosmological model. We then apply this formalism to a Little Sibling of the Big Rip (LSBR), an abrupt event milder than a Big Rip and that is known to be generic to several minimally coupled three-form fields for a variety of potentials. We obtain a set of analytical solutions of the Wheeler-DeWitt equation using different analytical approximations and explore the physical consequences of them. It turns out that there are quantum states where the wave function of the universe vanishes, i.e. the DeWitt condition is fulfilled for them. Given that this happens only for some subset of solutions of the Wheeler-DeWitt equation, this points out that the matter inducing the LSBR is equally important in the process as, it has been previously shown, a minimally coupled phantom scalar field feeding classically a LSBR is smoothed at the quantum level, i.e. all the quantum states lead to a vanishing wave function.