arXiv subjects

Eduardo Rodríguez

Publications and source records attributed to Eduardo Rodríguez.

At least 19 recordsLinked to original sources

arXiv2025

The shear Alfvén continuum of quasisymmetric stellarators

The shear Alfvén wave (SAW) continuum plays a critical role in the stability of energetic particle-driven Alfvén eigenmodes. We develop a theoretical framework to analyze the SAW continuum in three-dimensional quasisymmetric magnetic fields, focusing on its implications for stellarator design. By employing a near-axis model and degenerate perturbation theory, the continuum equation is solved, highlighting unique features in 3D configurations, such as the interactions between spectral gaps. Numerical examples validate the theory, demonstrating the impact of flux surface shaping and quasisymmetric field properties on continuum structure. The results provide insights into optimizing stellarator configurations to minimize resonance-driven losses of energetic particles. This work establishes a basis for incorporating Alfvénic stability considerations into the stellarator design process, demonstrated through optimization of a quasihelical configuration to avoid high-frequency spectral gaps.

physics.plasm-ph↗

arXiv2025

A comprehensive exploration of quasisymmetric stellarators and their coil sets

We augment the `QUAsi-symmetric Stellarator Repository' (QUASR) to include vacuum field stellarators with quasihelical symmetry using a globalized optimization workflow. The database now has almost 370,000 quasisaxisymmetry and quasihelically symmetric devices along with coil sets, optimized for a variety of aspect ratios, rotational transforms, and discrete rotational symmetries. This paper outlines a couple of ways to explore and characterize the data set. We plot devices on a near-axis quasisymmetry landscape, revealing close correspondence to this predicted landscape. We also use principal component analysis to reduce the dimensionality of the data so that it can easily be visualized in two or three dimensions. Principal component analysis also gives a mechanism to compare the new devices here to previously published ones in the literature. We are able to characterize the structure of the data, observe clusters, and visualize the progression of devices in these clusters. The topology of the data is governed by the interplay of the design constraints and valleys of the quasisymmetry objective. These techniques reveal that the data has structure, and that typically one, two or three principal components are sufficient to characterize it. The latest version of QUASR is archived at https://zenodo.org/doi/10.5281/zenodo.10050655 and can be explored online at quasr.flatironinstitute.org.

physics.app-ph↗

arXiv2022

Weakly Quasisymmetric Near-Axis Solutions to all Orders

We show that the equations satisfied by weakly quasisymmetric magnetic fields can be solved to arbitrarily high order in powers of the distance from the magnetic axis. This demonstration does not consider force balance. The existence of solutions requires an appropriate choice of parameters, most notably the toroidal current or rotational transform profiles. We do not prove that the expansion converges (it is likely divergent but asymptotic), and thus the demonstration here should not be taken as definitive proof of the existence of global solutions. Instead, we provide a systematic construction of solutions to arbitrarily high order.

physics.plasm-ph↗

arXiv2021

Islands and current singularities in quasisymmetric toroidal plasmas

The presence of current singularities in a quasisymmetric magnetic field is explored. Quasisymmetry is shown effective in isolating Pfirsch-Schlüter singularities, to leading order, to a single magnetic surface resonant with the helicity of the symmetry. The effects of pressure driven currents are analysed, indicating that exclusion of this surface from the plasma volume reduces the potential opening of islands, but does not generally eliminate them completely due to higher order asymmetric geometric effects. These three-dimensional effects are contained in quasisymmetry and indicate the complexity of finding consistent solutions and their potential sensitivity. The $δ$-function current singularities show a distinct quasisymmetric behaviour only when the higher-order Fourier content of $B$ is relevant for the resistive stability parameter $D_R$ (not included in leading-order near-axis expansions). In such scenarios, quasisymmetry proves advantageous, both in simplicity and avoidance of amplification by low-order rational surfaces.

physics.plasm-ph↗

arXiv2020

Wave Operators, Torsion, and Weitzenböck Identities

We offer a mathematical toolkit for the study of waves propagating on a background manifold with nonvanishing torsion. Examples include electromagnetic and gravitational waves on a spacetime with torsion. The toolkit comprises generalized versions of the Lichnerowicz-de Rham and the Beltrami wave operators, and the Weitzenböck identity relating them on Riemann-Cartan geometries. The construction applies to any field belonging to a matrix representation of a Lie (super) algebra containing an $\mathfrak{so}$ subalgebra. Using these tools, we analyze the propagation of different massless waves in the eikonal (geometric optics) limit in a model-independent way and find that they all must propagate at the speed of light along null torsionless geodesics, in full agreement with the multimessenger observation GW170817/GRB170817A. We also discuss how gravitational waves could be used as a probe to test for torsion.

gr-qc↗

arXiv2019

Luminal Propagation of Gravitational Waves in Scalar-tensor Theories: The Case for Torsion

Scalar-tensor gravity theories with a nonminimal Gauss-Bonnet coupling typically lead to an anomalous propagation speed for gravitational waves, and have therefore been tightly constrained by multimessenger observations such as GW170817/GRB170817A. In this paper we show that this is not a general feature of scalar-tensor theories, but rather a consequence of assuming that spacetime torsion vanishes identically. At least for the case of a nonminimal Gauss-Bonnet coupling, removing the torsionless condition restores the canonical dispersion relation and therefore the correct propagation speed for gravitational waves. To achieve this result we develop a new approach, based on the first-order formulation of gravity, to deal with perturbations on these Riemann-Cartan geometries.

gr-qc↗

arXiv2019

Linear and Second-order Geometry Perturbations on Spacetimes with Torsion

In order to study gravitational waves in any realistic astrophysical scenario, one must consider geometry perturbations up to second order. Here, we present a general technique for studying linear and quadratic perturbations on a spacetime with torsion. Besides the standard metric mode, a "torsionon" perturbation mode appears. This torsional mode will be able to propagate only in a certain kind of theories.

gr-qc↗

arXiv2017

Nonminimal couplings, gravitational waves, and torsion in Horndeski's theory

The Horndeski Lagrangian brings together all possible interactions between gravity and a scalar field that yield second-order field equations in four-dimensional spacetime. As originally proposed, it only addresses phenomenology without torsion, which is a non-Riemannian feature of geometry. Since torsion can potentially affect interesting phenomena such as gravitational waves and early Universe inflation, in this paper we allow torsion to exist and propagate within the Horndeski framework. To achieve this goal, we cast the Horndeski Lagrangian in Cartan's first-order formalism, and introduce wave operators designed to act covariantly on p-form fields that carry Lorentz indices. We find that nonminimal couplings and second-order derivatives of the scalar field in the Lagrangian are indeed generic sources of torsion. Metric perturbations couple to the background torsion and new torsional modes appear. These may be detected via gravitational waves but not through Yang-Mills gauge bosons.

gr-qc↗

arXiv2016

Randall-Sundrum brane universe as a ground state for Chern-Simons gravity

In stark contrast with the three-dimensional case, higher-dimensional Chern-Simons theories can have non-topological, propagating degrees of freedom. Finding those vacua that allow for the propagation of linear perturbations, however, proves to be surprisingly challenging. The simplest solutions are somehow "hyper-stable," preventing the construction of realistic, four-dimensional physical models. Here, we show that a Randall-Sundrum brane universe can be regarded as a vacuum solution of Chern-Simons gravity in five-dimensional spacetime, with non vanishing torsion along the dimension perpendicular to the brane. Linearized perturbations around this solution not only exist, but behave as standard gravitational waves on a four-dimensional Minkowski background. In the non-perturbative regime, the solution leads to a four-dimensional "cosmological function" $Λ\left( x \right)$ which depends on the Euler density of the brane. Interestingly, the fact that the solution admits nontrivial linear perturbations seems to be related to an often neglected property of the Randall-Sundrum spacetime: that it is a group manifold, or, more precisely, two identical group manifolds glued together along the brane. The gravitational theory is then built around this fact, adding the Lorentz generators and one scalar generator needed to close the algebra. In this way, a conjecture emerges: a spacetime that is also a group manifold can be regarded as the ground state of a Chern-Simons theory for an appropriate Lie algebra.

hep-th↗

arXiv2015

Supersymmetric 3D model for gravity with $SU(2)$ gauge symmetry, mass generation and effective cosmological constant

A Chern--Simons system in $2+1$ dimensions invariant under local Lorentz rotations, $SU(2)$ gauge transformations, and local $\mathcal{N}=2$ supersymmetry transformations is proposed. The field content is that of $(2+1)$-gravity plus an $SU(2)$ gauge field, a spin-1/2 fermion charged with respect to $SU(2)$ and a trivial free abelian gauge field. A peculiarity of the model is the absence of gravitini, although it includes gravity and supersymmetry. Likewise, no gauginos are present. All the parameters involved in the system are either protected by gauge invariance or emerge as integration constants. An effective mass and effective cosmological constant emerge by spontaneus breaking of local scaling invariance. The vacuum sector is defined by configurations with locally flat Lorentz and $SU(2)$ connections sporting nontrivial global charges. Three-dimensional Lorentz-flat geometries are spacetimes of locally constant negative --or zero--, Riemann curvature, which include Minkowski space, AdS$_3$, BTZ black holes, and point particles. These solutions admit different numbers of globally defined, covariantly constant spinors and are therefore good candidates for stable ground states. The fermionic sector in this system could describe the dynamics of electrons in graphene in the long wavelength limit near the Dirac points, with the spin degree of freedom of the electrons represented by the $SU(2)$ label. If this is the case, the $SU(2)$ gauge field would produce a spin-spin interaction giving rise to strong correlation of electron pairs.

hep-th↗

arXiv2013

A generalized action for $\left( 2 + 1 \right)$-dimensional Chern--Simons gravity

We show that the so-called semi-simple extended Poincaré (SSEP) algebra in $D$ dimensions can be obtained from the anti-de~Sitter algebra $\mathfrak{so} \left( D-1,2 \right)$ by means of the $S$-expansion procedure with an appropriate semigroup $S$. A general prescription is given for computing Casimir operators for $S$-expanded algebras, and the method is exemplified for the SSEP algebra. The $S$-expansion method also allows us to extract the corresponding invariant tensor for the SSEP algebra, which is a key ingredient in the construction of a generalized action for Chern--Simons gravity in $2+1$ dimensions.

gr-qc↗

arXiv2013

Effectively four-dimensional spacetimes emerging from d=5 Einstein-Gauss-Bonnet Gravity

Einstein-Gauss-Bonnet gravity in five-dimensional spacetime provides an excellent example of a theory that, while including higher-order curvature corrections to General Relativity, still shares many of its features, such as second-order field equations for the metric. We focus on the largely unexplored case where the coupling constants of the theory are such that no constant-curvature solution is allowed, leaving open the question of what the vacuum state should then be. We find that even a slight deviation from the anti-de Sitter Chern-Simons theory, where the vacuum state is five-dimensional AdS spacetime, leads to a complete symmetry breakdown, with the fifth dimension either being compactified into a small circle or shrinking away exponentially with time. A complete family of solutions, including duality relations among them, is uncovered and shown to be unique within a certain class. This dynamical dimensional reduction scenario seems particularly attractive as a means for higher-dimensional theories to make contact with our four-dimensional world.

hep-th↗

arXiv2012

Dirac Matrices for Chern-Simons Gravity

A genuine gauge theory for the Poincaré, de Sitter or anti-de Sitter algebras can be constructed in (2n-1)-dimensional spacetime by means of the Chern-Simons form, yielding a gravitational theory that differs from General Relativity but shares many of its properties, such as second order field equations for the metric. The particular form of the Lagrangian is determined by a rank n, symmetric tensor invariant under the relevant algebra. In practice, the calculation of this invariant tensor can be reduced to the computation of the trace of the symmetrized product of n Dirac Gamma matrices Γ_{ab} in 2n-dimensional spacetime. While straightforward in principle, this calculation can become extremely cumbersome in practice. For large enough n, existing computer algebra packages take an inordinate long time to produce the answer or plainly fail having used up all available memory. In this talk we show that the general formula for the trace of the symmetrized product of 2n Gamma matrices Γ_{ab} can be written as a certain sum over the integer partitions s of n, with every term being multiplied by a numerical coefficient α_{s}. We then give a general algorithm that computes the α-coefficients as the solution of a linear system of equations generated by evaluating the general formula for different sets of tensors B^{ab} with random numerical entries. A recurrence relation between different coefficients is shown to hold and is used in a second, "minimal" algorithm to greatly speed up the computations. Runtime of the minimal algorithm stays below 1 min on a typical desktop computer for up to n=25, which easily covers all foreseeable applications of the trace formula.

math-ph↗

arXiv2011

On eleven-dimensional Supergravity and Chern-Simons theory

We probe in some depth into the structure of eleven-dimensional, osp(32|1)-based Chern-Simons supergravity, as put forward by Troncoso and Zanelli (TZ) in 1997. We find that the TZ Lagrangian may be cast as a polynomial in 1/l, where l is a length, and compute explicitly the first three dominant terms. The term proportional to 1/l^9 turns out to be essentially the Lagrangian of the standard 1978 supergravity theory of Cremmer, Julia and Scherk, thus establishing a previously unknown relation between the two theories. The computation is nontrivial because, when written in a sufficiently explicit way, the TZ Lagrangian has roughly one thousand non-explicitly Lorentz-covariant terms. Specially designed algebraic techniques are used to accomplish the results.

hep-th↗

arXiv2009

Standard General Relativity from Chern-Simons Gravity

Chern-Simons models for gravity are interesting because they provide with a truly gauge-invariant action principle in the fiber-bundle sense. So far, their main drawback has largely been the perceived remoteness from standard General Relativity, based on the presence of higher powers of the curvature in the Lagrangian (except, remarkably, for three-dimensional spacetime). Here we report on a simple model that suggests a mechanism by which standard General Relativity in five-dimensional spacetime may indeed emerge at a special critical point in the space of couplings, where additional degrees of freedom and corresponding "anomalous" Gauss-Bonnet constraints drop out from the Chern-Simons action. To achieve this result, both the Lie algebra g and the symmetric g-invariant tensor that define the Chern-Simons Lagrangian are constructed by means of the Lie algebra S-expansion method with a suitable finite abelian semigroup S. The results are generalized to arbitrary odd dimensions, and the possible extension to the case of eleven-dimensional supergravity is briefly discussed.

hep-th↗

arXiv2009

Dual Formulation of the Lie Algebra S-expansion Procedure

The expansion of a Lie algebra entails finding a new, bigger algebra G, through a series of well-defined steps, from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite abelian semigroup S to accomplish this task. In this paper we put forward a dual formulation of the S-expansion method which is based on the dual picture of a Lie algebra given by the Maurer-Cartan forms. The dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra, which in turn is relevant for physical applications in, e.g., Supergravity. It also sheds new light on the puzzling relation between two Chern-Simons Lagrangians for gravity in 2+1 dimensions, namely the Einstein-Hilbert Lagrangian and the one for the so-called "exotic gravity".

hep-th↗

arXiv2008

Eleven-Dimensional Gauge Theory for the M Algebra as an Abelian Semigroup Expansion of osp(32|1)

A new Lagrangian realizing the symmetry of the M Algebra in eleven-dimensional space-time is presented. By means of the novel technique of Abelian Semigroup Expansion, a link between the M Algebra and the orthosymplectic algebra osp(32|1) is established, and an M Algebra-invariant symmetric tensor of rank six is computed. This symmetric invariant tensor is a key ingredient in the construction of the new Lagrangian. The gauge-invariant Lagrangian is displayed in an explicitly Lorentz-invariant way by means of a subspace separation method based on the extended Cartan homotopy formula.

hep-th↗

arXiv2007

The Extended Cartan Homotopy Formula and a Subspace Separation Method for Chern--Simons Theory

In the context of Chern--Simons (CS) Theory, a subspace separation method for the Lagrangian is proposed. The method is based on the iterative use of the Extended Cartan Homotopy Formula, and allows one to (1) separate the action in bulk and boundary contributions, and (2) systematically split the Lagrangian in appropriate reflection of the the subspace structure of the gauge algebra. In order to apply the method, one must regard CS forms as a particular case of more general objects known as transgression forms. Five-dimensional CS Supergravity is used as an example to illustrate the method.

hep-th↗