Affine Symmetry Research Articles

Affine symmetries for ABJM partition function and its generalization

Partially motivated by the fact that the grand partition function of the ABJM theory or its generalization is expressed by a spectral operator enjoying symmetries of the Weyl group, it was found that the grand partition function satisfies the q-Painlevé equation, which is constructed from the affine Weyl group. In this paper we clarify the affine symmetries of the grand partition function. With the affine symmetries, we find that the grand partition function extends naturally outside the fundamental domain of duality cascades and once the Painlevé equation holds in the fundamental domain, so does it outside.

Spherically symmetric teleparallel geometries

We are interested in the development of spherically symmetric geometries in F(T) teleparallel gravity which are of physical importance. We first express the general forms for the spherically symmetric frame and the zero curvature, metric compatible, spin connection. We then analyse the antisymmetric field equations (the solutions of which split into two cases, which we subsequently consider separately), and derive and analyse the resulting symmetric field equations. In order to further study the applications of spherically symmetric teleparallel models, we study 3 subcases in which there is an additional affine symmetry so that the resulting field equations reduce to a system of ordinary differential equations. First, we study static spherical symmetric geometries and solve the antisymmetric field equations and subsequently derive the full set of symmetric field equations. In particular, we investigate vacuum spacetimes and obtain a number of new solutions. Second, we consider an additional affine frame symmetry in order to expand the affine frame symmetry group to that of a spatially homogeneous Kantowski–Sachs geometry. Third, we study the special case of spherical symmetry with an additional fourth similarity affine vector.

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

Generalized teleparallel de Sitter geometries

Theories of gravity based on teleparallel geometries are characterized by the torsion, which is a function of the coframe, derivatives of the coframe, and a zero curvature and metric compatible spin-connection. The appropriate notion of a symmetry in a teleparallel geometry is that of an affine symmetry. Due to the importance of the de Sitter geometry and Einstein spaces within General Relativity, we shall describe teleparallel de Sitter geometries and discuss their possible generalizations. In particular, we shall analyse a class of Einstein teleparallel geometries which have a 4-dimensional Lie algebra of affine symmetries, and display two one-parameter families of explicit exact solutions.

Teleparallel Robertson-Walker Geometries and Applications

In teleparallel geometries, the coframe and corresponding spin connection are the principal geometric objects and, consequently, the appropriate definition of a symmetry is that of an affine symmetry. The set of invariant coframes and their corresponding spin connections that respect the full six dimensional Lie algebra of Robertson–Walker affine symmetries are displayed and discussed. We will refer to such geometries as teleparallel Robertson–Walker (TRW) geometries, where the corresponding derived metric is of Robertson–Walker form and is characterized by the parameter k=(−1,0,1). The field equations are explicitly presented for the F(T) class of teleparallel TRW spacetimes. We are primarily interested in investigating the k≠0 TRW models. After first studying the k=0 models and, in particular, writing their governing field equations in an appropriate form, we then study their late time stability with respect to perturbations in k in both the cases of a vanishing and non-vanishing effective cosmological constant term. As an illustration, we consider both quadratic F(T) theories and power-law solutions.

Bianchi type cosmological models in f(T) tele-parallel gravity

Symmetry assumptions on the geometrical framework have provided successful mechanisms to develop physically meaningful solutions to many problems. In tele-parallel gravity, invariance of the frame and spin-connection under a group of motions defines an affine symmetry group. Here, we assume there exists a three-dimensional group of affine symmetries acting simply transitively on a spatial hypersurface and that this group of symmetry actions defines our affine frame symmetry group. We determine the general form of the co-frame and spin connection for each spatially homogeneous Bianchi type. We then construct the corresponding field equations for f(T) tele-parallel gravity. We show that if the symmetry group is of Bianchi type A (I, II, VI 0, VII 0, VIII or IX) then there exists a co-frame/spin connection pair that is consistent with the antisymmetric part of the field equations of f(T) tele-parallel gravity. For those geometries having a Bianchi type B symmetry group (IV, V, VIh , VIIh ), we find that in general these geometries are inconsistent with the antisymmetric part of the f(T) tele-parallel gravity field equations unless the theory reduces to an analog of General Relativity with a cosmological constant.

Computing affine equivalences and symmetries of trigonometric curves in arbitrary dimension

We present a new and efficient algorithm to compute affine equivalences and symmetries between two trigonometric curves in an arbitrary dimension. The algorithm benefits from the power of invariance and polynomial gcd and factoring without solving any system of equations. The algorithm is implemented in MAPLE, and extensive experimentations demonstrating the efficiency of the method are given.

The Normalizer of a Lie Group: Applications and Challenges

Let G be a connected Lie group with Lie algebra g. This review is devoted to studying the fundamental dynamic properties of elements in the normalizer NG of G. Through an algebraic characterization of NG, we analyze the different dynamics inside the normalizer. NG contains the well-known left-invariant vector fields and the linear and affine vector fields on G. In any case, we show the shape of the solutions of these ordinary differential equations on G. We give examples in low-dimensional Lie groups. It is worth saying that these dynamics generate the linear and bilinear control systems on Euclidean spaces and the invariant and linear control systems on Lie groups. Moreover, the Jouan Equivalence Theorem shows how to extend this theory to control systems on manifolds.

Teleparallel geometry with a single affine symmetry

In teleparallel geometries, symmetries are represented by affine frame symmetries that constrain both the (co)frame basis and the spin-connection (which are the primary geometric objects). In this paper, we shall study teleparallel geometries with a single affine symmetry, utilizing the locally Lorentz covariant approach and adopting a complex null gauge. We first introduce an algorithm to study geometries with an affine frame symmetry, which consists of choosing coordinates adapted to the symmetry, constructing a canonical frame, and solving the equations describing the symmetry. All of the constraints on the geometry are determined in the case of a single affine symmetry, but there are additional constraints arising from the field equations for a given theory of teleparallel gravity. In particular, we find that in f(T) teleparallel gravity there will be severe constraints on the geometry arising from the antisymmetric part of the field equations.

Affine symmetries in quantum cohomology: corrections and new results

Affine symmetries in quantum cohomology: corrections and new results

Exploiting Different Symmetries for Trajectory Tracking Control with Application to Quadrotors*

High performance trajectory tracking control of quadrotor vehicles is an important challenge in aerial robotics. Symmetry is a fundamental property of physical systems and offers the potential to provide a tool to design high-performance control algorithms. We propose a design methodology that takes any given symmetry, linearises the associated error in a single set of coordinates, and uses LQR design to obtain a high performance control; an approach we term Equivariant Regulator design. We show that quadrotor vehicles admit several different symmetries: the direct product symmetry, the extended pose symmetry and the pose and velocity symmetry, and show that each symmetry can be used to define a global error. We compare the linearised systems via simulation and find that the extended pose and pose and velocity symmetries outperform the direct product symmetry in the presence of large disturbances. This suggests that choices of equivariant and group affine symmetries have improved linearisation error.

Symmetric teleparallel geometries

In teleparallel gravity and, in particular, in F(T) teleparallel gravity, there is a challenge in determining an appropriate (co-)frame and its corresponding spin connection to describe the geometry. Very often, the ‘proper’ frame, the frame in which all inertial effects are absent, is not the simplest (e.g, diagonal) (co-)frame. The determination of the frame and its corresponding spin connection for F(T) teleparallel gravity theories when there exist affine symmetries is of much interest. In this paper we present the general form of the coframe and its corresponding spin connection for teleparallel geometries which are invariant under a G 6 group of affine symmetries. The proper coframe and the corresponding F(T) field equations are also shown for these teleparallel Robertson–Walker geometries. Further, with the addition of an additional affine symmetry, it is possible to define a teleparallel de Sitter geometry.

Searching for gravity without a metric

Recently it has been explicitly shown how a theory with global $GL(d,\mathbb{R})$ coordinate (affine) invariance which is spontaneously broken down to its Lorentz subgroup will have as its Goldstone fields enough degrees of freedom to create a metric and a covariant derivative arXiv:1105.5848. Such a theory would constitute an effective theory of gravity. So far however, no explicit theory has been found which exhibits this symmetry breaking pattern, mainly due to the difficulty of even writing down a $GL(d,\mathbb{R})$ invariant actions in the absence of a metric. In this paper we explicitly construct an affine generalization of the Dirac action employing infinite dimensional spinorial representations of the group. This implies that it is built from an infinite number of spinor Lorentz multiplets. We introduce a systematic procedure for obtaining $GL(d,\mathbb{R})$ invariant interaction terms to obtain quite general interacting models. Such models have order operators whose expectation value can break affine symmetry to Poincar\'{e} symmetry. We discuss possible interactions and mechanisms for this symmetry breaking to occur, which would provide a dynamical explanation of the Lorentzian signature of spacetime.

All eight- and nine-dimensional string vacua from junctions

String and 5-brane junctions are shown to succinctly classify all known 8d ${\cal N}$=1 string vacua. This requires an extension of the description for ordinary $[p,q]$-7-branes to consistently include O7$^+$-planes, which then naturally encodes the dynamics of $\mathfrak{sp}_n$ gauge algebras, including their higher-form center symmetries. Central to this analysis are loop junctions, i.e., strings/5-branes which encircle stacks of 7-branes and O7$^+$'s. Loop junctions further signal the appearance of affine symmetries of emergent 9d descriptions at the 8d moduli space's boundaries. Such limits reproduce all 9d string vacua, including the two disconnected rank (1,1) moduli components.

On the symmetries of Siklos spacetimes

We consider the large class of Siklos spacetimes and investigate their relevant symmetries (homothetic and affine vector fields, Ricci, curvature, Weyl and matter collineations). We prove some general results and obtain complete classifications for homogeneous Siklos spacetimes.

First Cohomology of the Lie Algebra of Vector Fields on the Affine Real Line Relative to Affine Vector Fields with Coefficients in Bilinear Differential Operators on Weighted Densities

First Cohomology of the Lie Algebra of Vector Fields on the Affine Real Line Relative to Affine Vector Fields with Coefficients in Bilinear Differential Operators on Weighted Densities

Symmetries of four-dimensional Lorentzian Damek-Ricci spaces

We consider the four-dimensional Damek-Ricci spaces, equipped with theleft-invariant Lorentzian metric. We obtain a full classification of Killing andaffine vector fields as well as Ricci, curvature and matter collineations. Inparticular, we prove the non-existence of proper (that is non Killing) affinevector field

Translation covers of some triply periodic Platonic surfaces

We study translation covers of several triply periodic polyhedral surfaces that are intrinsically Platonic. We describe their affine symmetry groups and compute the quadratic asymptotics for counting saddle connections and cylinders, including the count of cylinders weighted by area. The mathematical study of triply periodic surfaces was initiated by Novikov, motivated by the study of electron transport. The surfaces we consider are of particular interest as they admit several different explicit geometric and algebraic descriptions, as described, for example, in the second author’s thesis.

Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics

Two Kahler metrics on a complex manifold are called c-projectively equivalent if their $J$-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kahler metrics of arbitrary signature and use this description to prove the classical Yano-Obata conjecture: we show that on a closed connected Kahler manifold of arbitrary signature, any c-projective vector field is an affine vector field unless the manifold is CPn with (a multiple of) the Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz conjecture for metrics of Lorentzian signature: we show that on a closed connected Lorentzian manifold, any projective vector field is an affine vector field.

Алгебры Ли дифференцирований линейных алгебр над полем

In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equa­tions. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of ap­plying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an exam­ple, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.