Symmetric Tensor Fields Research Articles

Gauge invariant and gauge fixed actions for various higher-spin fields from string field theory

We propose a systematic procedure for extracting gauge invariant and gauge fixed actions for various higher-spin gauge field theories from covariant bosonic open string field theory. By identifying minimal gauge invariant part for the original free string field theory action, we explicitly construct a class of covariantly gauge fixed actions with BRST and anti-BRST invariance. By expanding the actions with respect to the level N of string states, the actions for various massive fields including higher-spin fields are systematically obtained. As illustrating examples, we explicitly investigate the level N⩽3 part and obtain the consistent actions for massive graviton field, massive 3rd rank symmetric tensor field, or anti-symmetric field. We also investigate the tensionless limit of the actions and explicitly derive the gauge invariant and gauge fixed actions for general rank n symmetric and anti-symmetric tensor fields.

Tensor tomography on surfaces

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

Tomography of small residual stresses

We study the inverse problem of determining the residual stress in Man’s model using tomographic data. Theoretically, the tomographic data are obtained at the zeroth approximation of geometrical optics for Man’s residual stress model. For compressional waves, the inverse problem is equivalent to the problem of inverting the longitudinal ray transform of a symmetric tensor field. For shear waves, the inverse problem, after the linearization, leads to another integral geometry operator which is called the mixed ray transform. Under some restrictions on coefficients, we are able to prove the uniqueness results in these two cases.

Emergence of helicity ±2 modes (gravitons) from qubit models

We construct two quantum qubit models (or quantum spin models) on three-dimensional lattice in space, L-type model and N-type model. We show that, under a controlled approximation, all the low energy excitations of the L-type model are described by one set of helicity ±2 modes with ω∝k3 dispersion. We also argue that all the low energy excitations of the N-type model are described by one set of helicity ±2 modes with ω∝k dispersion. In both model, the low energy helicity ±2 modes can be described by a symmetric tensor field hμν in continuum limit, and the gaplessness of the helicity ±2 modes is protected by an emergent linearized diffeomorphism gauge symmetry hμν→hμν+∂μfν+∂νfμ at low energies. Thus the linearized quantum gravity emerge from our lattice models. It turns out that the low energy effective Lagrangian density of the L-type model is invariant under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω∝k3 helicity ±2 modes. In contrast, the low energy effective Lagrangian of the N-type model changes by a boundary term under the linearized diffeomorphism gauge transformation. Such a property protects the gapless ω∝k helicity ±2 modes. From many-body physics point of view, the ground states of the our two qubit model represent new states of quantum matter, whose low energy excitations are all described by one set of gapless helicity ±2 modes.

Symmetric tensor fields of bounded deformation

We introduce and study spaces of symmetric tensor fields of bounded deformation for tensors of arbitrary order, i.e., where the symmetrized derivative is still a Radon measure. A Sobolev–Korn type estimate, a boundary trace theorem and continuous as well as compact embedding properties into Lebesgue spaces are obtained, showing that these spaces can be regarded as a natural generalization of the spaces of bounded deformation to higher-order symmetric tensors.

Ordinary-derivative formulation of conformal low-spin fields

Conformal fields in flat space-time of even dimension greater than or equal to four are studied. Second-derivative formulation for spin 0,1,2 conformal bosonic fields and first-derivative formulation for spin 1/2,3/2 conformal fermionic fields are developed. For the spin 1,3/2,2 conformal fields, we obtain gauge invariant Lagrangians and the corresponding gauge transformations. Gauge symmetries are realized by involving Stueckelberg fields and auxiliary fields. Realization of global conformal boost symmetries is obtained. Modified Lorentz and de Donder gauge conditions are introduced. Ordinary-derivative Lagrangian of interacting Weyl gravity in 4d is obtained. In our approach, the field content of Weyl gravity, in addition to conformal graviton field, includes one auxiliary rank-2 symmetric tensor field and one Stueckelberg vector field. With respect to the auxiliary tensor field, the Lagrangian contains, in addition to other terms, the Pauli-Fierz mass term. Using the ordinary-derivative Lagrangian of Weyl gravity, we discuss interrelation of Einstein AdS gravity and Weyl gravity via breaking conformal gauge symmetries. Also, we demonstrate use of the light-cone gauge for counting on-shell degrees of freedom in higher-derivative conformal field theories.

Rayleigh dissipation from the general recurrence of metrics in path spaces

Given a pair (metric g, symmetric 2-covariant tensor field H though as a Rayleigh dissipation) on a path space (manifold M, semispray S), the family of nonlinear connections N such that H equals the dynamical derivative of g with respect to (S,N) is determined by using the Obata tensors. In this way, we generalize the case of metric nonlinear connections as well as that of recurrent metrics. As applications, we treat firstly the case of Finslerian (α,β)-metrics finding all nonlinear connections for which the associated Finsler–Sasaki metric is exactly the dynamical derivative of the Riemannian–Sasaki metric. Secondly, we apply our results for the case of Beil metrics used in Relativity and field theories.

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.

Topological Features in 2D Symmetric Higher‐Order Tensor Fields

AbstractThe topological structure of scalar, vector, and second‐order tensor fields provides an important mathematical basis for data analysis and visualization. In this paper, we extend this framework towards higher‐order tensors. First, we establish formal uniqueness properties for a geometrically constrained tensor decomposition. This allows us to define and visualize topological structures in symmetric tensor fields of orders three and four. We clarify that in 2D, degeneracies occur at isolated points, regardless of tensor order. However, for orders higher than two, they are no longer equivalent to isotropic tensors, and their fractional Poincaré index prevents us from deriving continuous vector fields from the tensor decomposition. Instead, sorting the terms by magnitude leads to a new type of feature, lines along which the resulting vector fields are discontinuous. We propose algorithms to extract these features and present results on higher‐order derivatives and higher‐order structure tensors.

ON MANIFOLDS ADMITTING THE CONSISTENT LAGRANGIAN FORMULATION FOR HIGHER SPIN FIELDS

We study a possibility of Lagrangian formulation for free massive higher spin bosonic totally symmetric tensor field on the background manifold characterizing by the arbitrary metric, vector and third-rank tensor fields in the framework of BRST approach. Assuming analytical dependence on the mass, curvatures and the other background fields in the Lagrangian and using the most general linearized ansatz for transversality condition, we prove that the consistent formulation is possible only in constant curvature space and that there is only a trivial possibility to include the vector and third-rank tensor in the theory. This result finally proves that the consistent Lagrangian formulation at the conditions under consideration is possible only in constant curvature Riemann space.

UNCONSTRAINED HIGHER SPINS IN FOUR DIMENSIONS

The relativistic symmetric tensor fields are, in four dimensions, the right candidates to describe Higher Spin Fields. Their highest spin content is isolated with the aid of covariant conditions, discussed within a group theory framework, in which auxiliary fields remove the lower intrinsic angular momenta sectors. These conditions are embedded within a Lagrangian Quantum Field theory which describes an Higher Spin Field interacting with a Classical background. The model is invariant under a (B.R.S.) symmetric unconstrained tensor extension of the reparametrization symmetry, which include the Fang–Fronsdal algebra in a well defined limit. However, the symmetry setting reveals that the compensator field, which restore the Fang–Fronsdal symmetry of the free equations of motion, is in the existing in the framework and has a relevant geometrical meaning. The Ward identities coming from this symmetry are discussed. Our constraints give the result that the space of the invariant observables is restricted to the ones constructed with the Highest Spin Field content. The quantum extension of the symmetry reveals that no new anomaly is present. The role of the compensator field in this result is fundamental.

6D conformal gravity

In the framework of an ordinary-derivative approach, conformal gravity in spacetime of dimension 6 is studied. The field content, in addition to conformal graviton field, includes two auxiliary rank-2 symmetric tensor fields, two Stueckelberg vector fields and one Stueckelberg scalar field. The gauge invariant Lagrangian with conventional kinetic terms and corresponding gauge transformations are obtained. One of the rank-2 tensor fields and the scalar field have a canonical conformal dimension. With respect to these fields, the Lagrangian contains, in addition to other terms, a cubic potential. Gauging away the Stueckelberg fields and excluding the auxiliary fields via equations of motion, the higher derivative Lagrangian of 6D conformal gravity is obtained. The higher derivative Lagrangian involves quadratic and cubic curvature terms. This higher derivative Lagrangian coincides with the simplest Weyl invariant density discussed in the earlier literature. Generalization of de Donder gauge conditions to 6D conformal fields is also obtained.

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.

Visualizing High‐Order Symmetric Tensor Field Structure with Differential Operators

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4.

Linear liftings of skew symmetric tensor fields of type (1, 2) to Weil bundles

The paper contains a classification of linear liftings of skew symmetric tensor fields of type (1, 2) on n-dimensional manifolds to tensor fields of type (1, 2) on Weil bundles under the condition that n ⩾ 3. It complements author’s paper “Linear liftings of symmetric tensor fields of type (1, 2) to Weil bundles” (Ann. Polon. Math. 92, 2007, pp. 13–27), where similar liftings of symmetric tensor fields were studied. We apply this result to generalize that of author’s paper “Affine liftings of torsion-free connections to Weil bundles” (Colloq. Math. 114, 2009, pp. 1–8) and get a classification of affine liftings of all linear connections to Weil bundles.

Superquadric Glyphs for Symmetric Second-Order Tensors

Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.

New equations of infinitesimal deformations of surfaces in E 3

We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E 3.

Coherent Structures of Characteristic Curves in Symmetric Second Order Tensor Fields

This paper generalizes the concept of Lagrangian coherent structures, which is known for its potential to visualize coherent regions in vector fields and to distinguish them from each other. In particular, we extend the concept of the flow map to generic mappings of coordinates. As the major application of this generalization, we present a semiglobal method for visualizing coherent structures in symmetric second order tensor fields. We demonstrate the usefulness by examples from DT-MRI, uncovering anatomical structures in linearly anisotropic regions not amenable to local feature criteria. To further exemplify the suitability of our concept, we also present its application to stress tensor fields. Last, an accelerated implementation utilizing GPUs is presented.

Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on [formula omitted

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of complex symmetric covariant tensor fields on the complex projective space P n ( C ) . The spaces of symmetric eigentensors are explicitly given.

Conformal linear gravity in de Sitter space

It has been shown that the theory of linear conformal quantum gravity must include a tensor field of rank-3 and mixed symmetry [Binegar et al., Phys. Rev. D 27, 2249 (1983)]. In this paper, we obtain the corresponding field equation in de Sitter space. Then, in order to relate this field with the symmetric tensor field of rank-2, Kαβ related to graviton, we will define homomorphisms between them. Our main result is that if one insists Kαβ to be a unitary irreducible representation of de Sitter and conformal groups, it must satisfy a field equation of order of 6, which is obtained.