Symmetric Tensor Fields Research Articles

Hall-like behaviour of higher rank Chern-Simons theory of fractons

Fracton phases of matter constitute an interesting point of contact between condensed matter and high-energy physics. The limited mobility property of fracton quasi-particles finds applications in many different contexts, including quantum information, spin liquids, elasticity, hydrodynamics, gravity and holography. In this paper we adopt a field theoretical approach to investigate the three dimensional action of a rank-2 symmetric tensor field invariant under the covariant fracton symmetry. The theory appears as a non-topological higher rank generalization of the ordinary Chern-Simons model, depending only on the traceless part of the tensor gauge field. After defining a field strength, a rank-2 traceless “electric” field and a “magnetic” vector field are identified, in analogy with the standard Chern-Simons ones. Once matter is introduced, a Hall-like behaviour with fractonic features emerges. In particular, our model shows a Hall-like dipole current, together with a vectorial “flux-attachment” relation for dipoles. This gives a possible starting point for a fracton-vortex duality. A gauge-fixing term is then introduced, from which propagators are computed and the counting of the degrees of freedom is performed. Finally, the energy-momentum tensor is shown to be conserved and the integrated energy density is proved to be zero, which reminds the topological nature of the standard Chern-Simons model.

Realization of planar and surface conformal mappings through stress-free growth of hyperelastic plates: Analytical formulas and numerical calculations

Conformal mapping is a well-known concept and has a long research history in mathematics. Accompanying the developments of computational science and 3D digital scanning technology, conformal mapping has also found wide applications in geometric modeling, computer graphics, medical imaging and other fields. In the virtual image or animation world, conformal mapping offers a convenient approach to achieve various shape changes of planar regions and 3D surfaces. In this work, we propose a promising approach, i.e., through the stress-free growth of hyperelastic plates, to realize arbitrary planar and surface conformal mappings in the physical world. The growth field in a hyperelastic plate can be represented by a symmetric tensor field. For any given planar or surface conformal mappings with explicit analytical expressions, the formulas of growth functions in the growth tensor are derived. In the case that the target surfaces of conformal mappings have no explicit analytical expressions, we further propose a numerical scheme to determine the growth data in the hyperelastic plates. The efficiency of the approach proposed in the current work is validated through several typical examples, where the 3D finite element simulations are conducted. The results demonstrate that, by incorporating the obtained growth functions or growth data, the target surfaces can be generated accurately through growth deformations of hyperelastic plates. As an application of the current work, we further propose an approach of shape-control for double-layer grid structures with plate forms. According to any given target shapes, the values of elongation or contraction of the bar elements are determined, then the desired deformations of the grid structures can be realized.

Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

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.

Global Topology of 3D Symmetric Tensor Fields.

There have been recent advances in the analysis and visualization of 3D symmetric tensor fields, with a focus on the robust extraction of tensor field topology. However, topological features such as degenerate curves and neutral surfaces do not live in isolation. Instead, they intriguingly interact with each other. In this paper, we introduce the notion of topological graph for 3D symmetric tensor fields to facilitate global topological analysis of such fields. The nodes of the graph include degenerate curves and regions bounded by neutral surfaces in the domain. The edges in the graph denote the adjacency information between the regions and degenerate curves. In addition, we observe that a degenerate curve can be a loop and even a knot and that two degenerate curves (whether in the same region or not) can form a link. We provide a definition and theoretical analysis of individual degenerate curves in order to help understand why knots and links may occur. Moreover, we differentiate between wedges and trisectors, thus making the analysis more detailed about degenerate curves. We incorporate this information into the topological graph. Such a graph can not only reveal the global structure in a 3D symmetric tensor field but also allow two symmetric tensor fields to be compared. We demonstrate our approach by applying it to solid mechanics and material science data sets.

Range characterization of ray transform on Sobolev spaces of symmetric tensor fields

Range characterization of ray transform on Sobolev spaces of symmetric tensor fields

Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric m-tensor fields, . Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in , .

Theory of a symmetric tensor field with boundary: Kac-Moody algebras in linearized gravity

Theory of a symmetric tensor field with boundary: Kac-Moody algebras in linearized gravity

Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

The operators of longitudinal, transverse and mixed ray transforms acting on two-dimensional symmetric tensor fields of arbitrary degree m in an unit disk are considered in the article. The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original spaces and image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions the polynomial expressions for the (pseudo)inverse and adjoint operators are obtained.

Iterative inversion of the tensor momentum x-ray transform

The momentum x-ray transform maps a symmetric tensor field f of order m in to its integrals over lines at + b with the weight tk , . In this article we propose an iterative approach to the reconstruction problem. Namely, a new concept of partial momentum transforms is introduced. These transforms are intermediaries between the momentum and the component-wise x-ray transforms. Moreover, one can use a differential operator to construct a sequence of partial momentum transforms that starts with the momentum and finishes with the component-wise transform. The latter can be inverted with the standard component-wise backprojection. The proposed method was also applied to obtain an analog of the Plancherel formula as well as the stability estimates in the Sobolev spaces that are valid for any m.

New characterizations of ruled real hypersurfaces in complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections. For any nonnull constant k and any symmetric tensor field of type (1, 1) L on M, we can define two tensor fields of type (1, 2) on M, L_F^{(k)} and L_T^{(k)}, related to both connections. We study the behaviour of the structure operator phi with respect to such tensor fields in the particular case of L=A, the shape operator of M, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.

Multipole analysis on stationary massive vector and symmetric tensor fields with irreducible Cartesian tensors

The multipole expansions for massive vector and symmetric tensor fields in the region outside spatially compact stationary sources are obtained by using the symmetric and trace-free formalism in terms of the irreducible Cartesian tensors, and the closed-form expressions for the source multipole moments are provided. The expansions show a Yukawa-like dependence on the massive parameters of the fields, and the integrals of the stationary source multipole moments are all modulated by a common radial factor. For stationary massive vector field, there are two types of ‘magnetic’ multipole moments, among which one is the generalization of that of the magnetostatic field, and another, being an additional set of multipole moments of the stationary massive vector field, can not be transformed away. As to the stationary massive symmetric tensor field, its multipole expansion is presented when the trace of its spatial part is specified, where besides the counterparts of the mass and spin multipole moments of massless symmetric tensor field, three additional sets of multipole moments also appear. The multipole expansions of the tensor field under two typical cases are discussed, where it is shown that if the spatial part of the tensor field is trace-free, the monopole and dipole moments in the corresponding expansion will vanish.

Universal spinning Casimir equations and their solutions

Conformal blocks are a central analytic tool for higher dimensional conformal field theory. We employ Harish-Chandra’s radial component map to construct universal Casimir differential equations for spinning conformal blocks in any dimension d of Euclidean space. Furthermore, we also build a set of differential “shifting” operators that allow to construct solutions of the Casimir equations from certain seeds. In the context of spinning four-point blocks of bulk conformal field theory, our formulas provide an elegant and far reaching generalisation of existing expressions to arbitrary tensor fields and arbitrary dimension d. The power of our new universal approach to spinning blocks is further illustrated through applications to defect conformal field theory. In the case of defects of co-dimension q = 2 we are able to construct conformal blocks for two-point functions of symmetric traceless bulk tensor fields in both the defect and the bulk channel. This opens an interesting avenue for applications to the defect bootstrap. Finally, we also derive the Casimir equations for bulk-bulk-defect three-point functions in the bulk channel.

Maxwell theory of fractons

We show that the main properties of the fracton quasiparticles can be derived from a generalized covariant Maxwell-like action. Starting from a rank-2 symmetric tensor field ${A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$, we build a partially symmetric rank-3 tensor field strength ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$ which obeys a kind of Bianchi identity. The most general action invariant under the covariant ``fracton'' transformation ${\ensuremath{\delta}}_{\text{fract}}{A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)={\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\nu}}\mathrm{\ensuremath{\Lambda}}(x)$ consists of two independent terms: one describing linearized gravity (LG) and the other referable to fractons. The whole action can be written in terms of ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, and the fracton part of the invariant Lagrangian writes as ${F}^{2}(x)$, in analogy with Maxwell theory. The canonical momentum derived from the fracton Lagrangian coincides with the tensor electric field appearing in the fracton literature, and the field equations of motion, which have the same form as the covariant Maxwell equations (${\ensuremath{\partial}}^{\ensuremath{\mu}}{F}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\mu}}(x)=0$), can be written in terms of the generalized electric and magnetic fields and yield two of the four Maxwell equations (generalized electric Gauss and Amp\`ere laws), while the other two (generalized magnetic Gauss and Faraday laws) are consequences of the ``Bianchi identity'' for the tensor ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, as in Maxwell theory. In the covariant generalization of the fracton theory, the equations describing the fracton limited mobility, i.e., the charge and dipole conservation, are not external constraints, but rather consequences of the field equations of motion, hence of the invariant action and, ultimately, of the fracton covariant symmetry. Finally, we increase the known analogies between LG and fracton theory by noting that both satisfy the generalized Gauss constraint which underlies the limited mobility property, which one would not expect in LG.

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes.

Tensor fields are useful for modeling the structure of biological tissues. The challenge to measure tensor fields involves acquiring sufficient data of scalar measurements that are physically achievable and reconstructing tensors from as few projections as possible for efficient applications in medical imaging. In this paper, we present a filtered back-projection algorithm for the reconstruction of a symmetric second-rank tensor field from directional X-ray projections about three axes. The tensor field is decomposed into a solenoidal and irrotational component, each of three unknowns. Using the Fourier projection theorem, a filtered back-projection algorithm is derived to reconstruct the solenoidal and irrotational components from projections acquired around three axes. A simple illustrative phantom consisting of two spherical shells and a 3D digital cardiac diffusion image obtained from diffusion tensor MRI of an excised human heart are used to simulate directional X-ray projections. The simulations validate the mathematical derivations and demonstrate reasonable noise properties of the algorithm. The decomposition of the tensor field into solenoidal and irrotational components provides insight into the development of algorithms for reconstructing tensor fields with sufficient samples in terms of the type of directional projections and the necessary orbits for the acquisition of the projections of the tensor field.

The elasticity complex: compact embeddings and regular decompositions

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.

The generalized Saint Venant operator and integral moment transforms

In this article, we work with a generalized Saint Venant operator introduced by Vladimir Sharafutdinov [Inverse and ill-posed problems series, VSP, Utrecht, 1994] to describe the kernel of the integral moment transforms over symmetric m m -tensor fields in n n -dimensional Euclidean space. We also provide an equivalence between the injectivity question for the integral moment transforms and generalized Saint Venant operator over symmetric tensor fields of Schwartz class.

Asymptotic structure with a positive cosmological constant

This is the second of two papers (Fernández-Álvarez F and Senovilla J M M 2021 Class. Quantum Grav. 39 165011) that study the asymptotic structure of space–times with a non-negative cosmological constant Λ. This paper deals with the case Λ > 0. Our approach is founded on the ‘tidal energies’ built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel–Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in , where the first element of the triplet is a three-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on , and the third element is a traceless symmetric tensor field on . The full set of physically relevant properties of the space–time cannot be characterised at infinity without taking D ab into consideration, and our radiation criterion takes this fully into account. We similarly propose a no-incoming radiation criterion based also on the triplet and on radiant supermomenta deduced from the rescaled Bel–Robinson tensor too. We search for news tensors encoding the two degrees of freedom of gravitational radiation and argue that any news-like object must be associated to, and depends on, two-dimensional cross-sections of . We identify one component of news for every such cross-section and present a general strategy to find the second component, which depends on the particular physical situation. We put in connection the radiation condition and the news-like tensors with the directional structure of the gravitational field at infinity and the criterion of no-incoming radiation. We also introduce the concept of equipped by endowing the conformal boundary with a selected congruence of curves which may be determined by the algebraic structure of the asymptotic Weyl tensor. We also define a group of asymptotic symmetries preserving the new structures. We consider the limit Λ → 0, and apply all our results to selected exact solutions of Einstein field equations in order to illustrate their validity.

Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

Let \(I_{m}\) denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f, and \(I_{m}^{*}\) be its formal \(L^2\) adjoint. We study a unique continuation result for the operator \(N_{m}=I_{m}^{*}I_{m}\). More precisely, we show that if \(N_{m}f\) vanishes to infinite order at a point \(x_0\) and if the Saint-Venant operator W acting on f vanishes on an open set containing \(x_0\), then f is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher-order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher-order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.

The theory of symmetric tensor field: From fractons to gravitons and back

We consider the theory of a symmetric tensor field in 4D, invariant under a subclass of infinitesimal diffeomorphism transformations, where the vector diff parameter is the 4-divergence of a scalar parameter. The resulting gauge symmetry characterizes the “fracton” quasiparticles and identifies a theory which depends on a dimensionless parameter, which cannot be reabsorbed by a redefinition of the tensor field, despite the fact that the theory is free of interactions. This kind of “electromagnetic gauge symmetry” is weaker that the original diffeomorphism invariance, in the sense that the most general action contains, but is not limited to, linearized gravity, and we show how it is possible to switch continuously from linearized gravity to a mixed phase where both gravitons and fractons are present, without changing the degrees of freedom of the theory. The gauge fixing procedure is particularly rich and rather peculiar, and leads to the computation of propagators which in the massive case we ask to be tachyonic-free, thus constraining the domain of the parameter of the theory. Finally, a closer contact to fractons is made by the introduction of a parameter related to the “rate of propagation”. For a particular value of this parameter the theory does not propagate at all, and we guess that, for this reason, the resulting theory should be tightly related to the fracton excitations.