Arbitrary Tensor Fields Research Articles

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.

Covariant series in the normal neighborhood of a submanifold

We consider the covariant series in a some normal neighborhood of a submanifold. Such a neighborhood is a generalization of the normal neighborhood of a point. We discuss how the coefficients of the covariant Taylor series of an arbitrary tensor field can be expressed in terms covariant derivatives of the torsion, Riemann curvature and the field under consideration. We also discuss the algorithm of calculating coefficients of a pseudo-Riemannian metric with respect to the corresponding metric connection without torsion. As an example, we calculate the covariant expansion of the Schwarzschild metric in the normal tubular neighborhood of a circular orbit up to fifth order using the Fermi coordinate system.

Quantization and injective submodules of differential operator modules

The Lie algebra of vector fields on Rm acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to slm+1, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor glm. We prove two results. First, we realize explicitly all injective objects of the parabolic category Oglm(slm+1) of glm-finite slm+1-modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e.,slm+1-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.

Natural operators on tensor fields

Natural operators on tensor fields of different types are classified. We abstain from any assumptions such as symmetry or additional structures and consider arbitrary tensor fields.

Solving partial differential equations numerically on manifolds with arbitrary spatial topologies

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as a set of non-overlapping cubic regions, plus a set of maps to identify the faces of adjoining regions. The differential structure on these manifolds is fixed by specifying a smooth reference metric tensor. Matching conditions that ensure the appropriate levels of continuity and differentiability across region boundaries are developed for arbitrary tensor fields. Standard numerical methods are then used to solve the equations with the appropriate boundary conditions, which are determined from these inter-region matching conditions. Numerical examples are presented which use pseudo-spectral methods to solve simple elliptic equations on multi-cube representations of manifolds with the topologies T3, S2×S1 and S3. Examples are also presented of numerical solutions of simple hyperbolic equations on multi-cube manifolds with the topologies R×T3, R×S2×S1 and R×S3.

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.

The heat kernel on AdS

We explicitly evaluate the heat kernel for the Laplacian of arbitrary spin tensor fields on the thermal quotient of (Euclidean) $AdS_N$ for $N\geq 3$ using the group theoretic techniques employed for $AdS_3$ in arXiv:0911.5085. Our approach is general and can be used, in principle, for other quotients as well as other symmetric spaces.

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.

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.

COVARIANT STAR PRODUCT ON SYMPLECTIC AND POISSON SPACE–TIME MANIFOLDS

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures that were recently defined on the algebra of (scalar-valued) differential forms. A covariant star product of arbitrary smooth tensor fields is obtained as a special case. Finally, we study covariant star products on a more general Poisson manifold with a linear connection, first for smooth functions and then for smooth tensor fields of any type. Some observations on possible applications of the covariant star products to gravity and gauge theory are made.

The heat kernel on AdS 3 and its applications

We derive the heat kernel for arbitrary tensor fields on S^3 and (Euclidean) AdS_3 using a group theoretic approach. We use these results to also obtain the heat kernel on certain quotients of these spaces. In particular, we give a simple, explicit expression for the one loop determinant for a field of arbitrary spin s in thermal AdS_3. We apply this to the calculation of the one loop partition function of N=1 supergravity on AdS_3. We find that the answer factorizes into left- and right-moving super Virasoro characters built on the SL(2, C) invariant vacuum, as argued by Maloney and Witten on general grounds.

The Formula of Grangeat for Tensor Fields of Arbitrary Order in n Dimensions

The cone beam transform of a tensor field of order m in n ≥ 2 dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of the n-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces for m = 0 and n = 3 to the well-known formula of Grangeat. In that sense, the paper contains a generalization of Grangeat's formula to arbitrary tensor fields in any dimension. We further briefly explain the importance of that formula for the problem of tensor field tomography. Unfortunately, for m > 0, an inversion method cannot be derived immediately. Thus, we point out the possibility to calculate reconstruction kernels for the cone beam transform using Grangeat's formula.

A weighted de Rham operator acting on arbitrary tensor fields and their local potentials

We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r -fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence of 2 r potentials for any tensor field, where r is its form-structure number. By specialising this result to symmetric double forms, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2, 3)-form potential for the Weyl tensor due to its tracelessness. This latter potential is the n -dimensional version of the double dual of the classical four-dimensional (2, 1)-form Lanczos potential. We also introduce a new concept of harmonic tensor fields, and demonstrate that the new weighted de Rham operator has many other desirable properties and, in particular, is the natural operator to use in the Laplace-like equation for the Riemann tensor.

The Normal Holonomy Group of Kähler Submanifolds

We study the (restricted) holonomy group Hol(∇⊥) of the normal connection ∇⊥ (shortened to normal holonomy group) of a Kähler submanifold of a complex space form. We prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible Hermitian symmetric space. In particular, it is a compact group and the complex structure J belongs to its Lie algebra.We prove that the normal holonomy group acts irreducibly if the submanifold is full (that is, it is not contained in a totally geodesic proper Kähler submanifold) and the second fundamental form at some point has no kernel. For example, a Kähler–Einstein submanifold of CPn has this property.We define a new invariant μ of a Kähler submanifold of a complex space form. For non‐full submanifolds, the invariant μ measures the deviation of J from belonging to the normal holonomy algebra. For a Kähler–Einstein submanifold, the invariant μ is a rational function of the Einstein constant. By using the invariant μ, we prove that the normal holonomy group of a not necessarily full Kähler–Einstein submanifold of CPn is compact, and we give a list of possible holonomy groups.The approach is based on a definition of the holonomy algebra hol(P) of an arbitrary curvature tensor field P on a vector bundle with a connection and on a De Rham type decomposition theorem for hol(P). 2000 Mathematics Subject Classification 53C40 (primary), 53B25 (secondary).

All-sky convolution for polarimetry experiments

We discuss all-sky convolution of the instrument beam with the sky signal in polarimetry experiments, such as the Planck mission which will map the temperature anisotropy and polarization of the cosmic microwave background (CMB). To account properly for stray light (from e.g. the galaxy, sun, and planets) in the far side-lobes of such an experiment, it is necessary to perform the beam convolution over the full sky. We discuss this process in multipole space for an arbitrary beam response, fully including the effects of beam asymmetry and cross polarization. The form of the convolution in multipole space is such that the Wandelt-G\'orski fast technique for all-sky convolution of scalar signals (e.g. temperature) can be applied with little modification. We further show that for the special case of a pure co-polarized, axisymmetric beam the effect of the convolution can be described by spin-weighted window functions. In the limits of a small angle beam and large Legendre multipoles, the spin-weight 2 window function for the linear polarization reduces to the usual scalar window function used in previous analyses of beam effects in CMB polarimetry experiments. While we focus on the example of polarimetry experiments in the context of CMB studies, we emphasize that the formalism we develop is applicable to anisotropic filtering of arbitrary tensor fields on the sphere.

Optimal distribution of material properties for an elastic continuum with structure-dependent body force

The simultaneous optimization of material properties and structural layout for an elastic continuum is formulated and analyzed. The objective is to obtain the maximum structural stiffness for prescribed surface loads and displacements, taking into account a body force that depends on the structural layout. Optimization with an account of self-weight or centrifugal forces are examples of these type of problems. Arbitrary elasticity tensor fields are considered as the problems variable and necessary conditions satisfied by the solution are established. The use of a spectral decomposition of the elasticity tensor is emphasized since it provides a simple geometrical interpretation. Typical examples which illustrate the effects of the structure-dependent body force are analyzed. It is found that a commonly used isoperimetric restriction (known as the resource constraint) is not necessarily active and that the optimal structure is locally stiffer in areas where the body force and the displacement field have opposite directions and locally weaker otherwise. This, interestingly, leads to a non-symmetrical distribution of material properties even for prismatic bodies under anti-symmetric surface loads.

Analytic continuation of tensor fields along geodesics by covariant Taylor series

It is shown that in a certain normal neighborhood of a submanifold—the analog of a normal neighborhood of a point—the covariant derivatives of all orders of an arbitrary tensor field and of the curvature and torsion along geodesics normal to the submanifold, taken at points of the submanifold, determine under conditions of analyticity the given tensor field by Taylor series with tensor coefficients. Explicit expressions are obtained that provide a recursive procedure for calculating the coefficients of the series in any order. Special cases of the expansion of the components of a pseudo-Riemannian metric with respect to a metric connection without torsion for a point and hypersurface are considered.

STRINGY RESTRICTIONS ON THE BACKGROUNDS IN THE HETEROTIC SIGMA MODEL

We show that for the heterotic string theory in the presence of arbitrary background gauge, gravitational and antisymmetric tensor fields, truncated by a general coordinate dependent compactification a la Scherk-Schwarz, the requirement of 2D conformal invariance is as restrictive as to inhibit supersymmetry breaking with vanishing cosmological constant.

Gauge group, degeneration, and degrees of freedom in the ECSK theory of gravity. I. The symplectic structure of the ECSK theory

This paper is the first of a two-article series in which the connections between the gauge group, degeneration, and degrees of freedom in the ECSK theory coupled to an arbitrary tensor field are discussed. In this paper a multisymplectic formulation of the ECSK theory is presented and the symplectic 2-formω, which plays a leading role in our considerations, is found.

On the quantization of tensor field. II

The Dirac problem is presented and studied for an arbitrary tensor field. Our previous work is corrected and quantum-mechanical operators on the Hilbert space are obtained by using the Schrodinger representation. The approach is applied to tensor fields of valences 5 and 6. The commutators of quantum-mechanical operators corresponding to tensor fields of valence 6 and scalar fields are computed.