Infinitesimal Conformal Transformations Research Articles

New conformal-like symmetry of strictly massless fermions in four-dimensional de Sitter space

We present new infinitesimal ‘conformal-like’ symmetries for the field equations of strictly massless spin-s ≥ 3/2 totally symmetric tensor-spinors (i.e. gauge potentials) on 4-dimensional de Sitter spacetime (dS4). The corresponding symmetry transformations are generated by the five closed conformal Killing vectors of dS4, but they are not conventional conformal transformations. We show that the algebra generated by the ten de Sitter (dS) symmetries and the five conformal-like symmetries closes on the conformal-like algebra so(2, 4) up to gauge transformations of the gauge potentials. The transformations of the gauge-invariant field strength tensor-spinors under the conformal-like symmetries are given by the product of γ5 times a usual infinitesimal conformal transformation of the field strengths. Furthermore, we demonstrate that the two sets of physical mode solutions, corresponding to the two helicities ±s of the strictly massless theories, form a direct sum of Unitary Irreducible Representations (UIRs) of the conformal-like algebra. We also fill a gap in the literature by explaining how these physical modes form a direct sum of Discrete Series UIRs of the dS algebra so(1, 4).

Infinitesimal Transformations of Riemannian Manifolds—The Geometric Dynamics Point of View

In the present paper, we study the geometry of infinitesimal conformal, affine, projective, and harmonic transformations of complete Riemannian manifolds using the concepts of geometric dynamics and the methods of the modern version of the Bochner technique.

A classification of conformal vector fields on the tangent bundle

UDC 514.7 Let ( M , g ) be a Riemannian manifold and T M be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian) lift metric derived from g . We give a classification of infinitesimal fibre-preserving conformal transformations on the tangent bundle.

Infinitesimal Transformations of Locally Conformal Kähler Manifolds

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.

Conformal invariance for Wilson actions

We discuss the realization of conformal invariance for Wilson actions using the formalism of the exact renormalization group. This subject has been studied extensively in the recent works of O. J. Rosten. The main purpose of this paper is to reformulate Rosten's formulas for conformal transformations using a method developed earlier for the realization of any continuous symmetry in the exact renormalization group formalism. The merit of the reformulation is simplicity and transparency via the consistent use of equation-of-motion operators. We derive equations that imply the invariance of the Wilson action under infinitesimal conformal transformations which are non-linearly realized but form a closed conformal algebra. The best effort has been made to make the paper self-contained; ample background on the formalism is provided.

On infinitesimal conformal transformations of the tangent bundles with the synectic lift of a Riemannian metric

The purpose of the present article is to investigate some relations between the Lie algebra of the infinitesimal fibre-preserving conformal transformations of the tangent bundle of a Riemannian manifold with respect to the synectic lift of the metric tensor and the Lie algebra of infinitesimal projective transformations of the Riemannian manifold itself.

Noncommutative instantons come in families

A construction of instantons in the context of noncommutative geometry, in particular SU(2) instantons on a noncommutative 4 sphere, has been recently reported. Firstly, a noncommutative principal fibration A(S4θ)↬ A(S7θ) which ‘quantizes’ the classical SU(2)-Hopf fibration over S4, has been constructed in [11] on the toric noncommutative four-sphere S4θ The generators of A4θ are the entries of a projection p which describes the basic instanton on A4θ. That is, p gives a projective module of finite type p[A(S4θ)]4 and a connection ▽ = p ° D on it which has a self-dual curvature and charge 1, in some appropriate sense; this is the basic instanton. In [12] infinitesimal instantons — ‘the tangent space to the moduli space’ -were constructed using infinitesimal conformal transformations, that is elements in a quantized enveloping algebra Uθ(so(5, 1)). In [10] we looked at a global construction and obtain generic charge 1 instantons by ‘quantizing’ the action of the Lie groups SL(2, ℌ) and SO(2) on the basic instanton which enter the classical construction [1]. We review all this here.

On conformal transformations in tangent bundles

Let $M$ be a complete, simply connected Riemannian manifold with positive constant scalar curvature, and $TM$ its tangent bundle with the complete lift metric. Assume that $TM$ admits an essential infinitesimal conformal transformation, then $M$ is isometric to the standard sphere.

On spherically symmetric perfect fluid solutions with shear. II

Infinitesimal conformal transformations and motion are investigated for a special type of spherically symmetric spacetime, and, by the application of the results, these transformations are determined for the spherically symmetric perfect fluid solutions with shear obtained in the author's earlier paper. Exact spherically symmetric non-accelerating perfect fluid solutions with shear are obtained in concrete forms, which give Kantowski--Sachs cosmological models, and several such examples are given. For these solutions, infinitesimal conformal transformations and motion are determined. The plane symmetric version is treated briefly.

On a class of exact locally conformal cosymlectic manifolds

An almost cosymplectic manifold M is a (2m + 1)‐dimensional oriented Riemannian manifold endowed with a 2‐form Ω of rank 2m, a 1‐form η such that Ωm Λ η ≠ 0 and a vector field ξ satisfying iξΩ = 0 and η(ξ) = 1. Particular cases were considered in [3] and [6].Let (M, g) be an odd dimensional oriented Riemannian manifold carrying a globally defined vector field T such that the Riemannian connection is parallel with respect to T. It is shown that in this case M is a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. Moreover T defines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal transformation of the curvature forms.The existence of a structure conformal vector field C on M is proved and their properties are investigated. In the last section, we study the geometry of the tangent bundle of an exact locally conformal cosymplectic manifold.

Modular invariance and two-dimensional critical systems

1. Critical systems or massless field theories are highly sensitive to geometric effects. The idea that one can implement a local (as opposed to global) scale invariance follows from the work of Belavin Polyakov and Zamolodchikov coming after numerous studies on finite size scaling. These authors applied the techniques developed in the framework of string theories to the study of two dimensional statistical systems, with a prominent role played by the energy momentum tensor, the generator of coordinate transformations. A natural role is played by the infinitesimal conformal transformations generating an infinite Lie algebra, the Virasoro algebra with a “quantum” anomaly -the so called central charge- for which a rich representation theory exists.

The Inhomogeneous Transformation Laws of Currents and the Schwinger Terms in 2D Conformal Field Theories

In the last couple of years there has been a growing interest in two dimensional field theories possessing conformal symmetry for their applications to two dimensional statistical systems at critical points)-7) and to (super)string theories. l) Of paramount importance in such theories is the role played by the energy momentum tensor. Infinitesimal conformal transformations are generated by the energy momentum tensor and the infinitesimal transformation law of the energy momentum tensor itself is equivalent to the algebra of conformal invariance, i.e., the Virasoro algebra. There is a natural question one can raise: How does the energy momentum tensor transform under not an infinitesimal but a finite conformal transformation? In the seminal paper of Belavin, Polyakov and Zamolodchikov,l) it has been noted that under a finite conformal change of coordinates z-> w=u(z) the energy momentum tensor Tzz(z) should transform as:

On the conformal transformations in the massless Thirring model

On the basis of solutions for the massless scalar field in two space-time dimensions obtained in a previous paper, the field satisfying the renormalized Thirring equation is constructed. Both infinitesimal and global transformations with respect to the two-dimensional conformal group for these field is obtained. The latter do not coincide with the standard ones. The renormalized Thirring equation is proved to be covariant under infinitesimal conformal transformations as well as under the global transformations belonging to the representations of the universal covering of the conformal group.

RIEMANNIAN MANIFOLDS ADMITTING A CONFORMAL TRANSFORMATION GROUP

Let M be a Riemannian manifold with constant scalar curvature K which admits an infinitesimal conformal transformation. A necessary and sufficient condition in order that it be isometric with a sphere is obtained.Inequalities giving upper and lower bounds for K are also derived.

Certain infinitesimal transformation of normal contact metric manifold

Introduction. In the previous paper [2], the author studied infinitesimal conformal and projective transformations of normal contact metric manifold. In the present paper, we study certain infinitesimal transformation of normal contact metric manifold and prove the following THEOREM 1. In a normal contact metric manifold, any curvature-preserving infinitesimal transformation is necessarily an infinitesimal isometry. If the above theorem is proved, we have obviously the following THEOREM 2. In a normal contact metric manifold, an infinitesimal affine transformation is necessarily an infinitesimal isometry. THEOREM 3. In a normal contact metric manifold, an infinitesimal homothetic transformation is necessarily an infinitesimal isometry.

On infinitesimal conformal and projective transformations of normal contact spaces

Introduction. In the previous paper [ 4 ]° the author discussed some properties of normal contact spaces. However, problems concerning infinitesimal transformations have not been studied. In the present paper such problems are concerned and some satisfactory answers are given. In § 1, we state the fundamental identities of normal contact spaces. In § 2, we shall give some preliminary facts concerning infinitesimal transformations for the later use. After these preparations, in § 3, v e shall prove that an infinitesimal conformal transformation in normal contact spaces is necessarily concircular. Thus we know that a connected complete normal contact space admitting an infinitesimal non-isometric conformal transformation is isometric with a unit sphere. It will be shown in § 4 that an infinitesimal projective transformation in a normal contact space has some analogous properties of the one in an Einstein space, for example, that any infinitesimal projective transformation in this space is decomposed as a sum of a Killing vector and an infinitesimal gradient projective transformation. In § 5, we shall define the notion of ^-Einstein spaces and discuss infinitesimal conformal and projective transformations in such spaces. Finally, we shall devote § 6 to show that one of Sasaki's examples of normal contact spaces is an example of ^-Einstein spaces. l (Φ>?> v> ^-structure and contact structure. On an n(= 2m + l)-dimensional real differentiable manifold M with local coordinate systems {x}, if there exist a tensor field φ/, contravariant and covariant vector fields ξ and ηt satisfying the relations (1. 1) ξ% = 1, (1. 2) rank | φ / | = n 1, (1. 3) φ/ξ> = 0, (1. 4) φ/Vί = 0, (15) Φ / Φ * ' = δ * < + *7*f,

The local and global covariant variations of differential forms under an infinitesimal conformal transformation.

The local and global covariant variations of differential forms under an infinitesimal conformal transformation.