Theory of the conformally invariant mass

By interpreting the conformal transformations as space-time-dependent change of units and introducing the concept of the conformally invariant mass and charge, we develop new conformally invariant Maxwell equations with source terms and equations of motion for massive particles. Although the usual equations of motion with mass terms break the conformal symmetry, it is shown that the Minkowski space is not the most general framework to describe physical processes and there exists a wider consistent dynamics in which conformal invariance is exact. New results also include the general transformation laws of the electromagnetic fields, of currents and force densities. The theory leads naturally to an affine connection and to the 21-parameter inhomogeneous conformal group, ISO(4, 2).     

Dynamical Conformal Transformation and Classical Euclidean Wormholes

We investigate the necessary condition for the existence of classical Euclidean wormholes in a conformally non-invariant gravitational model minimally coupled to an scalar field. It is shown that while the original Ricci tensor with positive eigenvalues does not allow the Euclidean wormholes to occur, under dynamical conformal transformations the Ricci tensor, with respect to the original metric, is dynamically coupled with the conformal field and its eigenvalues may become negative allowing the Euclidean wormholes to occur. Therefore, it is conjectured that dynamical conformal transformations may provide us with effective forms of matter sources leading to Euclidean wormholes in conformally non-invariant systems.     

On weyl's gauge field in a non-local field

In this paper, a non-local field (i.e. the (x, ψ)-field) is constructed by regarding the spinor (ψ) as the internal freedom attached to each point (x). Since this field is likened to a unified field between the (x)- and (ψ)-fields, the metric is given bydσψ=gλ dx ^λψ. Concerning this, some conformally equivalent relations are considered. Next, Weyl's gauge field is introduced into the concept of connection in order to consider the gauge invariance. Finally, some essential features underlying our non-local field are grasped by formulating some fundamental equations of the spin curvature tensors.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

Classical solutions to topologically massive Yang-Mills theory

Classical solutions to (2 + 1)-dimensional Yang-Mills theory in the presence of the Chern-Simons invariant are considered. The SO(3)-invariant solutions to the Euclidean field equations are complex, whereas the equations in Minkowski space-time possess real SO(2, 1)-invariant solutions. The field equations for time independent axially symmetric vector potentials are derived and some solutions are obtained. The behavior of general Euclidean spacetime solutions is discussed. It is also shown that, because of the gauge dependence of the Chern-Simons invariant, the reduced field equations cannot be uniquely obtained from the reduced action. Applications of the results to the infrared structure of finite temperature QCD are discussed; in particular, it is argued that the Chern-Simons invariant cannot be consistently incorporated as a gauge-invariant magnetic mass term in a three-dimensional effective long distance theory at high temperatures.     

Complex and quaternionic analyticity in chiral and gauge theories,I

A comparative and systematic study is made of 2-dimensional CP(n) σ-models and new 4-dimensional HP(n) σ-models and their respective embedded U(1) and Sp(1) holonomic gauge field structures. The central theme is complex versus quaternionic analyticity. A unified formulation is achieved by way of Cartan's method of moving frames adapted to the hypercomplex geometries of the harmonic symmetric spaces $\text{CP(n) ≈ SU(n + 1)}\text{SU(n) × U(1)}$ and $\text{HP(n) ≈ Sp(n + 1)}\text{Sp(n) × Sp(1)}$ respectively. Elements of complex Kähler manifolds are applied to a detailed analysis of the CP(n) σ-model and its instanton sector. Generalization to any Kählerian σ-model is manifest. On the basis of Cauchy-Riemann analyticity, Kählerian models are shown to have an infinite number of local continuity equations. In a parallel manner, new 4-dimensional conformally invariant HP(n) σ-models are constructed. Focus is on the latter's hidden local gauge invariance in their holonomy group Sp(n) × Sp(1) which allows a natural embedding of the Sp(1) ≈ SU(2) pure Yang-Mills theory. The associated quaternionic structure is discussed in light of both quaternionic quantum mechanics and Kählerian geometry. In this chiral setting, the SU(2) Yang-Mills duality equations are cast into quaternionic Cauchy-Riemann equations over S⁴ ≈ HP(1), the conformal spacetime. In analogy to the CP(n) case, their rational solutions are the most general (8n ? 3) parameter instantons where the associated algebraic nonlinear equations of the type of Atiyah, Drinfeld, Hitchin, and Manin are now expressed in a new conformally invariant form. Geometrically, the SU(2) instantons solve the Frenet-Serret equations for quaternionic holomorphic curves; they are conformal maps from HP(1) into HP(n) with n their second Chern index. Fueter's quaternionic analysis is presented, then applied: Fueter functions are particularly suited for the solutions of 't Hooft, of Jackiw, Nohl and Rebbi, and of Witten and Peng, as well as the self-dual finite action per unit time solution of Bogomol'nyi, Prasad and Sommerfield. Generalizing the latter, a new solution with unit Chern index and finite action per unit spacetime cell is found. It is expressed in terms of the quaternionic fourfold quasi-periodic Weierstrass Zeta function. Finally the essence of our method is revealed in terms of universal connections over Stiefel bundles; generalization to real, complex and quaternionic classifying Grassmanian σ-models with their embedded SO(m), SU(m) and Sp(m) gauge fields is outlined in terms of gauge invariant projector valued chiral fields. Other outstanding problems are briefly discussed.     

Conformal gravity and the bosonic string

We show that the Callan et al. effective equations of motion for the bosonic closed string in a non-trivial background may not be derived only from the bosonic part of theN=1 supergravity action of Manton and Chapline, but also from just a general conformally invariant theory of gravity. In our scheme the invariance under gauge transformations of the antisymmetric tensor field turns out to be a consequence of the metricity condition.     

Rational conformal correlation functions of gauge-invariant local fields in four dimensions

Global conformal invariance in Minkowski space and the Wightman axioms imply strong locality (Huygens principle) and rationality of correlation functions, thus providing an extension of the concept of a vertex algebra to higher (even) dimensions D. We (p)review current work on a model of a Hermitian scalar field L of scale dimension 4 (D = 4) which can be interpreted as the Lagrangian of a gauge field theory that generates the algebra of gauge-invariant local observables in a conformally invariant renormalization group fixed point.     

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR ², in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR ² the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Conformal field theory and the symmetries of string field theory

The two-dimensional conformal field theory representation of Witten's bosonic string field theory is discussed. The basic overlap equations, K_n symmetry and BRST invariance are proved directly, without the usual expansion in oscillators. The conformal field theory approach naturally provides local overlap identities which (when integrated over half the string) can be used to verify properties of the cubic action. In particular, a recently proposed diffeomorphism invariance is shown to be free of anomalies. Finally, a new class of symmetries, including generalizations of the K_n symmetries which are local in spacetime, are presented.     

Unique continuation results for Ricci curvature and applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed.     

Properties of SO(8) extended supergravity

The conjectured SU(8) invariance of the field equations of SO(8) extended supergravity is used to elucidate the general structure of the extended theories. Due to the representation content of the spinless fields this does not lead to a complete determination of the theory, as was the case for N = 4. The non-polynomial modifications by spinless fields are given in terms of a number of SU(8) covariant tensors, for which various identities are derived and discussed.     

Equations of motion for N = 1 massless super-Poincaré representations in D = 3,4 space-time

We derive the complete set of equations of motion for an arbitrary N = 1 massless super-Poincaré representation in D = 3 and D = 4 space-time. These equations are derived from the constraints imposed by the on-shell conformal invariance of a massless super-Poincaré representation.     

The integrability of N=16 supergravity

For the maximally extended N=16 supergravity theory in two dimensions, we explicitly construct a linear system whose integrability conditions are equivalent to the full nonlinear field equations of this theory. All the (on-shell) information contained in it can thus be encoded into a single E₈ matrix and its dependence on a spectral parameter; the invariance of the equations of motion under E₉ is manifest. Possible consequences and further developments are briefly discussed.     

On classification of conformally flat spaces

Classification of conformally flat n-dimensional pseudo-Riemannian spaces via Plebanski's method is discussed. It is based on embedding these spaces into a flat (n + 2)-dimensional space and on finding their minimal isometry groups which are subgroups of the conformal group. In particular, the case n = 4 is given in full detail and compared with incomplete results known in the literature. The found conformally flat spacetimes are identified with the associated solutions of the Einstein equations and with the spacetimes used in various cosmological considerations.     

Modular invariant partition functions for N = 2 superconformal field theories

We find the characters of all the unitary degenerate highest weight representations of the N = 2 superconformal algebras with c<3. The constraints of modular invariance on the field contents of the two-dimensional N = 2 superconformal field theories are explored. The constraints take the form of sum rules on the number of superconformal highest weight representations. Models which satisfy these constraints are classified. We also discuss the condition of unbroken supersymmetry in these models.     

Integrable Structures in Classical Off-Shell 10D Supersymmetric Yang–Mills Theory

The field equations of supersymmetric Yang–Mills theory in ten dimensions may be formulated as vanishing curvature conditions on light-like rays in superspace. In this article, we investigate the physical content of the modified SO(7) covariant superspace constraints put forward earlier [11]. To this end, group-algebraic methods are developed which allow to derive the set of physical fields and their equations of motion from the superfield expansion of the supercurl, systematically. A set of integrable superspace constraints is identified which drastically reduces the field content of the unconstrained superfield but leaves the spectrum including the original Yang–Mills vector field completely off-shell. A weaker set of constraints gives rise to additional fields obeying first order differential equations. Geometrically, the SO(7) covariant superspace constraints descend from a truncation of Witten's original linear system to particular one-parameter families of light-like rays. Received: 20 April 2000 / Accepted: 10 September 2000     

General relativity and gauge theories

The theory of general relativity is presented in the form of a gauge field theory by use of the group SL(2,C). The following topics are discussed: (1)Spinor representation of the group SL(2,C); (2)Connection between spinors and tensors; (3)Maxwell, Weyl and Riemann Spinors; (4)Classification of Maxwell spinor; (5)Classification of Weyl spinor; (6)Isotopic spin and gauge fields; (7)Lorentz invariance and the gravitational field; (8)SL(2,C) invariance and the gravitational field; (9)Gravitational field equations.     

Factorization of bosonic string amplitudes

We consider the bosonic string path integral over degenerating Riemann surfaces. We first review the factorization of conformal field theory on a degenerating surface. A careful treatment of the degeneration of the measure for moduli leads to a modification of the usual ghost insertions so as to assure covariance under a change of conformal frame. More generally, amplitudes with BRST invariant but conformally non-invariant operators are well defined with the covariant ghost insertions. As a detailed application we study the string modifications to the background field equations. We find to first order in the tadpole and all orders in string coupling that the ratio of the graviton source, dilaton source, and zero-point amplitude agrees with that found from general covariance and the soft-dilaton theorem in the low-energy field theory. We also discuss the unitarity of the bosonic string theory,