The Klein-Gordon operator in conformal space

We consider the Minkowski space M₄ as a local chart of a compact differentiable pseudo-Riemannian manifold M⁴_c, on which the whole conformal group $\text{O(2, 4)}\text{Z}{}_2$ acts continuously. We investigate the conditions under which functions or differential operators on the space M₄ can be uniquely continued to the conformal manifold M⁴_c. This is done by using methods well-known in the theory of differentiable manifolds. In particular, we show that the Klein-Gordon operator □+m² can be uniquely continued to the space M⁴_c and we discuss the conformal invariance of the Klein-Gordon equation on the manifold M⁴_c.     

The dynamical differential forms of the Klein—Gordon field

We classify the conserved currents for the Klein-Gordon field. We show that there are many which are not associated with invariances of the Lagrangian.     

Dilogarithm identities in conformal field theory and group homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin (equivalent to an identity found earlier by Lewin) can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2×2 real matrices viewed as adiscrete group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic K-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2×2 real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with a summary of a number of open conjectures on the mathematical side.To Professor C. N. Yang for his 70^th birthdayThis work was partially supported by grants from the Statens Naturvidenskabelige Forskningsraad and the Paul and Gabriella Rosenbaum Foundation     

The fifteen-parameter conformal group

The space of lines in a Hermitean quadric of signature (2, 2) in complex projective three-space is a quadric of signature (2, 4) in real projective five-space, the conformal compactification of Minkowski space. This geometric fact leads to the classical isomorphism ofPSU(2, 2) and the identity component ofPO(2, 4; ), the 15-parameter conformal group. In this paper it is shown how the geometry and the isomorphism, for all components ofPO(2, 4; ), arise naturally from a real form of the Clifford algebra, and its associated spin groups, of a certain complex vector space determined by skew-symmetric 4×4 matrices and having their Pfaffian as quadratic form.     

A new unified field theory based on the conformal group

We develop here a new unified theory of the electromagnetic and gravitational field, based on a six-dimensional generalization of Maxwell's equations; additional space-time coordinates are interpreted only as mathematical tools in order to obtain a linear realization of the four-dimensional conformal group.     

Gravitation of the Klein-Gordon scalar field

This paper presents an exact solution of the Einstein-Klein-Gordon equations in the static and spherically symmetric case and points out the differences between it and Yilmaz's solution. In addition, the essential difference between the exact solution and the post-Newtonian approximate solution is also shown.     

Hidden quantum group symmetry and integrable perturbations of conformal field theories

The hidden quantum group symmetry in the quantum Sine-Gordon model is found. This symmetry provides the possibility to restrict the operator algebra of the model to subalgebras. It is shown that these subalgebras are massive deformations of minimal conformal field theories.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

Gauging the conformal group

It is demonstrated that Kibble’s method of gauging the Poincaré group can be applied to the gauging of the conformal group. The action of the gauge transformations is the action of general spacetime diffeomorphisms (or coordinate transformations) combined with a local action of an 11-parameter subgroup of SO(4,2). Because the translational subgroup is not an invariant subgroup of the conformal group the appropriate generalisation of the derivative of a physical field is not a covariant derivative in the usual sense, but this does not lead to any inconsistencies.     

Classical origin of quantum group symmetries in Wess-Zumino-Witten conformal field theory

We elucidate the way by which the quantum group symmetries of the WZW models arise within the canonical formalism of the classical field theory.     

On differential equation on four-point correlation function in the conformal Toda field theory

The properties of completely degenerate fields in the conformal Toda field theory are studied. It is shown that a generic four-point correlation function that contains only one such field does not satisfy an ordinary differential equation, in contrast to the Liouville field theory. Some additional assumptions for other fields are required. Under these assumptions, we write such a differential equation and solve it explicitly. We use the fusion properties of the operator algebra to derive a special set of three-point correlation functions. The result agrees with the semiclassical calculations.     

The conformal factor in the SAS Einstein-Maxwell field equations and a central extension of a formal loop group

We consider a relation between the conformal factor in the stationary axisymmetric (SAS) Einstein-Maxwell field equations and a central extension of a formal loop group which is described by a group 2-cocycle on the formal loop group. The corresponding 2-cocycle on the Lie algebra of the formal loop group is the one which describes an affine Lie algebra. As a result, we see that the space of formal solutions with conformal factors is a homogeneous space of a central extension of the Hauser group.     

Stochastic Loewner evolution for conformal field theories with lie group symmetries

The stochastic Loewner evolution is a recent tool in the study of two-dimensional critical systems. We extend this approach to the case of critical systems with continuous symmetries, such as SU(2) Wess-Zumino-Witten models, where domain walls carry an additional spin-1/2 degree of freedom.     

On the integrability of the Lie algebra of the conformal group in quantum field theory

It is shown that the infinitesimal conformal symmetry implies (in any quantum field theory which satisfies the Wightman axioms without invoking locality and global Poincaré symmetry) that there exists a uniquely defined unitary representation of the universal (-sheeted) covering group of the Minkowskian conformal groupSO _e (4,2)/ ₂. Proof was obtained using sufficient conditions for the integrability of a representation of a Lie algebra given by [8].     

Quantum group structure of quantum Toda conformal field theories (I)

We consider the quantization of non-affine Toda field theories in the light-cone and lattice formalisms. The vertex operators are constructed and their braiding is found to be a consequence of the fundamental commutation relations satisfied by the monodromy matrix. For certain values of the coupling, which correspond to the minimal models, the truncation of the operator algebra is closely tied to the quantum group structure.     

The six-point families of exceptional representations of the conformal group

Six-point families of topological reducible (indecomposable) representations of the conformal group of space-time M ₄ are studies. The structure of invariant subspaces is analysed and the full set of equivalence relations among the suprepresentations is derived.     

Vertex operators in the conformal field theory on P 1 and monodromy representations of the braid group

Vertex operators (primary fields) are constructed for the conformal field theory on P ¹ by means of A ₁ ⁽¹⁾ modules. The commutation relations of vertex operators induce monodromy representations of the braid group on the spaces of vacuum expectations of compositions of vertex operators.     

Gauge theory of the conformal and superconformal group

We show that the locally scale invariant Weyl theory of gravity is the gauge theory of the conformal group. Proper conformal transformations are gauged by a non-propagating gauge field.A gauge theory for the superconformal group is obtained which is locally scale, Lorentz, and chiral invariant but not locally supersymmetric despite remarkable cancellations.     

Representing functionally the two-dimensional conformal group

Projective representations of the two-dimensional conformal group are explicitly constructed in terms of propagation kernels. The representation functionals are then used to study the effect of conformal transformations on the states of a free massless scalar field theory.