New quantum deformations ofD=4 conformal algebra

We consider new class of classicalr-matrices forD=4 conformal Lie algebra. These r-matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebras. In such a way we generalize the lowest order of known nonstandard quantum deformation ofsl(2) to the Lie algebrasl(4)so(6). As an exercise we interpret nonstandard deformation ofsl(2) as describing quantumD=1 conformal algebra with fundamental mass parameter. Further we describeD=4 conformal bialgebras with deformation parameters equal to the inverse of fundamental masses. It appears that forD=4 the deformation of the Poincaré algebra sector coincides with null plane quantum Poincaré algebra.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.Partially supported by the project 5270/95 of the Polish-French Scientific Cooperation.On leave of absence from the Institute of Theoretical Physics, University of Wrocaw, 50-204 Wrocaw, Poland.Two of the authors (J.L and M.M) would like to thank the University of Bordeaux I for the hospitality and financial support.     

On quantum deformations of d=4 conformal algebra

Three classes of classical r-matrices for sl(4, C) algebra are constructed in quasi-Frobenius algebra approach. They satisfy CYBE and are spanned respectively on 8,10,12 generators. The o(4, 2) reality condition can be imposed only on the eight dimensional r-matrices with dimension-full deformation parameters. Contrary to the Poincaré algebra case, it appears that all deformations with a mass-like deformation parameter (-deformations) are described by classical r-matrices satisfying CYBE.     

Nonstandard 2+1 conformal Lie bialgebras and their quantum deformations

The nonstandard and so(2, 2) Lie bialgebras are generalized to the so(3, 2) case in two natural ways by considering this algebra as the conformal algebra of the 2+1 Minkowskian spacetime. Lie bialgebra contractions are analyzed providing conformal bialgebras of the 2+1 Galilean and Carroll spacetimes. The corresponding quantum Hopf so(3, 2) algebras are presented and contractions are performed at the quantum level.     

Lichnerowicz-York equation and conformal deformations on maximal slicings in asymptotically flat space-times

The solvability of the Lichnerowicz-York equation is discussed on each sliceS _t=IR³ of a spacelike, asymptotically Euclidean maximal foliation {S _τ}. Following Cantor, the problem is reduced to a discussion of the properties of a smooth, time-dependent, family of conformal transformations,ø _t, relating the physical metrich _tofS _t to a metric ? _t =ø ⁴h_t, with vanishing scalar curvature. An estimate is provided for infø _t. This allows us to examine the properties of the scale geometry on eachS _twhen strong field regions are probed. It is shown that in such regions ? _t tends to become degenerate exponentially as a suitable average of the scalar curvature of (S _t, h _t ) increases. This is interpreted as representing the approach to a singular regime for (S _t, h _t ). An estimate is also provided for the lapse function-N _t defining {S _t}. This is found to be in agreement with a similar estimate suggested, on heuristic grounds, by Smarr and York. This latter result indicates that asymptotically flat maximal slicings in general (but not always) avoid reaching regions where the above singular regime is approached.     

Pair correlations in sandpile model: A check of logarithmic conformal field theory

We compute the correlations of two height variables in the two-dimensional Abelian sandpile model. We extend the known result for two minimal heights to the case when one of the heights is bigger than one. We find that the most dominant correlation logr/r⁴$\log r/r^4$ exactly fits the prediction obtained within the logarithmic conformal approach.     

Lagrangians for conformal gauge gravity and conformal simple supergravity

The connection, curvature, and Lagrangian for a conformal gauge gravity are obtained. A set of generators of the conformal simple supergroup is given, the commutation and anticommutation relations for the superalgebra are calculated, and a Lagrangian of the simple supergravity is established.     

Representations of conformal supersymmetry

The superalgebras of (generalized) conformal supersymmetry have some very interesting unitarizable representations that contain only massless representations of the conformal subalgebra, in spite of contrary claims that have recently been made , .

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Triple point of nuclear deformations

We show that the second-order phase transition between spherical and deformed shapes of atomic nuclei is an isolated point following from the Landau theory of phase transitions. This point can occur only at the junction of two or more first-order phase transitions which explains why it is associated with one special type of structure and requires the recently proposed first-order phase transition between prolate and oblate nuclear shapes. Finally, we suggest the first empirical example of a nucleus located at the isolated triple-point.     

Integrable deformations of Lotka-Volterra systems

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map Δ(2) that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.     

Quantum conformal superspace

For a compact connected orientablen-manifoldM, n 3, we study the structure ofclassical superspace ,quantum superspace ,classical conformal superspace , andquantum conformal superspace . The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry ofM is zero, thenS,S ₀,C, andC ₀ areilh orbifolds. The case of most importance for general relativity is dimensionn=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS ₀ and quantum conformal superspaceC ₀ are in factilh-manifolds. If, moreover,M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractible ilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS ₀ andC ₀. Our results are applicable to questions involving the problem of thereduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the spaceC ₀ will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.     

Extended conformal symmetries

We discuss various aspects of two-dimensional extended conformal symmetries, better known under the name “W-symmetries”. In particular, we discuss the gauging of W - symmetries and the construction of the so-called “W-gravity” theories.     

Commuting conformal and dual conformal symmetries in the Regge limit

In this paper we continue our study of the dual SL(2,C)$\mathit{SL}(2,C)$ symmetry of the BFKL equation, analogous to the dual conformal symmetry of N=4$N=4$ super-Yang–Mills. We find that the ordinary and dual SL(2,C)$\mathit{SL}(2,C)$ symmetries do not generate a Yangian, in contrast to the ordinary and dual conformal symmetries in the four-dimensional gauge theory. The algebraic structure is still reminiscent of that of N=4$N=4$ SYM, however, and one can extract a generator from the dual SL(2,C)$\mathit{SL}(2,C)$ close to the bi-local form associated with Yangian algebras. We also discuss the issue of whether the dual SL(2,C)$\mathit{SL}(2,C)$ symmetry, which in its original form is broken by IR effects, is broken in a controlled way, similar to the way the dual conformal symmetry of N=4$N=4$ satisfies an anomalous Ward identity. At least for the lowest orders it seems possible to recover the dual SL(2,C)$\mathit{SL}(2,C)$ by deforming its representation, keeping open the possibility that it is an exact symmetry of BFKL. Independently of a possible relation to N=4$N=4$ scattering amplitudes, this opens an avenue for explaining the integrability of BFKL in terms of two finite-dimensional subalgebras.     

Group deformations and relativity

A theory of deformation (homcomorphism but non-isomorphism) of topological groups is developed. In particular, a theory of deformation of subgroups structure is considered. The whole formalism is based on conceptions of holonomicity and relative geometry.A field is postulated to deform the symmetry group of a free physical system. It is shown that the classical fields deform the Poincare group. Thanks to this fact, gravitation appears as space-time curvature (non-holonomicity of the Lorentz subgroup mapping); and electromagnetism reveals itself by space-time torsion (non-holonomicity of the translation subgroup mapping). From physically evident premises it follows that space-time also has a torsion in the rotating and accelerated systems of reference.     

Towards conformal cosmology

Approximate de Sitter symmetry of inflating Universe is responsible for the approximate flatness of the power spectrum of scalar perturbations. However, this is not the only option. Another symmetry that can explain nearly scale-invariant power spectrum is conformal invariance. We give a short review of models based on conformal symmetry that lead to the scale-invariant spectrum of the scalar perturbations. We discuss also potentially observable features of these models.