Chiral deformations of conformal field theories

We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W_1+∞ algebra, that is treated in detail.     

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.     

Logarithmic conformal field theory through nilpotent conformal dimensions

We study logarithmic conformal field theories (LCFTs) through the introduction of nilpotent conformal weights. Using this device, we derive the properties of LCFTs such as the transformation laws, singular vectors and the structure of correlation functions. We discuss the emergence of an extra energy momentum tensor, which is the logarithmic partner of the energy momentum tensor.     

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.     

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.     

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.     

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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