There is no automorphism of the conformal superalgebra that induces PT (parity and time reversal) on space time. Those conformal transformations that induce PT must change the sign of the odd brackets. In particular, in any implementation of conformal supersymmetry, CP can preserve but CPT must (at best) reverse the sign of the odd brackets (C, charge conjugation).

The purpose of this article is to present a short review of local conformal symmetry in curved 4d space-time. Furthermore we discuss the conformal anomaly and anomaly-induced effective actions. Despite the conformal symmetry is always broken at quantum level, it may be a basis of useful and interesting approximations for investigating quantum corrections.

We give necessary and suucient conditions for a conformal foliation locally generated by conformal vector elds to produce harmonic morphisms. Natural constructions of harmonic maps and morphisms are thus obtained. Also we obtain reducibility results for harmonic morphisms induced by (innnitesimal) conformal actions on Einstein manifolds.

I give a short proof of the following algebraic statement: if V is a simple vertex algebra, then the underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra. This provides many examples of non-finite simple Lie conformal algebras, and should prove useful for classification purposes.

We give an algebraic characterization of the case when conformal Weyl and conformal Lyra connections have the same curvature tensor. It is determined a (1,3)-tensor field invariant to certain transformation of semi-symmetric connections, compatible with Weyl structures on conformal manifolds. It is studied the case when this tensor is vanishing. M.S.C. 2000: 53B05, 53B20, 53B21.

We outline the structure of boundary conditions in conformal field theory. A boundary condition is specified by a consistent collection of reflection coefficients for bulk fields on the disk together with a choice of an automorphism ω of the fusion rules that preserves conformal weights. Non-trivial automorphisms ω correspond to D-brane configurations for arbitrary conformal field theories.

We study the logarithmic conformal field theories in which conformal weights are continuous subset of real numbers. A general relation between the correlators consisting of logarithmic fields and those consisting of ordinary conformal fields is investigated. As an example the correlators of the Coulomb-gas model are explicitly studied.

We analyse conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmological singularity and prove a local extension theorem along a congruence of time-like conformal geodesics.

In logarithmic conformal field theory, primary fields come together with logarithmic partner fields on which the stress-energy tensor acts non-diagonally. Exploiting this fact and global conformal invariance of two-and three-point functions, operator product expansions of logarithmic operators in arbitrary rank logarithmic conformal field theory are derived.

In this article we study the Ahlfors regular conformal gauge of a compact metric space (X, d), and its conformal dimension dimAR(X, d). Using a sequence of finite coverings of (X, d), we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gaug...