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Conformal field theories and applications

During these last years, two main directions were followed in the study of conformal -field theories (CFT) in the LPTHE. The first one concerns the construction
of new parafermionic theories and the second one the construction of interfaces on lattice realizations of statistical models correspond to CFT and in analogy with the construction of SLE interfaces.

1) Parafermionic Theories. In the domain of conformal field theories, the parafermionic theories are likely to be amongst the most sophisticated ones. A first series of parafermionic theories, which correspond to the lowest dimensions of parafermionic fields allowed by the associativity and the discrete Z_N symmetry, is widely known nowdays and is used in various branches of physics : statistical physics, condensed matter, quantum Hall in particular, string theory etc. A second series, corresponding to the next (higher) allowed conformal dimensions of parafermionic fields, is much richer in its field theory content, as compared to the first series. It is also much more complicated. The parafermionic theories of the second series are much less known and used. They still await their applications.

The second series parafermionic field theories perturbed by two slightly relevant operators was studied. In the corresponding phase diagram, which is two-dimensional in this case, two new -fixed points were found with the renormalization group methods. These new fixed points were identified with the ’neighbouring’ parafermionic theories of the second type. Next, new methods for constructing new parafermionic theories were developed. With these methods, three new parafermionic theories were found, which are the first ones of the third series. These newly found theories are still richer in their fields content and in the interaction rules, as compared
to the second type ones. But they are also more complicated.

2) Interfaces. The fractal properties of interfaces for lattice models corresponding to
parafermions have been studied. Very good agreement has been obtained with analytical predictions for one type of spin interfaces for spin models with 4 or 5 states with a symmetry Z_4 and Z_5 and corresponding to theories of parafermions. These fractals dimensions have been compared to the ones of finite geometrical clusters. The identification of these fractal dimensions is not trivial since in these spin models, there exists different types of interface since spins with different values can interact.
The number of types of interfaces grows rapidly with N (the number of values that the spins can take). Thus it is diff-cult to identify these interfaces. A more complete study was performed in the case of the Ashkin-Teller model. Note that the parafermionic theory with a Z_4 symmetry is contained in the line of critical points of the Ashkin-Teller model. This model can then be seen as a generalization to a more general model of spins with 4 states. Here again there exists different types of interfaces. By
performing a systematic numerical study of all the possible type of interfaces, we were able to identify four types of fractal dimensions. In particular, one of these dimensions is associated to the dimension expected for a theory of free bosons compactified with c = 1.

Finally, in a last work, the fractal study of interfaces for the Potts model in presence of
disorder was considered. An excellent agreement was obtained between an analytical computations of the fractal dimension and a numerical computation. We also suggested that the duality between spin interfaces and the FK interfaces (similar to the Duplantier duality in the context of SLE) is preserved by the addition of disorder.