We report progress along the line of a previous article -nb. 1 of the series -by one of us (J.-L. G.). One main point is to include chiral operators with fractional quantum group spins (fourth or sixth of integers) which are needed to achieve the necessary correspondence between the set of conformal weights of primaries and the physical spectrum of Virasoro highest weights. This is possible by extending the study of the chiral bootstrap (recently completed by E. Cremmer, and the present authors) to the case of semi-infinite quantum-group representations which correspond to positive integral screening numbers. In particular, we prove the Bidenharn-Elliot and Racah identities for q-deformed 6-j symbols generalized to continuous spins. The decoupling of the family of physical chiral operators (with real conformal weights) at the special values C Liouville = 7, 13, and 19, is shown to provide a full solution of Moore and Seiberg's equations, only involving operators with real conformal weights. Moreover, our study confirms the existence of the strongly coupled topological models put forward earlier. The three-point functions are determined. They are given by a product of leg factors similar to the ones of the weakly coupled models. However, contrary to this latter case, the equality between the quantum group spins of the holomorphic and antiholomorphic components is not preserved by the local vertex operator. Thus the "c=1" barrier appears as connected with a deconfinement of chirality.

The low-energy sputtering of boron nitride, magnesium oxide, boron nitride and aluminum nitride ͑BNAlN͒, and boron nitride and silicon oxide (BNSiO 2 ) by xenon ions of bombarding energies 350, 500 eV, and 1 keV was studied experimentally. In order to measure the ion current without being significantly disturbed by slow ions, only planar probes were used during short duration sputtering experiments ͑of the order of 10 h͒. Moreover, slow ion current contribution was estimated by numerical simulations and subtracted from each ion current measurement. It was found that the ion-beam incidence effect on sputtering yields was not as important as for theoretical results or experimental results on quasinonrough solid surfaces, for which it is possible to observe a more pronounced angular dependence of the sputtering yield. This phenomenon is due to surface irregularities of ceramic materials and because of surface roughness the macroscopic sputtering yield should actually result from the convolution of the microscopic sputtering yield by the angular distribution of surface facet incidences. The irregular surface structure of ceramics like BNSiO 2 or BNAlN seems to be sputtered differently due to a differential erosion from grain to grain and from grain to surrounding matrix. This uneven erosion may be explained by the wide angular distribution of facet incidences of surface microprofiles and the various binding energies in the selvage of material. Finally, the dependence of sputtering yield at normal incidence on ion energy, in the range 0.35-1 keV, is almost a linear one.

No abstract

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to ∞ in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.

No abstract