The initial classification of fusion rules have shown that rational conformal field theory is very limited. In this paper we study the fusion rules of extended current algebras. Explicit formulas are given for the S matrix and the fusion rules, based on the full splitting of the fixed point fields. We find that in some cases sensible fusion rules are obtained, while in others this procedure leads to fractional fusion constants.

Light first generation leptoquarks are being hunted for in HERA and at FNAL and there are various proposals for further searches in future machines. Such leptoquarks are however problematic from a theoretical point of view: Low energy precision measurements imply strong constraints on the couplings of the leptoquarks, and up till now the fulfilment of these constraints seemed extremely unnatural. Here we show that horizontal symmetries, which are very conventional and widely used in the literature for completely different purposes, can suppress the unwanted couplings. Therefore light first generation leptoquarks can be natural.

For each lattice one can define a free boson theory propagating on the corresponding torus. We give an alternative definition where one employs any automorphism of the group M * /M. This gives a wealth of conformal data, which we realize as some bosonic theory, in all the 'regular' cases. We discuss the generalization to affine theories. As a byproduct, we compute the gauss sum for any lattice and any diagonal automorphism.

We present fermionic sum representation for the general Virasoro character of the unitary minimal superconformal series (N = 1). Example of the corresponding "finitizated" identities relating corner transfer matrix polynomials with fermionic companions is considered. These identities in the thermodynamic limit lead to the generalized Rogers-Ramanujan identities. * The expression for the identity character χ1,1 conjectured in ref.[2] is recovered from Eqs.(2.9-2.12).

In this work we build the trigonometric solutions of the Yang-Baxter equation that can not be obtained from quantum groups in any direct way. The solution is obtained using the construction suggested recently [1] from the rational conformal field theory corresponding to the WZW model on SO(3)4R = SU (2)4R/Z2. We also discuss the full elliptic solution to the Yang-Baxter equation whose critical limit corresponds to the trigonometric solution found below.