The W_N minimal model classification

We first rigourously establish, for any N, that the toroidal modular invariant partition functions for the (not necessarily unitary) W_N(p,q) minimal models biject onto a well-defined subset of those of the SU(N)xSU(N) Wess-Zumino-Witten theories at level (p-N,q-N). This permits considerable simplifications to the proof of the Cappelli-Itzykson-Zuber classification of Virasoro minimal models. More important, we obtain from this the complete classification of all modular invariants for the W_3(p,q) minimal models. All should be realised by rational conformal field theories. Previously, only those for the unitary models, i.e. W_3(p,p+1), were classified. For all N our correspondence yields for free an extensive list of W_N(p,q) modular invariants. The W_3 modular invariants, like the Virasoro minimal models, all factorise into SU(3) modular invariants, but this fails in general for larger N. We also classify the SU(3)xSU(3) modular invariants, and find there a new infinite series of exceptionals.Comment: 25 page

On Fusion Algebras and Modular Matrices

We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix $S$, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the $A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$ and level $k$, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most $r$ and $k$. We also find new, systematic sources of zeros in the modular matrix $S$. In addition, we obtain a formula relating the entries of $S$ at fixed points, to entries of $S$ at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.Comment: 28 pages, plain Te