WZW Commutants, Lattices, and Level 1 Partition Functions

A natural first step in the classification of all `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of Bauer and Itzykson on the commutant from the $SU(N)$ case they consider to the case where the underlying algebra is any semi-simple Lie algebra (and the levels are arbitrary). We then use this analysis to show that the partition functions associated with even self-dual lattices span the commutant. This proves that the lattice method due to Roberts and Terao, and Warner, will succeed in generating all partition functions. We then make some general remarks concerning certain properties of the coefficient matrices $N_{LR}$, and use those to explicitly find all level 1 partition functions corresponding to the algebras $B_n$, $C_n$, $D_n$, and the 5 exceptionals. Previously, only those associated to $A_n$ seemed to be generally known.