Partition Functions for Heterotic WZW Conformal Field Theories

Thus far in the search for, and classification of, `physical' modular invariant partition functions \sum N_{LR}\,\c_L\,\C_R the attention has been focused on the {\it symmetric} case where the holomorphic and anti-holomorphic sectors, and hence the characters \c_L and \c_R, are associated with the same Kac-Moody algebras $\g_L=\g_R$ and levels $k_L=k_R$. In this paper we consider the more general possibility where $(\g_L,k_L)$ may not equal $(\g_R,k_R)$. We discuss which choices of algebras and levels may correspond to well-defined conformal field theories, we find the `smallest' such {\it heterotic} (\ie asymmetric) partition functions, and we give a method, generalizing the Roberts-Terao-Warner lattice method, for explicitly constructing many other modular invariants. We conclude the paper by proving that this new lattice method will succeed in generating all the heterotic partition functions, for all choices of algebras and levels.