Heterotic Modular Invariants and Level--Rank Duality

New heterotic modular invariants are found using the level-rank duality of affine Kac-Moody algebras. They provide strong evidence for the consistency of an infinite list of heterotic Wess-Zumino-Witten (WZW) conformal field theories. We call the basic construction the dual-flip, since it flips chirality (exchanges left and right movers) and takes the level-rank dual. We compare the dual-flip to the method of conformal subalgebras, another way of constructing heterotic invariants. To do so, new level-one heterotic invariants are first found; the complete list of a specified subclass of these is obtained. We also prove (under a mild hypothesis) an old conjecture concerning exceptional $A_{r,k}$ invariants and level-rank duality.