We study the heterotic string on the simplest (nontrivial) Calabi-Yau manifold K3 and its orbifold limits. We set the background gauge connection equal to the spin connection. The massless spectrum of the field theory that is the low-energy limit of the E8@E8 string is derived on K3. Some terms in the effective Lagrangian for the massless modes are shown to be determined by the topology of K3. Several orbifold limits of K3 are described as T 4 / z , , after the singular points are removed and replaced by well-behaved spaces. Massless spectra of the full string theory on these orbifolds are obtained. Orbifold and manifold results are compared. Knowing how to blow up the orbifolds determines much of the full string massless spectrum, including information about the spectra of the individual twisted sectors.

We relate in a novel way the modular matrices of GKO diagonal cosets without fixed points to those of WZNW tensor products. Using this we classify all modular invariant partition functions of su(3) k ⊕su(3) 1 /su(3) k+1 for all positive integer level k, and su(2) k ⊕ su(2) ℓ /su(2) k+ℓ for all k and infinitely many ℓ (in fact, for each k a positive density of ℓ). Of all these classifications, only that for su(2) k ⊕ su(2) 1 /su(2) k+1 had been known. Our lists include many new invariants. *

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