Heterotic/heterotic duality in D = 6,4

We consider E8 × E8 heterotic compactifications on K3 and K3 × T2. The idea of heterotic/heterotic duality in D = 6 has difficulties for generic compactifications since for large dilaton values some gauge groups acquire negative kinetic terms. Recently Duff, Minasian and Witten (DMW) suggested a solution to this problem which only works if the compactification is performed assuming the presence of symmetric gauge embeddings on both E8ʹs. We consider an alternative in which asymmetric embeddings are possible and the wrong sign of kinetic terms for large dilaton value is a signal of spontaneous symmetry breaking. Upon further toroidal compactification to D = 4, we find that the duals in the DMW case correspond to N = 2 models in which the β-function of the different group factors verify βα = 12 whereas the asymmetric solutions that we propose have βα = 24. We check the consistency of these dualities by studying the different large T,S limits of the gauge kinetic function. Dual N = 1, D = 4 models can also be obtained by the operation of appropriate freely acting twists, as shown in specific examples.