Heterotic String Theory on Calabi-yau - Manifolds in the Green-schwarz Formalism*

The a-model describing the propagation of the heterotic string in a CalabiYau background is written down in the Green-Schwarz formulation. This model has manifest N=l space-time supersymmetry, however only the SO(2) subgroup of the four dimensional Lorentz group is realized linearly. The criteria for the existence of the full SO(3,l) L orentz symmetry in this model are studied. These requirements turn out to be identical to those obtained by demanding the existence of space-time supersymmetry in the Neveu-Schwarz-Ramond formulation of the theory, where the SO(3,l) L orentz symmetry is manifest, but space-time supersymmetry is not. The analysis is easily generalized to more general backgrounds, which give rise to (2,0) su erconformal field theories on the world sheet. p Finally it is shown that if the requirements which give unbroken N=l space-time supersymmetry and full SO(3,l) L orentz invariance are satisfied, then the unbroken gauge group after the compactification of the &3 x J?& heterotic string theory on Calabi-Yau spaces iS Es X Es. =Submitted to Nuclear Physics B * Work supported by the Department of Energy, contract DE AC03 76SF00515.