Heterotic string theory on Calabi-Yau manifolds in the Green-Schwarz formalism

Abstract

The σ-model describing the propagation of the heterotic string in a Calabi-Yau background is written down in the Green-Schwarz formulation. This model has manifest N=1 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, 1) Lorentz 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, 1) Lorentz symmetry is manifest, but space-time supersymmetry is not. The analysis is easily generalized to more general backgrounds, which give rise to (2, 0) superconformal field theories on the world sheet. Finally it is shown that if the requirements which give unbroken N=1 space-time supersymmetry and full SO(3, 1) Lorentz invariance are satisfied, then the unbroken gauge group after the compactification of the E₈×E₈ heterotic string theory on Calabi-Yau spaces is E₈×E₆.