Superstring theory on smooth manifolds with a non-abelian lie group as covering space

Abstract

In this paper we develop superstring theory on target spaces M target = M 4 ⊗ G/B where G is a non-abelian Lie-group and B ⊂ G is a suitable discrete subgroup. These target spaces, different from orbifolds, are smooth differentiable manifolds. Nontrivial choices of B give rise to twisted Kač-Moody algebras providing the mechanism which allows the existence of massless fermions in the string spectrum notwithstanding the non-abelian character of G. Actually we show that there is a unique choice of the group G compatible with the requirement of massless fermion existence, two-dimensional conformal invariance and finally with N = 1 target supersymmetry. It is G = SU(2) 3. We discuss modular invariance and Goddard-Nahm-Olive fermionization. We show that at the quantum level we can describe the SU(2) 3 theory by means of 18 free fermions belonging to the adjoint representation of SU(2) 6. This enables us to make contact with the free fermion approach. However our group interpretation provides additional constraints on the permissible boundary conditions for free fermion theories admitting a geometrical interpretation as σ-models on a smooth manifold: the G/B space. Finally the choice of B is related to the number of space-time supersymmetries.