Critical Dimension of String Theories in Curved Space

The critical dimension of string theories in which the background metric is a product of Minkowski space and an $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ or $\mathrm{SO}\mathrm{}\left(N\right)\mathrm{}$ group manifold is derived. A consistent string theory can be constructed only in the presence of a Wess-Zumino term associated with the compactified dimension. This implies that the compactified radius is quantized in units of the string tension. A generalization to the supersymmetric case is discussed.