LAUSR

The interpolation from supersymmetric to non-supersymmetric heterotic theories is studied, via the Scherk-Schwarz compactification of supersymmetric $6D$ theories to $4D$. A general modular-invariant Scherk-Schwarz deformation is deduced from the properties of the $6D$ theories at the endpoints, which significantly extends previously known examples. This wider class of non-supersymmetric $4D$ theories opens up new possibilities for model building. The full one-loop cosmological constant of such theories is studied as a function of compactification radius for a number of cases, and the following interpolating configurations are found: two supersymmetric $6D$ theories related by a $T$-duality transformation, with intermediate $4D$ maximum or minimum at the string scale; a non-supersymmetric $6D$ theory interpolating to a supersymmetric $6D$ theory, with the $4D$ theory possibly having an AdS minimum; a ``metastable'' non-supersymmetric $6D$ theory interpolating via a $4D$ theory to a supersymmetric $6D$ theory.