What we learn on the heterotic string vacua from anomaly-free supergravity

The recently constructed D = 10 anomaly-free supergravity (AFS) has been argued to contain the full effective theory of the heterotic string. The solutions of the effective theory must be solutions of AFS, while the converse is not necessarily true since string theory might specify the boundary conditions for the AFS torsion equation. We show that Calabi-Yau spaces are exact solutions of AFS, while compact group and coset manifolds are not. This is due to a positivity argument, which is the extension to anomaly-free supergravity of the “ten into four won't go” theorem of Freedman, Gibbons and West for the Chapline-Manton theory.