Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory

We show that six-dimensional backgrounds that are T2T2 bundle over a Calabi–Yau two-fold base are consistent smooth solutions of heterotic flux compactifications. We emphasize the importance of the anomaly cancellation condition which can only be satisfied if the base is K  3 while a T4T4 base is excluded. The conditions imposed by anomaly cancellation for the T2T2 bundle structure, the dilaton field, and the holomorphic stable bundles are analyzed and the solutions determined. Applying duality, we check the consistency of the anomaly cancellation constraints with those for flux backgrounds of M-theory on eight-manifolds.