Supersymmetry-breaking compactifications on Riemann-flat manifolds

We consider compactifications of higher-dimensional supergravities on Riemann-flat manifolds of dimension d (d ≤ 7) that fully break supersymmetry at the classical level on a resulting D-dimensional Minkowski space. We systematically discuss consistency conditions, the Kaluza-Klein (KK) spectrum and harmonics, and the resulting one-loop effective potential V1, focusing for illustration on maximal supergravity and d = 3, in particular on the T3/ℤ3 and on the Hantzsche-Wendt manifolds. We show how the KK spectrum can be re-organized in multiplets of the broken D-dimensional supersymmetry, derive new universal supertrace mass relations valid at each redefined KK level and obtain an analytic finite expression for V1 after resumming the contributions of all KK levels. In all examples V1 is negative definite and scales with inverse powers of some internal radii. We extensively comment, when applicable, on the relation with the Scherk-Schwarz mechanism and with supersymmetry-breaking string compactifications on freely acting symmetric orbifolds. We also finally clarify the assumptions and constraints under which Scherk-Schwarz reductions correspond to twisted tori compactifications.