Supersymmetry-breaking compactifications on Riemann-flat manifolds

A bstract 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 V 1 , focusing for illustration on maximal supergravity and d = 3, in particular on the 𝕋 3 /ℤ 3 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 V 1 after resumming the contributions of all KK levels. In all examples V 1 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.