Preserving flat directions during inflation

Supersymmetry is generally broken by the non-vanishing vacuum energy density present during inflation. In supergravity models, such a source of supersymmetry breaking typically makes a contribution to scalar masses of order m̃2 ∼ H2, where H2 ∼ VMp2 is the Hubble parameter during inflation. We show that in supergravity models which possess a Heisenberg symmetry, supersymmetry breaking makes no contribution to scalar masses, leaving supersymmetric flat directions flat at tree-level. One-loop corrections in general lift the flat directions, but naturally give small negative squared masses ∼ −g2H2(4π)2 for all flat directions that do not involve the stop. No-scale supergravity of the SU(N,1) type and the untwisted sectors from orbifold compactifications are special cases of this general set of models. We point out the importance of the preservation of flat directions for baryogenesis.