Effective action in multidimensional gauged supergravities and spontaneous compactification

The off-shell one-loop effective action is obtained for d = 5, N = 2 and 8 gauged supergravities on the gravitational background R4 x S1 (R4 and S1 are four-dimensional flat space and the one-dimensional sphere) with a constant scalar field. The structure of the gauge dependence of the effective action is studied. The effective action for d = 4, N = 4 gauged supergravity on the space R3 x S1 with nontrivial topology is also found. The question of the possibility of spontaneous compactification induced by quantum effects in d = 5 gauged supergravities is discussed