Duality and dilation

This paper reviews and elaborates on the issue of the dilaton transformation under the usual τ → α'/τ target space duality and its non-static generalization (or σ-model duality). It is found that the transformation law τ → α'/τ, φ → φ In (τ/α') which guarantees duality at the one-loop σ-model level should be modified at two (and higher) loop order. The non-static duality is illustrated on the example of cosmological solutions in D ≥ 2 with time-dependent radii of space torus