Duality and Dilaton

We review and elaborate on the issue of the dilaton transformation under the usual $r\to {\alpha }^{\mathrm{\prime }}\mathrm{/}r$ target space duality and its non-static generalization (or $\sigma$-model duality). It is found that the transformation law $r\to {\alpha }^{\mathrm{\prime }}\mathrm{/}r$, $\phi \to \phi -\ln (r\mathrm{/}\sqrt{{\alpha }^{\mathrm{\prime }}})$ which guarantees duality at the one-loop $\sigma$-model level should be modified at two (and higher) loop order. The non-static duality is illustrated on the example of cosmological solutions in $D\ge 2$ with time-dependent radii of space torus.