Duality and Dilaton

Abstract

We review and elaborate on the issue of the dilaton transformation under the usual $r \rightarrow \alpha'/r$ target space duality and its non-static generalization (or $\sigma$-model duality). It is found that the transformation law $r \rightarrow \alpha'/r$, $\phi \rightarrow \phi - \ln(r/\sqrt{\alpha'})$ 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.