Duality and Dilaton

Abstract

We review and elaborate on the issue of the dilaton transformation under the usual $r\to \hat{I}{\pm }^{\prime }/r$ target space duality and its non-static generalization (or $\ddot{I}ƒ$-model duality). It is found that the transformation law $r\to \hat{I}{\pm }^{\prime }/r$, $\ddot{I}•\to \ddot{I}•-\ln (r/\sqrt{\hat{I}{\pm }^{\prime }})$ which guarantees duality at the one-loop $\ddot{I}ƒ$-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.

11 pages