Duality and Dilaton

Abstract

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

11 pages

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