Duality and Dilaton

TL;DR

This paper explores the transformation properties of the dilaton field under target space duality, highlighting modifications needed at higher loop orders and illustrating the concepts with cosmological solutions.

Contribution

It clarifies how dilaton transformations under duality change beyond one-loop and extends the understanding of non-static dualities in cosmological models.

Findings

The standard $r ightarrow rac{ ext{constant}}{r}$ duality law requires modification at two-loop order.

Non-static duality is demonstrated with time-dependent cosmological solutions.

The dilaton transformation law is context-dependent and varies with loop order.

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.