A Problem with Non-Abelian Duality

TL;DR

This paper explores duality transformations in non-Abelian symmetric backgrounds, revealing limitations in generating new conformal models through dilaton transformations and suggesting potential solutions.

Contribution

It generalizes duality transformations to non-Abelian isometries in cosmological models and analyzes their impact on conformal invariance.

Findings

No new conformal backgrounds from dilaton transformations in the studied models

Generalized duality transformations for non-Abelian isometries derived

Identifies challenges in extending Abelian duality results to non-Abelian cases

Abstract

We investigate duality transformations in a class of backgrounds with non-Abelian isometries, i.e. Bianchi-type (homogeneous) cosmologies in arbitrary dimensions. Simple duality transformations for the metric and the antisymmetric tensor field, generalizing those known from the Abelian isometry (Bianchi I) case, are obtained using either a Lagrangian or a Hamiltonian approach. Applying these prescriptions to a specific conformally invariant $\s$-model, we show that no dilaton transformation leads to a new conformal background. Some possible ways out of the problem are suggested.