Dual Non-Abelian Duality and the Drinfeld Double

TL;DR

This paper generalizes non-Abelian duality in string theory to models with non-commutative conserved charges, linking them to Lie bialgebras and Drinfeld doubles, and interprets duality as a symplectomorphism exchanging algebraic structures.

Contribution

It introduces a generalized non-Abelian duality framework for sigma-models associated with Lie bialgebras, providing explicit formulas and interpreting duality as a symplectomorphism.

Findings

Duality exchanges roles of Lie algebra and its dual in the Drinfeld double.

Explicit formulas for non-Abelian duality transformations.

The duality acts as a symplectomorphism on phase spaces.

Abstract

The standard notion of the non-Abelian duality in string theory is generalized to the class of $\si$-models admitting `non-commutative conserved charges'. Such $\si$-models can be associated with every Lie bialgebra $(\cg ,\cgt)$ and they possess an isometry group iff the commutant $[\cgt,\cgt]$ is not equal to $\cgt$. Within the enlarged class of the backgrounds the non-Abelian duality {\it is} a duality transformation in the proper sense of the word. It exchanges the roles of $\cg$ and $\cgt$ and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-Abelian duality transformation for any $(\cg,\cgt)$. The non-Abelian analogue of the Abelian modular space $O(d,d;{\bf Z})$ consists of all maximally isotropic decompositions of the corresponding Drinfeld double.