Dual Non-Abelian Duality and the Drinfeld Double

TL;DR

The paper extends non-Abelian duality to the setting of Lie bialgebras via the Drinfeld double $D$, showing the dual backgrounds are related by a canonical, symplectic transformation on reduced phase spaces and providing explicit duality formulas. It identifies the modular space of dual theories with maximally isotropic decompositions of $D$, connects the construction to known Abelian duality as a special case, and discusses consequences for integrability, dressing transformations, and potential exact CFT interpretations. The work lays a rigorous geometric framework for non-Abelian duality, with implications for quantization and connections to quantum groups in string theory.

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.