Dual non-Abelian duality and the Drinfeld double

Abstract

The standard notion of the non-Abelian duality in string theory is generalized to the class of σ-models admitting 'non-commutative conserved charges'. Such σ-models can be associated with every Lie bialgebra (G,G) and they possess an isometry group iff the commutant [G,G] is not equal toG. Within the enlarged class of the backgrounds the non-Abelian duality is a duality transformation in the proper sense of the word. It exchanges the roles of G andG 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 (G,G). The non-Abelian analogue of the Abelian modular space O(d, d; Z) consists of all maximally isotropic decompositions of the corresponding Drinfeld double.

Poisson-Lie duals of the η deformed symmetric space sigma model

“…The η deformation is conjectured to be related by Poisson-Lie duality, a generalisation of non-abelian duality to sigma models with Poisson-Lie symmetry [31,32], to an analytic continuation of the corresponding λ deformation [16]. A duality of this type was considered in [33], which relates an η s deformation (the superscript denoting it is based on a split solution of the modified classical Yang-Baxter equation) to the corresponding λ deformation.…”

“…Two sigma models are said to be Poisson-Lie dual if they are described by the same set of equations after appropriate non-local field and parameter redefinitions [31,32]. Quantities computed within the framework of one sigma model thus have an equivalent in the dual theory.…”

Target space duality I: general theory

“…The most intricate target space duality discovered thus far is the Poisson-Lie duality of Klimcik and Severa [23,24,25]. In this example we see a very nontrivial geometrical structure playing a central role.…”

Target space duality II: applications

“…The metrics and antisymmetric tensors constructed in this manner correspond to the Poisson-Lie duality of Klimcik and Severa [12]. The explicit duality transformation was obtained by Sfetsos [15].…”

“…To write down the symplectic structure in a convenient way we introduce some notation slightly different than the one given in [12]. Let g ∈ G then the adjoint representation on g is given by…”