Poisson-Lie T-duality and loop groups of Drinfeld doubles

Abstract

A duality invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both of the pair of σ-models related by Poisson-Lie T-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant σ-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shansky symplectic form on the Drinfeld double and it gives directly a generating function of the canonical transformation relating the σ-models from the dual pair.

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

“…Starting from the model on the Drinfel'd double D [37,38] with D ≡ G C we identified a class of subgroups G 0 ⊂ G with respect to which it is possible to dualise. The corresponding Lie algebras g 0 = Lie(G 0 ) are constructed by picking a subset of the Cartan generators of G C along with the associated roots.…”

“…In order to answer these questions we work with the first-order action on a Drinfel'd double D [37,38] and its generalisation to coset spaces [42,43]. As a vector space, the Lie algebra of the Drinfel'd double d = Lie(D) can be decomposed as…”

“…We start from the first-order action on the Drinfel'd double [10,37,38,41]. In the standard decomposition of the Drinfel'd double d = g +g, both g andg are Lie algebras.…”

“…The first-order action on the Drinfel'd double that underlies the Poisson-Lie duality is [37,38] 20) where the dynamical field l ∈ D and…”

Untitled

“…Eq. (8) can be explicitely solved in that case [6,11] for the matrix E ij . If G is connected and simply connected, any cocycle on G (the Lie algebra of G) with values in G ⊗ G can be integrated to a cocycle in G with values in G ⊗ G. Given the cocommutator δ K on G, we denote by Π R : G → G ⊗ G the corresponding cocycle in G. It satisfies the relation…”