Dressing cosets

Abstract

The account of the Poisson-Lie T-duality is presented for the case when the action of the duality group on a target is not free. At the same time a generalization of the picture is given when the duality group does not even act on σ-model targets but only on their phase spaces. The outcome is a huge class of dualizable targets generically having no local isometries or Poisson-Lie symmetries whatsoever.

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

“…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…”

“…The backgrounds for cases (i) and (ii) were constructed in [36,42] for a matrix E 0 depending on two parameters. In [16] it was shown that in the special case 57) in the ǫ → 0 limit, these backgrounds correspond to those of the η and λ * deformations respectively.…”

Quantum equivalence ofσmodels related by non-Abelian duality transformations

“…The interest of the O(3) sigma model and its non Abelian dual stems from the fact that the fields of the former parametrize a coset space while usually non Abelian duality is formulated for sigma models defined on group manifolds; see however ref. [25] for Poisson Lie duality in case of cosets.…”

“…The problem with this is that the generating functional, (17), is independent of g, yet for g = −1 the ψ field of the O(3) and the α field of the dual models decouple. In the Hamiltonian formalism, these decouplings can be handled in general by the procedure described in [25]. In the present case, the canonical transformation connecting these two models can be described by the following generating functional…”

“…(This particular non Abelian dual is a special case of the coset examples discussed in [25]). The only manifest symmetry of the dual Lagrangian, Eq.…”