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

Abstract

Poisson-Lie dualising the η deformation of the G/H symmetric space sigma model with respect to the simple Lie group G is conjectured to give an analytic continuation of the associated λ deformed model. In this paper we investigate when the η deformed model can be dualised with respect to a subgroup G 0 of G. Starting from the first-order action on the complexified group and integrating out the degrees of freedom associated to different subalgebras, we find it is possible to dualise when G 0 is associated to a subDynkin diagram. Additional U 1 factors built from the remaining Cartan generators can also be included. The resulting construction unifies both the Poisson-Lie dual with respect to G and the complete abelian dual of the η deformation in a single framework, with the integrated algebras unimodular in both cases. We speculate that extending these results to the path integral formalism may provide an explanation for why the η deformed AdS 5 × S 5 superstring is not one-loop Weyl invariant, that is the couplings do not solve the equations of type IIB supergravity, yet its complete abelian dual and the λ deformed model are.

Para-Hermitian geometries for Poisson-Lie symmetric σ-models

“…(6.30) from the generalized metric in (3.42). The resulting expressions match nicely with [89] once we remember that the deformation parameter η = e −p .…”

Integrable sigma models and 2-loop RG flow

“…A potentially useful application of our results is to the η-model of [24,25], the one-loop renormalisability of which was studied in [62] (see also [63]). Up to analytic continuation, the η-model and λ-model are related by limits and T-duality [10,28] or by Poisson-Lie duality [10,27,28]. These connections may be used to investigate both the renormalizability of the ηmodel at higher loops and corrections to non-abelian and Poisson-Lie duality.…”

“…The latter generalises the bosonic ηmodel of [24,25]. More precisely, the λ-model and η-model are related by the Poisson-Lie duality [26] (which is a generalisation of non-abelian duality) and a particular analytic continuation [10,27,28]. While both models describe a string propagating in a type II supergravity background [29], much remains to be understood about their structure.…”

Integrable 2d sigma models: Quantum corrections to geometry from RG flow

“…Furthermore, the η-model and the λ-model are, in general, related [58,53,59,60,61] by the Poisson-Lie (PL) duality [62] (and analytic continuation). Therefore, the same observations made above for NAD should apply also to PL duality, which (with suitable quantum corrections) should be a symmetry not only at 1-loop order [63,64], but also at higher loops.…”