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

TL;DR

This paper explores Poisson-Lie duals of eta deformed symmetric space sigma models, revealing conditions for dualisation and unifying duality frameworks, with implications for superstring theory and supergravity.

Contribution

It identifies when eta models can be dualised with respect to subgroups and unifies Poisson-Lie and abelian dualities within a single framework.

Findings

Dualisation possible for subgroups associated with sub-Dynkin diagrams.

Inclusion of U_1 factors from remaining Cartan generators.

Unified framework for Poisson-Lie and abelian dualities.

Abstract

Poisson-Lie dualising the eta 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 lambda deformed model. In this paper we investigate when the eta 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 sub-Dynkin 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 eta 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 eta deformed AdS_5 x 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 lambda deformed model are.