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

Abstract

The doubled target space of the fundamental closed string is identified with its phase space and described by an almost para-Hermitian geometry. We explore this setup in the context of group manifolds which admit a maximally isotropic subgroup. This leads to a formulation of the Poisson-Lie σ-model and Poisson-Lie T-duality in terms of para-Hermitian geometry. The emphasis is put on so called half-integrable setups where only one of the Lagrangian subspaces of the doubled space has to be integrable. Using the dressing coset construction in Poisson-Lie T-duality, we extend our construction to more general coset spaces. This allows to explicitly obtain a huge class of para-Hermitian geometries. Each of them is automatically equipped which a generalized frame field, required for consistent generalized Scherk-Schwarz reductions. As examples we present integrable λ-and η-deformations on the three-and two-sphere.

Generalised cosets

“…There are few examples known [9,10,18] but a systematic construction was missing until recently and instead considered on a case by case basis. In generalised geometry a complete construction of generalised parallelisable spaces was worked out in a series of papers [19,20] in which the right coset 3 G\D is identified with the internal manifold M , where D is a…”

‐Corrected Poisson‐Lie T‐Duality

“…Frame fields E A I that satisfy (1), can be constructed systematically on the coset H∖G, if H is a maximally isotropic subgroup of G. [15] Isotropy is defined in terms of an O(D, D) invariant pairing ⟨⋅, ⋅⟩ on . It is equivalent to AB , once an appropriate set of 2D linearly independent generators t A ∈ is chosen.…”

“…under the standard Lie derivative L. It matches the vector part of (1) and therefore it is natural to identify E A i = k A i . To complete the construction, we also need the corresponding one form part [15]…”

Consistent S3 reductions of six-dimensional supergravity

“…This amended potential turns out to support a stable supersymmetric AdS 3 vacuum [15], corresponding to the supersymmetric AdS 3 × S 3 solution of the D = 6 theory. The non-linear Kaluza-Klein Ansätze can be confirmed by direct computation.More recently, new techniques have emerged for a more systematic understanding of consistent truncations within exceptional field theory (ExFT) and generalized geometry [16][17][18][19][20][21], see also [22,23] in the context of double field theory. Using the reformulation of D = 6, N = (1, 0) supergravity as an exceptional field theory based on the group SO(4,4) [24], the non-linear Kaluza-Klein Ansätze from [12,15] can straightforwardly be reproduced from the generalized Scherk-Schwarz twist matrices U in this framework.…”