Effective Actions for σ‐Models of Poisson–Lie Type

Abstract

(Quasi‐)Poisson–Lie T‐duality of string effective actions is described in the framework of generalized geometry of Courant algebroids. The approach is based on a generalization of Riemannian geometry in the context of Courant algebroids, including a proper version of a Levi‐Civita connection. In our approach, the dilaton field is encoded in a Levi‐Civita connection and its form is determined by the Courant algebroid geometry. Explicit examples of background solutions are provided using the approach developed in the paper.

Hitchhiker's Guide to Courant Algebroid Relations

“…gives (43). Conversely, using (43) on local coordinate functions of M 2 around ϕ(m 1 ) proves (45) for the basis, hence for every α 2 by linearity. This finishes the proof.…”

“…Example 4.31 (Poisson-Lie T-duality). The Courant algebroid morphism of this subsection is central for the geometrical interpretation of Poisson-Lie T-duality [12,15] and also the related work [44,45]. One adds the following requirements:…”

“…In particular, if q = 0, to a given Courant algebroid connection ∇ ′ on E ′ , there is a unique ∇ 0 on E ′ 0 such that ∇ 0 ∼ R(H) ∇ ′ . This is crucial for the compatibility of supergravity with Poisson-Lie T-duality, see [45,15].…”

Poisson-Lie T-duality of WZW model via current algebra deformation

“…The new generators span the affine algebra associated with the Lie algebra sb(2, C), as can be easily checked by computing their Poisson brackets, which read as: 46) with f ij k = ij 3k the structure constants of sb(2, C) and…”

“…The latter represents a genuine generalisation, since it does not require isometries at all, while Abelian and non-Abelian cases can be obtained as particular instances. Recent results on Poisson-Lie T-duality and its relation with para-Hermitian geometry and integrability, as well as low-energy descriptions, can be found in [39][40][41][42][43][44][45][46][47][48].…”