Non-Abelian T-duality of $AdS_{d\le3}$ families by Poisson-Lie T-duality

Abstract

We proceed to investigate the non-Abelian T-duality of AdS 2 , AdS 2 × S 1 and AdS 3 physical backgrounds, as well as two-sphere S 2 from the point of view of Poisson-Lie (PL) T-duality. To this end, we reconstruct these metrics of the AdS families as backgrounds of non-linear σ-models on two-and three-dimensional Lie groups. By considering the Killing vectors of these metrics and by taking into account the fact that the subgroups of isometry Lie group of the metrics can be taken as one of the subgroups of the Drinfeld double (with Abelian duals) we look up the PL T-duality. To construct the dualizable metrics by the PL T-duality we find all subalgebras of Killing vectors that generate subgroup of isometries which acts freely and transitively on the AdS 2 , S 2 , AdS 2 × S 1 and AdS 3 manifolds. We then obtain the dual backgrounds for these families of AdS in such a way that we apply the usual rules of PL T-duality without further corrections. Meanwhile, we have shown that all original backgrounds (AdS families) are conformally invariant up to two-loop order, while this is not the case for dual solutions. Finally, by using the T-duality rules proposed by Kaloper and Meissner (KM) we calculate the Abelian T-duals of BTZ black hole up to two-loop by dualizing on the coordinates ϕ and t. In this way, only the corresponding dual metrics and dilaton fields receive α ′ -corrections. When the dualizing is implemented by the shift of direction ϕ, we show that the horizons and singularity of the dual spacetime are the same as in charged black string derived by Horne and Horowitz without α ′ -corrections, whereas in dualizing on the coordinate t we find a new three-dimensional black string whose structure and asymptotic nature are clearly determined. For this case, we show that the Abelian T-duality transformation of KM changes the asymptotic behavior from AdS 3 to flat.

Poisson-Lie T-plurality for dressing cosets

“…Other examples are the PL T -duality of coset spaces, AdS2 and S 2 . Their PL T -dualities are discussed in this paper by using the general framework, but a recent paper [12] found that some two-dimensional group acts freely and transitively on a certain coordinate patch of AdS2 or S 2 . Then, we can perform the standard PL T -duality with respect to the two-dimensional isometry group.…”

“…It will be instructive to begin by considering a simple case where both (k â b ) = kā b , kib and ( kâ b ) = kā b , kib span the linear subspace g . 12 In this case, there is a matrix R ab (which gives a map g → g), that realizes 13…”