A d S 3 × S 3 $AdS_3 \times S^3$ Background From Poisson–Lie T-Duality

Abstract

The author proceed to construct a dual pair for the $Ad{S}_3\times S^3$ background by applying non-Abelian T-duality (here as Poisson–Lie [PL] T-duality on a semi-Abelian double). By using a certain parametrization of the 4-dimensional Lie group $A_2\otimes 2A_1$ and by a suitable choice of spectator-dependent matrices the original $\sigma$-model including the $Ad{S}_3\times S^3$ metric and a non-trivial $B$-field are constructed. The dual background constructed by means of the PL T-duality with the spectators is an asymptotically flat one with a potential black hole interpretation supported by a non-trivial $H$-flux whose metric contains the true singularity with a single horizon. The question of classical integrability of the non-Abelian T-dual $\sigma$-models under consideration is addressed, and their corresponding Lax pairs are found, depending on some spectral parameters. Finally, the conformal invariance conditions of the models are checked up to two-loop order, and it has been concluded that the resulting model is indeed a solution of supergravity.