D-branes in $Ad{S}_2\times H^2\times H^2$ under the non-Abelian T-duality

Abstract

We proceed to construct a non-Abelian dual pair for the $Ad{S}_2\times H^2\times H^2$ background by applying the non-Abelian T-duality (here as Poisson-Lie T-duality on a semi-Abelian double). By using a certain parameterization of the $6$-dimensional Lie group $A_2\otimes A_2\otimes A_2$ we construct the original $\sigma$-model including the $Ad{S}_2\times H^2\times H^2$ metric in the absence of $B$-field. It is shown that the dual background constructed by means of the Poisson-Lie T-duality is supported by a $B$-field and a metric whose contains a physical singularity. By studying the behavior of the dual spacetime at small $(r,y,u)$ coordinates, we show that the dual metric with a zero field strength and a non-trivial dilaton field make up a solution for the vanishing of the one-loop beta-function equations. Furthermore, at large $(r,y,u)$, it is shown that the $Ad{S}_2\times H^2\times H^2$ solution is preserved under the non-Abelian T-duality. Finally, using the duality map obtained from the canonical transformation description of the Poisson-Lie T-duality for the gluing matrix which locally defines the properties of the D-brane, we find seven different cases of the gluing matrices for the $Ad{S}_2\times H^2\times H^2$ $\sigma$-model and its dual pair. In this way, it is found a symmetric duality action on the branes linking together in a duality chain.

23 pages