T-duality/plurality of BTZ black hole metric coupled to two fermionic fields

We ask the question of classical super (non-)Abelian T-duality for BTZ black hole metric coupling to two fermionic fields. Our approach is based on super Poisson-Lie (PL) T-duality in the presence of spectator fields. In order to study the Abelian T-duality of the metric we dualize over the Abelian Lie supergroups of the types $(1|2)$ and $(2|2)$, in such a way that it is shown that both original and dual backgrounds of the models are conformally invariant up to one-loop order in the presence of field strength. Then, we study the non-Abelian T-duality of the BTZ vacuum metric coupling to two fermionic fields. The dualizing is performed on some non-Abelian Lie supergroups of the type $(2|2)$, in such a way that we are dealing with semi-Abelian superdoubles which are non-isomorphic as Lie superalgebras in each of the models. In the non-Abelian T-duality case, it is interesting to mention that the models can be conformally invariant up to one-loop order in both cases of the absence and presence of field strength. In addition, starting from the decomposition of semi-Abelian Drinfeld superdoubles generated by some of the ${\cal C}^3 \oplus {\cal A}_{1,1}$ Lie superbialgebras we study the super PL T-plurality of the BTZ vacuum metric coupled to two fermionic fields. However, our findings are interesting in themselves, but at a constructive level, can prompt many new insights into supergravity and manifestly have interesting mathematical relationships with double field theory.