On String theory on deformed BTZ and $T\bar{T}+J\bar{T}+T\bar{J}$

Aspects of superstring theory on black-strings backgrounds, corresponding to deformed BTZ black holes, formed near $k$ NS5 branes by $p$ fundamental strings, and single-trace $T\bar{T}+J\bar{T}+T\bar{J}$ holography, are presented. It is shown that for a particular asymptotic value of the $B$-field, the excitation energy of a long string plus its contribution to the energy of the black hole is that in $T\bar{T}+J\bar{T}+T\bar{J}$ deformed $CF{T}_2$ with $c=6k$. The excitation energy of a winding $w>1$ long string plus its contribution to the background, a $w\mathrm{/}p$ fraction of the black-hole energy, evolves according to that in a $Z_w$ twisted sector of a $p$-fold symmetric product of the latter.