To $B$ or not to be in single-trace $T\bar T$ holography

Superstring theory on black-strings backgrounds, corresponding to deformed, rotating BTZ black holes, formed by $p$ fundamental strings or negative strings, is inspected. For non-rotating black strings, in the positive case, it was shown in [1] that the excitation energy of a probe long string, plus its contribution to the black-string background, is in harmony with single-trace $T\bar T$ holography, if the $B$-field vanishes at the origin. Here, we investigate the superstring on deformed, rotating BTZ black holes, for both the positive and negative cases, and find that for a particular value of the $B$-field, the harmony with single-trace $T\bar T$ holography persists.