To B or not to be in single-trace TT¯ holography

Superstring theory on black-strings backgrounds, corresponding to deformed, rotating BTZ black holes, formed near k NS5 branes 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, 1/p fraction of the black-string energy, is the same as that in TT¯ deformed CFT₂ with c=6k, if the B-field vanishes at the origin. This supports the claim that the black string can be thought of as a state in a symmetric product of p CFT₂'s with central charge c=6k, where the black-string energy is split equally among all p factors. In [1], it was not verified that this property persists in other cases. Here, we generalize the investigation of [1] to the superstring on deformed, rotating BTZ black holes, for both the positive and negative cases. We find the particular value of the B-field at the origin, for which the total energy of a probe long string, consisting of its excitation energy plus its contribution to the 1/p fraction of the black-string energy, is the same as that in TT¯ deformed CFT₂ with c=6k. Interestingly, the B-field required for this harmony with single-trace TT¯ holography, does not vanish at the origin in the rotating cases.