Warped AdS$_3$ black hole thermodynamics and the charged Cardy formula

1. The referee’s observation is fully consistent with our calculations, and indeed aligns with the broader literature. The additional phase factor appearing in the S-transform of the vacuum character in warped CFTs, explicitly shown in eq.~(2.25), is precisely the same as the one mentioned by the referee. This factor originates from the extra Kac-Moody level, which we first introduced in Section 2.2, independently of the warped CFT context. To highlight this standard interpretation—which we fully agree with—we have added further comments on the S-transform in Sections 2.1, 2.3, and 3.1, and cited additional references ([38–42]) elaborating on this point. The subtlety in the warped case, namely the interchange between spacetime and internal symmetry, is clearly visible in the black hole solution. We have emphasized this aspect in a new paragraph at the top of p.~13. These clarifications are supported by the sharper distinction we now make between the discussion of the asymptotic vacuum and its symmetries on the one hand, and the black hole solutions on the other.

2.–3. We have corrected the typos indicated by the referee, specifically below eqs.~(2.3) and (2.15), where we had omitted the explicit expressions for $\bar h$ and $p$, all of which vanish for the unique vacuum state under our assumption (standard in a holographic theory).

1. We agree with the referee’s comment on the calculation in Appendix A: it is indeed performed via an ad-hoc substitution of quantities such as the central charges. We have not carried out the crucial analysis of the consistency of the boundary conditions, which has been addressed in the newly added reference [57]. In this sense, our gravitational calculation should be regarded only as an indication that an alternative boundary condition might also be consistent. Thus, we see no conflict with [57], given that further work is required before a direct comparison can be made. We have clarified this important point in the last paragraph of Appendix A, stressing that our results there are not conclusive.

2. We have added a new subsection within Section 2.3, entitled “Microcanonical ensemble proposal”. This subsection highlights the point at which our gravitational results suggest a divergent field theory interpretation. We emphasize once more that our grand-canonical results remain holographically indistinguishable from those in the existing literature. Only upon transforming to the microcanonical ensemble does a difference of approach become apparent. In particular, eq.(2.30) differs from its analogue in reference [7], even though both reproduce the gravitational entropy, eq.(4.3), holographically. We have also underscored this observation in the introduction.