BBBW on the spindle

We thank the referees for their suggestions. We corrected the typos that found by the first referee and implemented the suggestions from the second referee. It follows a detailed list of changes.

Conceptual
- First bullet: We are grateful to the referee for finding the typos in the coefficient of the anomaly polynomial. We corrected them and checked that they give the right results
- Second bullet: We are extremely grateful to the referee for pointing out a potential misleading interpretation of our results. The deep reason why we referred to the analytic solution in (6.1) as the “universal twist” was hidden, in the previous version of the paper, in the discussion after formula (2.30). Indeed, the requirement of rationality of the exact R-symmetry imposes some constraints that boil down to a restricted set of choices for the parameter p,q that are exactly the ones that the referee refers to in the report. In this way, indeed, restricting to the gravity sector on the holographic dual side corresponds to a twist along the exact R-symmetry with the values of p,q, fixed consistently. This is the usual universal twist and as a check we can see that indeed the AdS5 vacuum is recovered in our construction.
- Third bullet: We quoted reference [13] before formula (6.1) in order to clarify the structure of the solution presented in this section. Furthermore, the dependence from the parameter \mathbb{z} of the central charge is now clarified by the comment after formula (6.9) in terms of W_crit, or equivalently R_AdS5 


Exposition
- First bullet: We added a comment after equation (23).
- Second bullet: We corrected the typos pointed out by the referee and adjusted the formulas. The anomaly in R2F was removed as suggested by the referee.
- Third bullet: we choose to differentiate between the two epsilons by adding a subscript 2d,4d which should make clearer the discussion.
- Fourth bullet: As commented on the first bullet of the conceptual errors, there were some typos in the anomaly polynomial which we now have fixed
- Fifth bullet: as pointed out, the a_{4d} was not needed and we removed it.
- Sixth bullet: as required by the referee, we added a comment on the limit. Indeed, in the sphere limit the U(1) symmetry enhances to SU(2) and therefore cannot mix with the R-symmetry anymore. As a consequence the mixing parameter goes to zero, as expected.
- Seventh bullet: we have inserted the missing definitions. In this way the paper should be self-contained, we thank the referee for pointing out this lack of clarity.
- Eighth bullet: We have inserted the proper citation.
- Ninth bullet: We have cited the paper where (6.1) was originally derived.
- Tenth bullet: We have added a reference to the definition of W_{crit} given in an earlier section of the paper
- Eleventh bullet: as pointed out by the referee, the sentence was misleading and we changed it.
- Twelfth bullet: As noted by the referee, we did not compute the finite-N contributions, so we deleted the highlighted sentence.

Typographic
- First bullet: We think that the explicit formulas are better suited rather than a simple comment about the n_S - >-n_S
- Second bullet: The layout was modified to one similar to the field theory result
- Third bullet: We added a comment in the caption of the figure to highlight to which parameters of Table 1 the plots are referenced to.