Bubbling saddles of the gravitational index

We thank the referee for the interesting comments. In the following we address the points raised in the report, specifying the corresponding changes that we have made in the paper:

1 - “Could the authors please expand on the competition between the different saddles to the gravitational path integral? Is it possible that solutions with different topologies contribute to the same index? If so, what is the expectation based on the action?”

Yes, we expect that solutions with different topologies can in principle contribute to the same index. For a given value of the chemical potentials, the dominant saddle should be determined by comparing their on-shell actions. However, not all possible candidate saddles are expected to contribute to the index, only those satisfying appropriate allowability criteria should compete. Clarifying this likely requires a careful analysis of the integration contour for the path integral. This is a complicated analysis, and goes beyond the scopes of the present work. Therefore, a detailed study of the competition between different topologies should be delayed to future work. We have added a comment on this point in the last paragraph of page 55.

2 - “Some additional minor points a) is the coordinate r_a used in (2.6) defined in (2.7)? b) is the expression (2.9) still locally defined? If so, where? c) the discussion about the "cap" above (2.17) seems a bit confusing, since later on the authors also discuss solutions without "horizon" rods. Should (2.17) be ∀a? d) (2.23) what is r ? e) p. 14 "the Gibbons-Hawking fibre ∂ψ" perhaps would be more properly characterized as something along the lines of "the vector ∂ψ along the Gibbons-Hawking fibre" f) p. 16 and p. 42 it should be "a Euclidean" rather than "an Euclidean”.”

a. Yes, we added a comment to clarify this under (2.6).
b. The one-form is singular on the vertical z-axis. In section 3.2 we showed that this singularity is, however, just a coordinate singularity that can be removed by performing suitable coordinate transformations involving the Euclidean time coordinate. Imposing that the coordinate transformations required to remove the singularity along the whole z-axis are compatible with each other leads to a set of quantization conditions, that we discussed.
c. Yes, Eq. (2.17) (now (2.18)), should be applied to each center, both for “horizon centers” and “bubbling centers”. We added “for each center” to clarify this below (2.18).
d. r is the radial direction in a system of spherical coordinates in R^3 centered at its origin. We added Eq. (2.7) to clarify this.
e. We implemented the suggested change.
f.  We implemented the corrections.

We thank the referee for their positive assessment of our work, and for their suggestions for improvement. Below we detail the changes we have made in the manuscript and provide answers to address the points raised in the report.

• Minor nitpick: the coordinate r (without the subscript "a" or the tilde) is never properly defined prior to (2.23), where it first appears; even if it is not hard to guess what this coordinate represents, it would be better if it was clarified.

We have fixed this by introducing all the sets of spherical coordinates used in the paper in page 9.

• Section 3.2 would be more complete if the case of vanishing f and H was commented, even very briefly. In analogy with lemma 1 of arxiv.org/abs/1712.07092, do the authors expect that the non-vanishing of K^2 + H L is sufficient to ensure that the metric is smooth and invertible? Or are situations where f and H vanish censored by the fact that saddles cap off at the horizon and the h_a coefficient are complex?

It is not clear to us whether critical surfaces where H and f vanish are part of the solutions we study. The answer to this question will certainly depend on the complexification we assume. For instance, in the two-center case analyzed in section 4, one can choose a Euclidean section in which the metric is real, simplifying the regularity analysis. In this case, we show that the positivity of the combination K^2 + H L (which ensures that the metric remains real and positive-definite) is equivalent to requiring the coefficients of the harmonic function H — and thus H itself — to be positive, thereby forbidding the presence of critical surfaces where both H and f vanish. In the more general setting in which the parameters of the harmonic functions can be complex, the presence of such critical surfaces would also require a complexification of the coordinates. Thus, it is not entirely clear to us how the analysis of 1712.07092 should be extended in this context. Nevertheless, since the local form of the solutions is the same as in the Lorentzian case analyzed in 1712.07092, we expect that the non-vanishing of K^2 + H L must be a necessary condition to be imposed so that the metric is finite and invertible at these potential critical surfaces. We have added footnote 9 to mention this issue.

• In sections 4 and 5, the authors discuss horizonless solitonic solutions by starting with saddles with a horizon and taking the limit of beta going to zero, and are thus able to reproduce the physical properties of Lorentzian topological solitons. In section 6, the authors mention that it is not clear whether horizonless solutions are true saddles of the grand-canonical index, given that they have purely imaginary action. The question that I have for the authors is the following: could it be that the purely imaginary action is an artifact of the beta-to-zero limit, and a finite-beta soliton may have an action that is not purely imaginary? Or do the authors have an argument against this possibility?

As far as we understand there is no way we can obtain an action which is not purely imaginary for the topological solitons. Indeed, before assuming a concrete complexification, the on-shell action and chemical potentials of the finite-beta soliton are generically complex. However, such grand-canonical quantities cannot be associated a priori to a physically relevant Lorentzian solution (note that the Lorentzian solution obtained after Wick rotation is not even real for generic complex values of the chemical potentials). As discussed in detail in section 5.4, there are two limits (beta-to-zero and beta-to-infinity) yielding physically sensible Lorentzian solutions, if a concrete complexification, which turns out to be different for each limit, is assumed.  n the case of the beta-to-zero limit, the complexification we need in order to make sense of the Wick-rotated solution implies that the chemical potentials and the action become purely imaginary.

We would like to thank the referee for the careful reading and the useful suggestions. In the following we address the four main remarks in the report.

1 - The usage of the term “finite temperature” is confusing when referring to the periodicity of a Euclidean circle that cannot be Wick rotated to Lorentzian signature. The underlying solutions do not appear to admit a natural interpretation as thermal horizons, since real time is absent. It may be clearer to use terminology such as “fictitious temperature” or simply “finite circle size.”

At page 5, in the third paragraph, we have specified what we mean by finite temperature, emphasizing that the size of the circle, beta, may be complex in addition to being finite.

2 - Is there a meaningful sense in which each class of Euclidean rod structure could be argued to connect smoothly to a Lorentzian solution by contracting the rod or segments of it to a point, or could there be cases where this is obstructed?

This is an interesting question. In the case where there is just one Euclidean horizon rod, it should indeed always be possible to smoothly contract it to a point and in this way obtain a Lorentzian solution. In particular, our eqs. (4.7) and (5.9) show (for the two-center and three-center solutions, respectively) that when the length delta of the horizon rod goes to zero, the inverse temperature beta goes to infinity, meaning that the solution becomes extremal. However, in general a solution may contain multiple horizon rods, in which case we do not have a definite answer. Indeed one should work out the precise map between the length of each of these rods and beta. Studying the bubble equations, which impose consistency conditions between the lengths of the different rods, it should be possible to see whether the different rods can be contracted independently, or only simultaneously. We believe that the second case would give rise to a good Lorentzian solution. Since we feel we do not have a sharp statement to make, we have chosen not to comment on this issue in the text. This may be addressed when we will study the multi-horizon case in detail, which we are planning to do in the future.

3 - The admissibility of Euclidean saddles in the gravitational index is presented as an open question. This is somewhat puzzling, since already in the second paragraph of the introduction the authors state that they have defined the boundary conditions for the index. Given their detailed analysis, one would expect a more definitive statement on admissibility, from a purely mathematical perspective. It would be also useful to discuss what changes in a different ensemble (fixing the asymptotic charges).

The boundary conditions respected by all the solutions we study are indeed those for the index. However, a field configuration extremizing the action and satisfying the assigned boundary conditions (i.e., a candidate saddle) does or does not contribute to the path integral (i.e., it is or is not a true saddle) depending on whether the integration contour passes through it. A precise definition of the integration contour and the determination of the true saddles in the context of the gravitational path integral is a hard question which goes beyond the scope of our work. We have slightly expanded the third paragraph in section 6 emphasizing this issue.

4 - Related to the above, the discussion of a “microscopic evaluation of the index in string theory” is left quite vague. The authors should be able to comment more concretely, in particular on whether the boundary conditions they define are compatible with a string-theoretic setting. In addition, I believe there already exist microscopic calculations for some of the black hole systems under consideration (though perhaps only in the extremal Lorentzian case?). If so, a direct comparison would be very valuable.

The reason why we postpone a concrete comparison with a microscopic evaluation of the index in string theory is that we believe there are some questions to be answered beforehand. The microscopic computations we are aware of rely on indices defined in the near-horizon region of a supersymmetric and extremal solution, which a priori depend on different variables than the index in our paper, which is defined in the asymptotically flat region. A precise map between these two partition functions has not been established yet (some progress has been recently made in arXiv:2501.17909). Since this analysis is quite different from the main focus of our paper, we have decided to postpone it to future work. No changes have been made in the text.

Also, we have replaced “well-definite” with “well-defined” everywhere in the text.