Hyperbolic compactification of M-theory and de Sitter quantum gravity

We thank the referee for the careful analysis of our work and comments. We have addressed the points in the new version, and here we reply explicitly to each point.

First let us make a brief comment on what is new here compared to previous negative-curvature compactifications, related to the referee's remark that ``the main new ingredient is the Casimir energy supported near small closed geodesics.''

The negative contribution (a role now played by Casimir energy) is indeed crucial and new. But we would like to stress that the role of rigidity of generic negatively curved spaces -- in addition to their positive contribution to the potential energy useful for dS and for moduli stabilization -- is also new to this class of examples, as is the direct use of the stabilizing warp factor dynamics. (Another simplification arises from the 11d SUGRA starting point with no dilaton.)

Regarding point 1), we have highlighted in a new Sec. 4.1 the explanation of how to obtain $a \ll 1$; see also the Appendix A. The parameters enabling this tuning are generically available in the setup of our work.

2) One useful toy example of the effects of the rigidity is the following. Consider first a partial cusp, \begin{equation} dy^2 + e^{-2y/\ell} d\vec x^2, ~~~~ 0\le y \le y_c \end{equation} with $\vec x$ living on a torus. This space has moduli (the moduli of the torus which preserve the volume of the space). However, a smooth finite-volume hyperbolic manifold has no such moduli. In the standard construction of such spaces as gluings of polygons, the cusps are connected to the bulk of the space along a totally geodesic submanifold, a particular region of $\mathbb{H}_{n-1}$ (as in figures 8, 10, 11), and there are no continuous parameters available in this construction; moreover, the Hessian is fully positive (see e.g. the Besse reference). As a result, the would-be moduli in the cross-sectional shape of the torus are lifted: the boundary conditions disallow those modes, and the allowed modes have positive gradient energy. This effect extends to our backreacted cusp in the sense that our numerical results in the near-cusp region have not revealed a tachyonic mode, and gluing this to the bulk will tend to increase all masses due to rigidity effects.

Although the ingredients in our model and the available tuning parameters are explicit, constructing a more detailed solution for the internal fields will require, as we have explained in the main text, numerically solving PDEs in 7 variables. Indeed, the stabilizing rigidity effects come from gluing the near-cusps region with the bulk via totally geodesic faces that bring in dependence in all internal variables. A numerical approach based on neural networks is currently under investigation (and briefly presented in Sec. 8).

At present we do not have a complete proof of the second order stability of our construction (and we have not claimed this), however, we expect on the basis of the extensive results summarized in section 5.4.2  that future work will establish this.

3) This is an important question and related to point 1) above. As we explain in the material now better highlighted in the newly created Sec. 4.1, given the wide availability of geometric and flux parameters, we present explicit discrete moves that are available among these parameters, enabling a tuning of our $a$ parameter. This, along with small tadpoles estimated in section 5, indicates that it is very likely that it is possible to parametrically control all the length scales in the full construction. Another hint for this parametric control comes from the parameter $c$ that acts on the bakcreacted cusp solution presented in Sec.~5.3. This acts both on the cusp geometry and on the flux by rescaling the length scales and the flux quanta in a way compatible with the general parametric estimates in Eq. (2.7)-(2.9) - as is also discussed below Eq.~(5.43) in the context of the auxiliary Schr\"odinger problem. This parameter of the local solution will ultimately be related to the available parameters of the core manifold to which the cusp region is glued.

We thank the referee for his careful and encouraging report. In the new version we have corrected the typos found by the referee. We have also addressed the bullet points as follows.

Regarding the first point on $B_\star$, we should stress that the parameter that controls it, $y_\star$, is a variational parameter. It should be chosen in order to minimize the wavefunction Hamiltonian (which maximizes the effective potential). In this sense there is no tuning here; finding the variational wavefunction that gives the largest effective potential puts the strongest bound on the exact solution. We have emphasized this in the new footnote 13.

Regarding the second bullet point, we stress two aspects. First, in order to avoid potential confusions and emphasize that our mechanism uses a large $K/a$, we have introduced the parameter $a$ from the beginning in Sec. 2. We refer there to the later Sec. 4 where we argue that it is possible to obtain $a\ll1$ with explicit discrete parameters in our construction. This decreases the relative strength of the curvature contribution, and allows the Casimir energy to compete without requiring sub-Planckian Casimir circles. This can be also confirmed in the backreacted cusp solution presented in Sec. 5.3. As explained around equation (5.68), it is possible to obtain a competitive Casimir energy with a large Casimir circle in Planck units, $\hat{R}_c \gg 1$. Since the geometry is smooth, higher-derivative curvature invariants are suppressed.

Our treatment of Axion Monodromy indeed leaves out the exit dynamics, as noted by the sentence `One aspect that requires further study is the exit from inflation.' We agree with the referee that this is a very interesting question, and that as they say it may be possible to incorporate additional couplings. One point to note intrinsic to the existing setup is that the axion $\int_{\Sigma_I} C_3$ and 4-form flux $\int_{\Sigma_4}F_4$ may be supported on cycles that are localized compared to the volume form. Although topologically $C_3\wedge F_4$ and $F_7$ are the same (deformable into each other since there is only one element of 7-cohomology), it is not clear to us that there isn't an energetic barrier to fully cancelling the $F_7$ using the $C_3\wedge F_4$ term -- deforming one into the other may require activating KK modes that cost energy. Be this as it may, we are not prepared to present a complete exit mechanism. So although we completely agree with the referee that this is important, we would like to defer this to future work (as indicated in the paper) to be able to give it the attention it deserves.

We have added in the conclusion section some comments on possible research directions for obtaining a realistic matter content. Since many of the existing constructions of this in other classes of compactifications are local, it appears possible to incorporate them here as well as potentially new mechanisms related to the geometry of hyperbolic manifolds and orbifolds. Not having delved into this in great detail yet, we still leave it as a future direction in this paper.