Extremal AdS Black Holes as Fluids: A Matrix Large-Charge EFT Approach

1: The paper targets the universality relating extremal (zero-temperature), large-charge rotating AdS black holes to rigidly rotating, zero-temperature conformal fluids on the boundary, framed within large-charge EFT

2: The main narrative emphasizes reproducing the zero-temperature conformal fluid energy/angular momentum/charge densities and the boundary stress tensor structure, consistent with the fluid/gravity paradigm.

1-The manuscript argues that other contributions are suppressed by
1/Q at large charge, hence mean-field should work, but provides little quantitative error control or a clear “parameter window” for reliability，especially important if thermodynamic matching is meant to be precise

2-The text gives an intuitive statement that projecting to singlets does not affect the dynamics, but a more careful discussion of constraints/gauge-field effects (particularly at T=0 with macroscopic occupation) would strengthen credibility

This manuscript proposes a “matrix large-charge EFT” to describe rigidly rotating, extremal AdS black holes in the large-charge limit. By promoting the complex scalar of large-charge EFT to an N×N adjoint matrix field and solving a self-consistent mean-field equation, the authors obtain a rotating condensate and compute energy, angular momentum, and charge densities, aiming to match the corresponding zero-temperature conformal-fluid (and fluid/gravity) structures. The idea is simple and appealing, the computations are reproducible, and the paper engages directly with the long-standing puzzle of macroscopic entropy at zero temperature by leveraging O(N2) degrees of freedom.

At the same time, the strength of the conclusions is limited by
(i) the lack of quantitative control over the mean-field approximation
(ii) the fact that achieving the gravitational scaling S∼Q appears to require additional assumptions (e.g. disorder interactions) that are not developed into a closed argument in the current version.

1-Provide a more quantitative parameter-control and error estimate (e.g. which terms are neglected in 1/Q, 1/\mu, and derivative expansions, and how they scale relative to the retained terms). At minimum, clarify to what order the thermodynamic matching is trustworthy.
2-Since the present mean-field model yields too little entropy, explicitly separate what is explained vs. what remains open. If SYK-like disorder is the proposed path to S∼Q, outline a more testable route (even a concrete calculation plan or quantities to compare).

Ask for minor revision

In this article, the author introduces a four-dimensional matrix model that generalizes the large-charge EFT of a complex scalar field, closely following the spirit of Refs. [10] and [27]. The analysis is, however, novel in that the system is studied in a different regime (large angular momentum) and using different techniques (the mean-field approximation). The main result is that the semiclassical behavior coincides with that of a rotating conformal superfluid and is, in turn, consistent with conformal boundary dynamics in holography, as anticipated in Ref. [6].

There are two main issues that I would like the author to address.

1. The author should show explicitly which parameter controls the validity of the mean-field approximation. While there is a hint in footnote 4, it would be much clearer to spell out the precise limit (possibly a double-scaling limit) in which the approximation is justified.

2. The analysis in Ref. [27] is problematic because, as already noted in the original paper and emphasized again here, the large-charge EFT solution has zero entropy at zero temperature and therefore cannot be dual to a finite-size horizon black hole. It is unclear to me whether this tension is resolved in the present work. The entropy computed in this example appears to be temperature-dependent and to vanish as (T \to 0) (see also the comment following Eq. (3.45)), which differs from the explicit black hole example in Eq. (A.3). Does the system proposed here possess a non-zero entropy at zero temperature?

A further, possibly less crucial, issue concerns Section 4 and BPS black holes. In the large-charge literature, it has been observed that BPS-like ground states with (E = Q) can arise in theories with extended supersymmetry (e.g. (\mathcal{N}=2) in four dimensions), and that their behavior is qualitatively different from theories with the more typical ground-state scaling (E \sim Q^{d/(d-1)}). In light of this, it is not clear to me whether the analysis of the previous section can be straightforwardly extended to this case.

Ask for minor revision

In this manuscript, the author promotes the usual large charge and large spin EFT model to a complex $N \times N$ adjoint scalar theory to capture the zero-temperature $O(N^2)$ modes of an AdS black hole. Using the mean-field approximation, the author shows that the energy, charge density and angular momenta of the EFT ground state reproduce the boundary stress tensor of large extremal AdS black holes. The classical thermodynamic behavior of extremal AdS black holes is reproduced by treating the ground state as a rotating, zero-temperature conformal fluid in the $Q \ll J \ll Q^{\frac{d-1}{d-2}}$ limit.

The author addresses the limitations of the EFT model. In particular, the EFT model fails to capture the universal chaotic spectrum and level repulsion of large$-c$ holographic CFTs and hence, there is a large discrepancy in the counting of microscopic degrees of freedom. The author does propose on including potential terms that involve random couplings between matrix modes, which aligns with the study of large$-N$ SYK model on reproducing holographic results in low-dimensional gravity.

The author does list out several interesting directions as future work, including: generalize the whole construction to unequal spins and general dimensions, include finite coupling or higher derivative corrections to Einstein gravity to account for $1/Q,1/N$ expansions, additional fields, excitations around ground state and quantum corrections by including Schwarzian-like modes.

However, here are a few questions/suggestions that would be nice to be discussed in the manuscript:

1. The author did suggest on working out the mean-field approximation of the EFT model for the $d=3$ case in the future. Since the conformal fluid and large-charge, large-spin EFT exhibit universal behavior, are there any interesting insights in relating to Virasoro symmetry for extremal BTZs $(d=2)$? This is because extremal BTZs with one moving-sector being zero-temperature and the other being excited are exactly solvable and are governed by boundary gravitons.

2. Based on the large-charge, large-spin EFT model developed, the correlation functions of the EFT model are expected to reproduce the expected classical behavior of holographic correlators of stress-energy tensors and Green's functions, which can be obtained by studying graviton fluctuations and quasinormal modes in the bulk. What are the restrictions or technical difficulties in establishing this connection?

1 - Include discussion for the $d=2$ case.
2 - Include discussion on the relation to holographic correlators.

Please refer to the report for details.

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