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One-loop determinants for black holes in 4d gauged supergravity

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The one-loop determinant for a vector multiplet on H2 x Sigma_g equals (v1 chi_V)^(1/4), and physical hypermultiplets contribute its inverse.

desk verdict Solid vector-multiplet determinants with three internal checks; the hypermultiplet result and the final entropy formula rest on assumptions that need referee attention. read the letter →

arxiv 1908.05696 v2 pith:LAK7EHOZ submitted 2019-08-15 hep-th

Kiril Hristov , Ivano Lodato , Valentin Reys This is my paper · ORCID
classification hep-th
keywords quantumentropyfunctiongaugedsupergravityone-loopdeterminantsupersymmetriclocalizationequivariantindexlogarithmiccorrectionsvectormultipletshypermultiplets
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish the one-loop determinants that control the logarithmic corrections to the Bekenstein-Hawking entropy of supersymmetric black holes in four-dimensional $\mathcal{N}=2$ gauged supergravity. Working on the non-compact near-horizon geometry $\mathbb{H}_2 \times \Sigma_g$, it derives that each abelian vector multiplet contributes $Z_{\text{vec}}^{1\text{-loop}} = (v_1\chi_V)^{1/4}$, each physical hypermultiplet contributes the inverse, and the combined one-loop factor entering the localized quantum entropy function is $(v_1\chi_V)^{\frac{1}{4}(n_V+1-n_H)+a_0}$. Three independent methods---an explicit mode analysis, the equivariant Atiyah-Singer index theorem, and the Atiyah-Bott fixed-point theorem with refinement---give the same determinant. If the claims hold, the entropy expansion is $\log d = \frac{A_H}{4G_N} - \frac{1}{2}(n_V-n_H)\log\frac{A_H}{4G_N} + (1-g)\left(\frac{1}{4}(n_V+1-n_H)+a_0\right)\log\frac{1}{g^2 G_N} + \cdots$, where $a_0$ stands for the not-yet-computed Weyl and Kaluza-Klein multiplet contributions. The result matters because it turns the localization program from reproducing the area law into a concrete prediction for the first subleading correction that holographic counting can test.

What carries the argument

The load-bearing object is the differential operator $D_{10}$ that appears when the localizing supercharge is put in cohomological form: for a vector multiplet it maps the bosonic set $X_0 = \{\sigma, \widetilde{W}_\mu\}$ to the fermionic and ghost set $X_1 = \{c,b,\lambda_{ij}\}$ through the symbol shown in (2.23), and the one-loop determinant is $\sqrt{\det \mathrm{Coker}\,D_{10} / \det \mathrm{Ker}\,D_{10}}$ evaluated on the eigenvalues $2in/\sqrt{v_1}$ of $\widehat{Q}^2$. Three methods fix the equivariant index of $D_{10}$: direct solution of the kernel and cokernel radial ODEs under the paper's boundary conditions, the Atiyah-Singer index theorem applied to the fixed codimension-two locus $\mathbb{S}^2$ (where the symbol reduces to the self-dual complex), and the Atiyah-Bott fixed-point theorem after refining the Hamiltonian so the fixed points are isolated. All three agree on cokernel multiplicities $m^{(1)}_n = 1$ for $n\neq 0$ and $2$ for $n=0$, with the two $n=0$ constant ghost modes discarded, which produces the product $\prod_{n\geq 1}(4n^2/v_1)^{1/2}$ and, after zeta-function regularization, $(v_1\chi_V)^{1/4}$. For physical hypermultiplets the same symbol at the fixed point is the anti-self-dual complex, explaining the inverse determinant.

What would settle it

Solve the radial ODE system (B.14)-(B.16) numerically on a regulated version of H2 x Sigma_g with the stated boundary and smoothness conditions; any normalizable kernel mode with n or l non-zero would change the multiplicities in (2.44) and therefore the v1 power in (2.86). Alternatively, repeat the n=0 analysis with a compact regulator that keeps the constant ghost modes, and check whether ghost-for-ghost zero-modes cancel them exactly.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the one-loop determinant of a single abelian vector multiplet on $\mathbb{H}_2 \times \Sigma_g$, with the paper's normalizable boundary conditions and gauge choice, is exactly $Z_{\text{vec}}^{1\text{-loop}} = (v_1\chi_V)^{1/4}$ for a spherical horizon, generalizing to $(v_1\chi_V)^{\chi(\Sigma_g)/8}$ for a genus-$g$ horizon; the compensating hypermultiplet contributes a trivial factor, while each physical hypermultiplet contributes the inverse $(v_1\chi_V)^{-1/4}$. Assembling these pieces with the classical attractor action, and assuming the localization measure scales as $\Lambda^0$ in the large-charge limit, the localized quantum entropy function gives $\log d = \frac{A_H}{4G_N} - \frac{1}{2}(n_V-n_H)\log\frac{A_H}{4G_N} + (1-g)\left(\frac{1}{4}(n_V+1-n_H)+a_0\right)\log\frac{1}{g^2 G_N} + \cdots$. The coefficient $k_1$ is fixed by the Legendre/Hessian conversion between canonical and microcanonical ensembles, while the newly computed $k_2$ is traced to the one-loop determinants and carries the horizon topology through the factor $(1-g)$; the unknown $a_0$ parametrizes the still-missing Weyl and massive Kaluza-Klein contributions.

Load-bearing premise

The load-bearing premise is that index theorems proven for compact manifolds stay valid on the non-compact H2 x Sigma_g once the paper's boundary and smoothness conditions are imposed, and that the localization measure is scale-free in the large-charge limit.

Editorial extensions

If this is right

  • If the central claim is right, the logarithmic term in the entropy depends on the horizon genus through $(1-g)$: spherical horizons receive the full one-loop contribution, a torus horizon ($g=1$) receives none, and higher-genus horizons flip its sign.
  • The coefficient of the $\log(1/(g^2G_N))$ term is set by the vector/hypermultiplet count ($n_V+1-n_H$) and the undetermined $a_0$, so matter content alone decides whether this correction is positive, negative, or zero.
  • The ensemble-conversion coefficient $k_1 = -\tfrac12(n_V-n_H)$ is independent of horizon topology, while the one-loop coefficient $k_2$ is not, so the two logarithms can be told apart by measuring their dependence on the AdS$_4$ scale versus the horizon area.
  • For higher-genus horizons the vector-multiplet determinant becomes $(v_1\chi_V)^{\chi(\Sigma_g)/8}$, making the one-loop factor a topological quantity through the Euler characteristic.
  • If future work fixes $a_0$ by computing the Weyl multiplet determinant, equation (4.16) becomes a complete macroscopic prediction for the subleading entropy that can be compared directly with the topologically twisted index on the field-theory side.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension beyond the paper: the same vector and hypermultiplet determinants should hold for rigidly supersymmetric four-dimensional $\mathcal{N}=2$ theories placed on $\mathbb{H}_2\times\Sigma_g$, since these multiplets are the basic matter building blocks; a direct rigid localization would test the $\pm 1/4$ exponents independently of gravity.
  • Extension beyond the paper: the fact that the vector and hypermultiplet contributions are inverses suggests a supersymmetric matter content with $n_H = n_V + 1$ would make the full one-loop factor topology-independent; checking whether any holographic dual realizes this spectrum would either sharpen or falsify the cancellation pattern.
  • Extension beyond the paper: the refinement trick used in Section 2.6 points to rotating near-horizon geometries as a regulator; one could test whether the unreffined determinant is independent of the order in which the $q_2$ expansion is taken, which would give a consistency condition on the regularization.
  • Extension beyond the paper: because $k_2$ multiplies $\log(1/(g^2G_N))$ while $k_1$ multiplies $\log(A_H/4G_N)$, a holographic test could separate the two coefficients by computing the entropy at fixed charges while varying the AdS radius, a distinction that does not exist for asymptotically flat black holes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript computes one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact near-horizon geometry H2 x Sigma_g in 4d N=2 gauged supergravity, as required for the quantum entropy function of BPS black holes. For a vector multiplet the determinant is obtained by three independent methods: an explicit mode analysis with the boundary conditions of Section 2.4, an equivariant Atiyah-Singer index computation, and an Atiyah-Bott fixed-point computation. All three give Z_vec = (v1 chi_V)^{1/4} for a spherical horizon and (v1 chi_V)^{chi(Sigma_g)/8} for higher genus. A physical hypermultiplet is argued, from the symbol of its D10 operator and its reduction to the anti-self-dual complex, to contribute the inverse factor, while the compensating hypermultiplet is argued to be trivial. Combining these results, the full one-loop determinant is written as (v1 chi_V)^{1/4(n_V+1-n_H)+a0}, with a0 parametrizing the uncomputed Weyl-multiplet and Kaluza-Klein contributions. The paper then derives logarithmic corrections to the black hole entropy with coefficients k1 = -(n_V - n_H)/2 and k2 = (1-g)(1/4(n_V+1-n_H)+a0), assuming a charge-independent localization measure.

Significance. If the assumptions are justified, this is a substantial step forward: it provides the first systematic one-loop determinant computations on H2 x Sigma_g in gauged supergravity, gives a three-way cross-check for the vector multiplet, and produces concrete logarithmic corrections that can be compared with topologically twisted index results and with 11d supergravity calculations. The separation into k1 and k2 and the topology dependence through chi(Sigma_g) are interesting and physically well motivated. The paper is also commendably transparent about the unresolved a0 and measure issues. However, the central entropy formula is conditional on two unproven inputs: the extension of index theorems from compact to non-compact spaces, and the assumed Lambda^0 scaling of the localization measure. Since the hypermultiplet result lacks an independent mode-level check, the n_H-dependence of the final coefficient is not yet established to the same standard as the vector multiplet result.

major comments (4)
  1. [§2.5 and footnote 2] The central technical assumption appears in footnote 2 and in the paragraph below Eq. (2.55): the Atiyah-Singer index theorem, stated for smooth compact manifolds, is applied to the non-compact space H2 x Sigma_g with the assertion that the boundary and smoothness conditions of Section 2.4.1 make the space effectively compact. No proof, reference, or counterexample is given for this replacement. For the vector multiplet the mode analysis of Section 2.4 provides an independent check of the resulting determinant, so the vector result (2.86) is robust. For the physical hypermultiplet (3.23) and the higher-genus formulas (2.88) and (4.16), there is no independent mode-level check, and a boundary correction to the equivariant index would change the n_H-dependent coefficient in (4.16). This assumption should either be proved or the hypermultiplet result should be derived by a method that does not rely on it.
  2. [§3.2, Eq. (3.23)] Eq. (3.23) is derived entirely from the symbol of D_hyp, whose reduction at eta=0 is identified with the ASD complex. In contrast to Section 2.4, no mode expansion of the hypermultiplet fields with the normalizable boundary conditions (2.29)-(2.33) is presented; the paper states in the last paragraph of Section 3.2 that such an analysis 'should still be possible in principle' but is not given. Since the equivariant index on H2 x Sigma_g is only known under the compactness assumption flagged above, the inverse vector determinant for each physical hypermultiplet is not established to the same standard as the vector result. Because n_H enters k2 in (4.16), this is a load-bearing gap, not a mere presentation issue.
  3. [Appendix B.1, case (n≠0, l≠0)] The asymptotic argument intended to prove kernel emptiness does not close. From (B.31), the requirement that u*(eta) not grow at infinity only forces c1=c3=0; the remaining terms with c2 and c4 decay and are compatible with the normalizable boundary conditions (2.29). The paper does not show that the smoothness conditions at eta=0 force c2=c4=0. The sentence 'A numerical analysis (which we will not present here) hints at the absence...' is not a substitute for a proof. This gap affects the completeness of Method I, though not the final vector determinant, which is independently supported by Methods II and III.
  4. [§4, below (4.12)] Eq. (4.14) and the final formula (4.16) depend on the assumption Z_measure ~ Lambda^0 below (4.12). The paper acknowledges this and notes that a nonzero exponent a_m would add a term to the coefficient of log(1/(g^2 G_N)). The cited results for ungauged supergravity ([5,9]) do not automatically apply to the gauged theory considered here. Since the log-correction formula is the main physics output, this assumption is load-bearing; the entropy formula should be stated as conditional on the measure scaling, or the scaling should be derived.
minor comments (5)
  1. [§2.4.4 and (2.58)] The n=0 constant ghost and anti-ghost modes are discarded because they are not normalizable on the non-compact space. This is plausible, but since the index theorem is being assumed to hold after a boundary-condition replacement, it would be useful to spell out why these modes are not part of the determinant under those boundary conditions and how they are removed in the Atiyah-Singer and Atiyah-Bott treatments.
  2. [§2.7, Eq. (2.81)] The zeta-function regularization step should state the explicit values zeta(0)=-1/2 and -zeta'(0)=1/2 log(2pi) (or cite the convention), since the exponent 1/4 depends on the regularization of sum_{n>=1} 1.
  3. [§2.6, around (2.75)-(2.76)] The replacement sum_{n2>=0} 1 = zeta_R(0) = -1/2 is a regularization choice rather than a topological input; this should be flagged as such, because different q2-expansions with the same zeta-function convention would give different finite pieces.
  4. [§4, Eq. (4.16)] The phrase 'extracted a factor of (1-g) from the unknown coefficient a0' is ambiguous; clarify whether a0 in (4.16) denotes the genus-zero coefficient rescaled by (1-g) or a genus-independent constant.
  5. [§3.2, Eq. (3.20)] The symbol matrix (3.20) uses the notation X^P_0, X^P_1 that was defined for vector multiplets in (2.47); redefine these for hypermultiplets to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the one-loop determinants are derived by explicit mode analysis and index-theorem computations, not by fitting or renaming their inputs.

full rationale

The central claim Z_vec^{1-loop}=(v1 chi_V)^{1/4} follows from solving the kernel/cokernel equations for D_vec^{10} under explicit normalizable boundary conditions in Section 2.4, and the same product is reproduced by the Atiyah-Singer computation (2.55)-(2.60) and the Atiyah-Bott refinement (2.65)-(2.79). The hypermultiplet result Z_hyp=(v1 chi_V)^{-1/4} in (3.23) is obtained from the symbol of D_hyp^{10}, which at eta=0 reduces to the ASD complex, giving the opposite index; this is an index-theoretic derivation rather than an assumed input. The final entropy formula (4.16) contains the explicitly undetermined parameter a0 for the uncomputed Weyl/KK sectors and assumes Z_measure~Lambda^0 with references to [5,9]; neither a0 nor the measure assumption is a fitted quantity disguised as a prediction. Self-citations to [1] supply the localization locus and classical action, but the one-loop determinant is not read off from [1]; it is derived independently and cross-checked by three methods. The acknowledged replacement of compactness by boundary conditions below (2.55) is an unresolved correctness assumption for the hypermultiplet and higher-genus sectors, but it is a mathematical gap, not circular reduction of outputs to inputs.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The determinant computations rely on index theorems on a non-compact space, on prior localization results from [1], and on the chosen reality conditions. The final entropy formula additionally assumes a scale-independent localization measure and parametrizes the uncomputed gravity and Kaluza-Klein contributions by a0.

free parameters (2)
  • a0 = unknown (placeholder)
    Introduced in Section 4 (Eq. (4.1) and below) to parametrize the uncomputed one-loop contribution of the Weyl and Kaluza-Klein multiplets. The final log coefficient k2 depends linearly on it, and no computation or bound is given.
  • am = 0 (assumed)
    The scaling exponent of the localization measure Zmeasure is assumed to vanish in the large-charge limit (Section 4 below Eq. (4.12)). The paper notes that a nonzero am would add an undetermined term to the coefficient of log(1/(g^2 G_N)).
assumptions (8)
  • ad hoc to paper Atiyah-Singer and Atiyah-Bott index theorems extend to the non-compact space H2 x Sigma_g once the boundary and smoothness conditions of Section 2.4.1 are imposed.
    Explicitly assumed in footnote 2 and Section 2.5 (below Eq. (2.55)) and Section 2.6. The theorems are stated for compact manifolds.
  • domain assumption The localization locus, classical action, Q^2 algebra, and Killing spinors derived in reference [1] are correct.
    Section 1.1 and Appendix A import these results from the authors' previous paper [1803.05920] without re-derivation.
  • ad hoc to paper The localization measure Zmeasure(phi+, phi-) scales as Lambda^0 in the large-charge limit (am = 0).
    Assumed below Eq. (4.12). Used to convert the determinant scaling into the final logarithmic coefficient.
  • ad hoc to paper Uncomputed Weyl and Kaluza-Klein multiplet contributions are summarized by a single coefficient a0 that multiplies the same (v1 chi_V) factor and, for genus g, carries a prefactor (1-g).
    Stated in Section 4 around (4.1) and (4.16). This is a parametrization, not a derivation.
  • domain assumption Physical hypermultiplets impose n_H delta-function constraints on the localized integral for general n_H.
    Expected in Section 4 before Eq. (4.16): the paper says it expects the BPS conditions to produce n_H additional delta-function constraints. This enters the k1 coefficient.
  • ad hoc to paper The rotated reality conditions for vector multiplet fields (2.13) and the real versus pseudo-imaginary contours for hypermultiplet scalars (3.12) and (3.11) are the correct integration contours for the path integral.
    Stated in Sections 2.3 and 3.1-3.2. These choices are taken to match the contour used in [1] and determine whether the compensating hypermultiplet gives a trivial determinant while physical hypermultiplets give the inverse.
  • ad hoc to paper The n = 0 constant ghost and anti-ghost zero modes are discarded from the one-loop determinant.
    Discussed in Section 2.4.4 below Eq. (2.44) and in Section 2.5 after Eq. (2.58). If these modes contributed, the determinant would change.
  • standard math Zeta-function regularization gives finite values to the divergent products used, including sum_{n>=1} n^0 = -1/2.
    Used in Section 2.7 and footnote 9. It is a standard regularization choice.

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Pith. "Pith review of One-loop determinants for black holes in 4d gauged supergravity." pith.science (2026). https://pith.science/paper/LAK7EHOZ

@misc{pith2026190805696,
  author       = {Pith},
  title        = {Pith review of: One-loop determinants for black holes in 4d gauged supergravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAK7EHOZ}},
  note         = {Machine review of arXiv:1908.05696}
}
abstract

We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d ${\cal N} = 2$ gauged supergravity, initiated in [1803.05920]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space $\mathbb{H}_2 \times \Sigma_{\rm g}$ with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.

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Forward citations

Cited by 1 Pith paper

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