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arxiv: 2606.01167 · v2 · pith:SVV5Z2NFnew · submitted 2026-05-31 · ✦ hep-th

One-loop divergences for KK theories on AdStimes S spaces; a reanalysis of AdS₄ times S⁷\,big/ ABJM precision holography

Federico Arrighi , Lorenzo Casarin This is my paper

Pith reviewed 2026-06-28 16:50 UTC · model grok-4.3

classification ✦ hep-th
keywords one-loop divergencesKaluza-Klein reductionAdS x S spacesABJM theorysupergravityzeta-function regularizationholographic free energyghost modes
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The pith

The logarithmic divergence in one-loop 11d supergravity on AdS4 x S7 arises solely from the 2-form ghost mode after Kaluza-Klein reduction.

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

The paper develops a method to extract the logarithmically divergent piece of one-loop determinants on AdS times sphere spaces by expanding higher-dimensional operators in spherical harmonics and converting the problem into an infinite sum of lower-dimensional AdS determinants. These are then handled with zeta-function techniques, with explicit subtraction of sphere zero-mode contributions. When applied to the full 11d supergravity multiplet on AdS4 x S7, the calculation shows that every other mode cancels and only the 2-form in the ghost sector survives, reproducing the 1/4 log N term in the ABJM free energy that supersymmetric localization had already predicted. A reader cares because the result supplies an independent check, performed entirely in four-dimensional language, that the holographic dictionary for this theory is consistent at the one-loop level.

Core claim

By expanding the kinetic operators of 11d supergravity fields in S7 spherical harmonics, the (11-dimensional) spectral problem reduces to an infinite tower of 4d AdS4 determinants whose logarithmic divergences are isolated via zeta-function regularization; after the Kaluza-Klein sum and zero-mode subtraction, the only surviving log divergence comes from the 2-form AdS mode in the ghost sector, yielding exactly the 1/4 log N correction to the ABJM free energy.

What carries the argument

Spherical-harmonic expansion of higher-dimensional operators that converts the AdS x S spectral problem into a sum of AdS determinants, followed by zeta-function extraction of the log divergence with explicit zero-mode handling.

If this is right

  • The same reduction applies to one-loop determinants on AdS_dA x S_dS spaces of any dimension.
  • All bosonic and fermionic modes in the 11d supergravity multiplet on AdS4 x S7 cancel except the indicated ghost 2-form.
  • The resulting 1/4 log N matches the value obtained by supersymmetric localization on the boundary.
  • The framework supplies a four-dimensional language for precision holography checks that avoids direct eleven-dimensional regularization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique could be applied to other compactifications such as AdS5 x S5 to check one-loop corrections in N=4 SYM.
  • If the zero-mode subtraction procedure remains consistent at two loops, the method might constrain higher-order terms in the holographic free energy.
  • The isolation of the divergence to a single ghost mode suggests that gauge-fixing choices in the bulk may control the entire logarithmic piece in related theories.

Load-bearing premise

The zero-mode contributions on the sphere can be isolated as extra AdS determinants and subtracted after the Kaluza-Klein sum without leaving behind scheme-dependent pieces or missing finite terms.

What would settle it

An explicit computation of the full Kaluza-Klein sum that produces a non-zero logarithmic divergence from any mode other than the 2-form ghost, or that leaves a residual scheme-dependent log term after zero-mode subtraction.

read the original abstract

We provide a systematic framework for computing the logarithmically divergent part of one-loop partition functions on product spaces $\mathrm{AdS}_{d_A} \times S^{d_S}$ of arbitrary dimension. By expanding the higher-dimensional kinetic operators in spherical harmonics, we reduce the ($d_A+d_S$)-dimensional spectral problem to an infinite tower of $d_A$-dimensional determinants, which are then represented via spectral $\zeta$-function methods. We isolate the logarithmic divergences arising from the interplay between the individual AdS determinants and the infinite Kaluza-Klein sum, carefully accounting for the contributions of zero modes on the sphere that produce additional AdS determinants. We test this framework on different fields and apply it to the complete multiplet of 11-dimensional supergravity on $\mathrm{AdS}_4 \times S^7$. We recover in a 4d language the result of arXiv:1210.6057, namely that the only non-vanishing logarithmic divergence originates entirely from the 2-form AdS mode in the ghost sector, reproducing the well-known $\frac{1}{4}\log N$ correction to the ABJM free energy predicted by supersymmetric localization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a systematic framework for extracting the logarithmically divergent part of one-loop partition functions on AdS_{d_A} × S^{d_S} spaces. It expands higher-dimensional operators in S^{d_S} harmonics to reduce the problem to an infinite tower of AdS_{d_A} determinants, evaluates these via spectral zeta functions, and subtracts zero-mode contributions on the sphere. Applied to the complete 11d supergravity multiplet on AdS_4 × S^7, the paper concludes that the only surviving logarithmic divergence arises from the 2-form mode in the ghost sector, reproducing the 1/4 log N correction to the ABJM free energy.

Significance. If the zero-mode handling is robust, the work supplies a concrete 4d language for one-loop KK computations on AdS × S backgrounds and provides an independent check of the supersymmetric localization result for ABJM. The explicit reduction to AdS determinants and the isolation of a single contributing mode are potentially reusable for other precision holography calculations.

major comments (2)
  1. [§4.2] §4.2 (zero-mode subtraction procedure): the claim that additional AdS determinants from sphere zero modes can be subtracted without residual scheme-dependent log terms or finite pieces after the KK sum is load-bearing for the central assertion that every multiplet except the ghost 2-form cancels. No explicit check is given that the subtraction commutes with the log-extraction step for different regulators (e.g., hard cutoff on KK level versus analytic continuation of the spectral zeta).
  2. [§5.3] §5.3 (application to 11d SUGRA multiplet): while the final coefficient is stated to match 1/4, the manuscript supplies no tabulated breakdown of the log-divergent contributions from each field in the multiplet (or even a summary of which cancel). This omission makes it impossible to verify that the cancellation is complete and that the ghost 2-form alone produces the quoted result.
minor comments (2)
  1. [Abstract] The abstract states the framework applies to 'arbitrary dimension' yet the only concrete application is d_A=4, d_S=7; a brief remark on the obstacles for other (d_A, d_S) pairs would clarify the scope.
  2. [§3] Notation for the AdS determinants after KK reduction (e.g., the precise definition of the shifted mass parameters m_{k,ℓ}) is introduced without a compact summary table, forcing the reader to hunt through multiple equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments, which help clarify the robustness of our framework. We address the two major points below and will incorporate revisions to improve transparency and verification.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (zero-mode subtraction procedure): the claim that additional AdS determinants from sphere zero modes can be subtracted without residual scheme-dependent log terms or finite pieces after the KK sum is load-bearing for the central assertion that every multiplet except the ghost 2-form cancels. No explicit check is given that the subtraction commutes with the log-extraction step for different regulators (e.g., hard cutoff on KK level versus analytic continuation of the spectral zeta).

    Authors: We agree that an explicit cross-check between regulators would strengthen the central claim. While the zeta-function approach is analytic and the logarithmic terms are expected to be regulator-independent once zero modes are properly subtracted, we will add a short appendix in the revised manuscript that performs the comparison with a hard KK cutoff (truncated at large level and extrapolated). This will explicitly confirm that no additional scheme-dependent logarithms survive after subtraction and that the procedure commutes with the log extraction. revision: yes

  2. Referee: [§5.3] §5.3 (application to 11d SUGRA multiplet): while the final coefficient is stated to match 1/4, the manuscript supplies no tabulated breakdown of the log-divergent contributions from each field in the multiplet (or even a summary of which cancel). This omission makes it impossible to verify that the cancellation is complete and that the ghost 2-form alone produces the quoted result.

    Authors: We accept that a tabulated breakdown would make the cancellation pattern verifiable at a glance. In the revised manuscript we will insert a new table (or compact summary table) in §5.3 that lists the logarithmic coefficient for every field in the 11d supergravity multiplet, grouped by type (metric, 3-form, gravitino, etc.), showing the individual contributions before and after zero-mode subtraction and highlighting that all cancel except the ghost 2-form mode. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no load-bearing reductions to inputs

full rationale

The paper develops a general framework reducing (dA+dS)-dimensional operators to towers of dA-dimensional determinants via spherical harmonics, then extracts log divergences with zeta functions while isolating zero-mode AdS determinants. This procedure is applied to the full 11d supergravity multiplet on AdS4×S7, yielding the result that only the ghost 2-form contributes. The reproduction of the known 1/4 log N (from arXiv:1210.6057) is presented as an output of the calculation rather than an input; no equations or steps are shown to be equivalent to the target by construction, and the zero-mode handling is described as part of the systematic subtraction without reference to fitting the final coefficient. The framework is tested on multiple fields independently of the ABJM case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard spectral zeta regularization and spherical-harmonic decomposition; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Spectral zeta-function regularization correctly isolates the logarithmic divergences after the KK sum.
    Invoked to extract the log part from the infinite tower of AdS determinants.
  • domain assumption Zero modes on the sphere generate additional AdS determinants whose divergences can be subtracted cleanly.
    Stated as a key step that must be accounted for carefully.

pith-pipeline@v0.9.1-grok · 5765 in / 1288 out tokens · 25616 ms · 2026-06-28T16:50:31.138892+00:00 · methodology

discussion (0)

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Reference graph

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