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A spool for every quotient: One-loop partition functions in AdS$_3$ gravity
by Robert Bourne, Jackson R. Fliss, Bob Knighton
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Jackson R. Fliss · Robert Knighton |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2507.05364v1 (pdf) |
| Date submitted: | Aug. 24, 2025, 5:38 p.m. |
| Submitted by: | Jackson R. Fliss |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The Wilson spool is a prescription for expressing one-loop determinants as topological line operators in three-dimensional gravity. We extend this program to describe massive spinning fields on all smooth, cusp-free, solutions of Euclidean gravity with a negative cosmological constant. Our prescription makes use of the expression of such solutions as a quotients of hyperbolic space. The result is a gauge-invariant topological operator, which can be promoted to an off-shell operator in the gravitational path integral about a given saddle-point. When evaluated on-shell, the Wilson spool reproduces and extends the known results of one-loop determinants on hyperbolic quotients. We motivate our construction of the Wilson spool from multiple perspectives: the Selberg trace formula, worldline quantum mechanics, and the quasinormal mode method.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
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The generalization of the `Wilson spool' approach of writing partition functions in 3 d gravity to arbitrary massive spin fields in all smooth cusp-free solutions of Euclidean gravity with negative cosmological constant.
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There is a test by comparting the resulting expression (eq 3.3) against the Selberg Trace formula for fields of spin=0, 1, 2,
Weaknesses
- There is no test of the in formula in 3.3 fo spins >2 using another approach say the Selberg trace formula. For the case of the BTZ case or the solid torus, the expression for the one loop determinant from the Wilson spool approach was tested against the Selberg trace for arbitrary massive spinning fields in https://arxiv.org/pdf/2507.05358.
It would have been good if there had been some tests for at least some other quotient of H_3.
- The general program of the Wilson spool approach in 3 d gravity gives empahsis to the Chern-Simons formulation over the metric formulation. For fields, massive or massless in 3d, one standard quantity which is evaluated in the conventional metric formalism are bulk 2 point functions or in the case of $AdS_3$ bulk boundary propagators. It would be worthwhile if the authors can comment on how these can be reformulated in the Wilson-spool formulation. Do, they correspond to expressions similar to 3.3, but with open Wilson lines.
The authors can just point that such questions do remain open in this effort at reformulating 3d gravity including external (massive and massless/spinning) fields in the Wilson-spool approach.
Report
one loop partition functions of massive spinning fields in 3 d gravity on smooth
cusp free solutions of 3d gravity. The earlier work in particular ref 7 carries out the analysis for dS_3 and AdS_3.
The paper is based on sound principles, the final result 3.3, is a natural
generalisation of earlier expressions say 1.4 of ref 7, to eq 3.3, in which there is a sum over unoriented non-contractable loops.
The generalisation is expected, it would have been more useful if the authors carried out some non-trivial tests for spins>2
Requested changes
I request the authors to comment on the use of the Wilson spool approach
to evaluate 2 point function either bulk 2 point functions or bulk boundary propagators of fields (even scalars)
in 3 d gravity.
It is important to point out the Wilson spool approach needs to be generalised to this cases and possibly can be generalised.
Also, I request the authors to comment on the possibility of using this
approach to evaluate the one loop corrections to entanglement entropy
on the lines of https://arxiv.org/pdf/1306.4682.
Note that the Selberg trace formula was used in eq 36 of this paper,here quaotients of H_3 involved the Schottky group.
Is it possibile to generalise the Wilson loop formualism to such quotients ?
There there would be a possibility of testing the expressions even for massive spining fields of s>2.
Recommendation
Ask for minor revision
Report
The manuscript proposes an extension of the “Wilson spool” framework for coupling free massive fields to $AdS_3$ gravity. The authors generalize the construction, previously developed for Euclidean $dS_3$ and rotating BTZ black holes, to all smooth, cusp-free hyperbolic 3-manifolds. They verify the prescription by reproducing known results for massive scalars and vectors on such spaces. One highlight is that this prescription enables the authors to extend these results to massive fields of arbitrary integer spin. The authors also motivate their prescription from the perspectives of the Selberg trace formula, the worldline path integral, and the quasinormal mode method.
The manuscript is well written and the analysis is clearly presented. While the advance is mainly technical, the construction is useful for researchers working on $AdS_3$ gravity. I recommend publication once the authors address the following minor points:
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In the spool expressions (1.3) and (1.4), are the relevant representations and connections those of $SL(2,\mathbb R)$? If so, please clarify why this choice is appropriate in the Euclidean setting, where $PSL(2,\mathbb C)$ (or $SO(1,3)$) would appear to be the natural isometry group.
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Just before equation (2.9), the symbol $\tau$ seems to be a typo and should likely read $\vartheta$.
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In Footnote 12, it might be worth noting that obtaining the full one-loop partition function for a massless spin-2 field requires dividing by the corresponding ghost determinant, in addition to taking the massless limit of the massive spin-2 result.
Recommendation
Ask for minor revision
We thank the referee for their careful reading of our manuscript and for their positive assessment on its suitability for being published in SciPost Physics. Below let us address directly their minor points:
1) The Euclidean group is indeed $PSL(2,\mathbb C)$ and the appropriate Chern-Simons connection should be regarded as a complex connection valued in this algebra. We addressed this point briefly in the paragraph below equation (2.5) stating we are regarding $A_L$ and $A_R$ as components of a real form of $\mathfrak{sl}(2,\mathbb C)$. This is not strictly necessary, however streamlines the discussion of including matter as we will classify matter into the single-particle representation theory of the Lorentzian isometry algebra which does split: $\mathfrak{so}(2,2)\simeq \mathfrak{sl}(2,\mathbb R)\oplus\mathfrak{sl}(2,\mathbb R)$. We have since clarified this point in the aforementioned paragraph. In any case, it easy to see that all quantities computed in terms of $A_L$ and $A_R$ (e.g. holonomies) come in complex conjugate pairs as appropriate for the components of a complex connection of $\mathfrak{sl}(2,\mathbb C)$. We have expanded the paragraph under equation (2.5) to clarify why we write the connection in a split manner while working in Euclidean signature.
2) We thank the referee for spotting this typo and it has since been amended.
3) We thank the referee for pointing this out. The expression for the spool at $\Delta=s=2$ is indeed the spin-2 massless symmetric transverse traceless determinant as can be checked in comparison to Giombi-Maloney-Yin, however the referee is indeed correct that the determinant of a physical linearized graviton includes the additional determinants over the vector-ghost and the scalar mode. We have included this comment in footnote 12.
Author: Jackson Fliss on 2026-01-09 [id 6218]
(in reply to Report 2 on 2025-11-28)We thank the referee for their thorough read of our manuscript and especially for their insightful questions. Below we reply to the specific weaknesses mentioned by the referee as well as the changes they requested.
In response to weakness 1: the spool evaluated on a BTZ background was checked for massive fields with arbitrary spins in https://arxiv.org/abs/2407.09608; this result matches the known literature (http://arxiv.org/abs/1112.4619, http://arxiv.org/abs/0804.1773,http://arxiv.org/abs/0911.5085,http://arxiv.org/abs/1707.06245). Our formula (3.3) reproduces this result as we comment on and so already provides the check that the referee is requesting. For general quotients there are no general results for spins s>2 to which we can compare our result. In this sense, the spool provides a novel result for these cases.
In response to weakness 2: A generalization to open Wilson lines is natural, in particular for the study of quantum corrections to CFT correlators; a natural first step in this direction was already provided in http://arxiv.org/abs/2001.09998. However, this research direction has been discussed in previous Wilson spool papers (see e.g. page 19 of https://arxiv.org/abs/2503.08657 and page 46 of https://arxiv.org/pdf/2302.12281) by one of the authors. Because we do not have additional insight towards this direction and to avoid repeating the same discussion points across multiple papers, we chose to omit this discussion from this paper.
In response to the requested change regarding of entanglement entropies, (bulk) one-loop corrections to CFT Rényi entropy calculated through a bulk Wilson spool on a Schottky quotient should be entirely possible to due the loxodromic nature of the quotients. This would be a non-trivial use of the spool proposal in light of FLM corrections to the Ryu-Takayanagi formula. We thank the referee for suggesting this and we have added a paragraph in the discussion section briefly discussing this direction.