Chaos and moduli space volumes in unorientable JT gravity

Jarod Tall; Juan Diego Urbina; Klaus Richter; Torsten Weber

REVIEW 2 major objections 6 minor 51 references

Chaos and moduli space volumes in unorientable JT gravity

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The τ-scaled spectral form factor of unorientable JT gravity agrees with the GOE random-matrix prediction through genus one, using new residue-based formulas for unorientable moduli volumes.

desk verdict Strong computation with a real but fixable gap: the tau-scaled SFF match is credible, but the streamlined volume formulas rest on an unproved cancellation of z1=-z2 residues. read the letter →

arxiv 2411.08129 v3 pith:4FEGIFSU submitted 2024-11-12 hep-th

Jarod Tall , Torsten Weber , Juan Diego Urbina , Klaus Richter This is my paper · ORCID

classification hep-th

keywords volumesunorientablegenusgravitylimitmatrixmodelboundaries

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

Jackiw-Teitelboim (JT) gravity is a simple model of quantum gravity in two dimensions, describing rubbery surfaces with fixed negative curvature. Normally these surfaces are orientable, like a sphere or donut, but a version with time-reversal symmetry must also include unorientable pieces such as Möbius strips. In such a theory, the natural volumes that count ways to build a surface from building blocks blow up because small crosscaps create divergences. This paper computes those volumes by borrowing a trick from random matrix theory: the same surface-counting data appears in the correlation functions of a matrix model, and the loop equations of an orthogonal matrix model reduce the volume problem to taking residues at infinitely many points. The authors streamline the computation so that many unwanted terms can be dropped as known cancellations, and they obtain explicit volumes with one and two boundaries through genus one. Using these volumes, they compute the spectral form factor, a quantity that measures how likely two copies of the system are to have similar energy levels. This is the standard probe of quantum chaos. They compare it to the prediction of the Gaussian orthogonal ensemble, the random-matrix ensemble for time-reversal symmetric systems. The two calculations agree term by term up to the cubic order in rescaled time, after a subtle Airy-type correction that had been worked out in earlier work by the same group. The matching is a signature that the time-reversal symmetric gravity theory is quantum chaotic in the same universal way as random matrices.

Extended reading notes

Core claim

The τ-scaled spectral form factor of unorientable JT gravity, computed from the unorientable moduli volumes, equals the universal GOE random-matrix result order by order in τ through τ³ (equations (1.26) and (1.30) agree in all non-Airy infinite-series terms), with the Airy τ³ coefficient reconciled in the authors' prior work [19]. If correct, this establishes the BGS quantum-chaos signature for the time-reversal symmetric version of JT gravity and shows the divergent parts of the unorientable volumes cancel in the SFF.

Load-bearing premise

The streamlined volume formulas (2.57)-(2.58) and (2.36) are derived by discarding, without a full proof, all O(b_i^{-1}) contributions and all residues at z1 = −z2, on the grounds that these must cancel in the full sum (Sections 2.3 and 2.4, especially the paragraph around eq. (2.35)). If any of those discarded terms carried a finite part, every volume computed in the paper, and hence the SFF match, would be off.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a careful read. The paper computes the tau-scaled spectral form factor for unorientable JT gravity through genus one and matches it to the GOE random-matrix prediction, using the loop equations of the sin(2πz) orthogonal matrix model. Along the way it produces the first analytic unorientable volumes at (g,n)=(1,1) and (1,2), including v1,0(b) which was previously numerical, and conjectures a multiple-polylog/zeta structure for the volumes. The central SFF matching is real: the non-Airy terms come from two genuinely independent computations — volume integrals versus a Laplace transform of the GOE form factor — with no fitted parameters, and the divergence cancellation that leaves a finite, log(p)-independent result is an actual check on the machinery. The low-genus volumes also agree with the existing results in [12]. Credit where due: this is substantial and mostly credible.

The soft spot is exactly the one flagged in the stress test. The streamlined formulas (2.57)-(2.58) drop all O(b_i^{-1}) residues and all z1=-z2 residues on the assertion that they cancel. The O(b^{-1}) argument is fine: simple poles give only b^{-1} or b^{-1}e^{-bk/2}, and no combination of those can survive in a finite volume. The z1=-z2 claim is weaker. The paper argues any such residue has the form (2.35) with a θ(b2-b1), and since its O(b2^{-1}) part must cancel, the whole expression cancels. That does not exclude a surviving finite c0 part if only the c_{-1}/b2 piece cancels against something else. The authors do not prove this. I think the conclusion is probably right, because the final results pass several internal checks, but the derivation has a real gap that a referee should ask them to close — ideally by proving the cancellation or by numerically evaluating the dropped contributions at finite b.

One more caveat: the advertised agreement up to genus one is only literal after importing a pseudo-renormalization from the authors' own [19] for the Airy τ^3 coefficient. They are transparent about this, so it is not a hidden flaw, but it does mean the headline result leans on prior work for one term.

Bottom line: this is for people working on 2D gravity, matrix models, or spectral form factors. It deserves a serious referee, and I would probably cite it, but I would not treat the streamlined volume formulas as fully proven until the cancellation question is settled.

Referee Report

2 major / 6 minor

Summary. The paper computes moduli-space volumes of unorientable JT gravity from the loop equations of an orthogonal matrix model with spectral curve y(z) ∝ sin(2πz), regularized via the (2,2p+1) minimal string model. The authors introduce 'streamlined formulas' that reduce volume computations to sums of residues, and use them to obtain explicit one- and two-boundary unorientable volumes up to genus one, including the three-crosscap volume v_{1,0}(b1,b2). They then compute the τ-scaled spectral form factor (SFF) of unorientable JT gravity up to genus one and compare it with the SFF obtained from the universal GOE microcanonical form factor. The two expressions are claimed to agree in all non-Airy infinite-series terms through τ^3, with the Airy τ^3 coefficient reconciled in the authors' prior work [19]. The paper also conjectures a multiple-zeta/multiple-polylogarithm structure for the polynomial parts of the unorientable volumes.

Significance. If the central claim is correct, the paper provides the first direct check of the BGS quantum-chaos signature for time-reversal-symmetric JT gravity, showing that the τ-scaled SFF matches universal GOE random-matrix theory order by order in the topological expansion. The computation of unorientable volumes up to genus one goes beyond previous results, and the streamlined residue formulas, if valid, are a substantial technical advance over Stanford's recursion. The paper includes several independent checks: the one-boundary volumes reproduce [12], the two-boundary volume v_{1/2,0} matches Stanford's recursion (2.40), and the long formula (2.52) passes a numerical symmetry check. However, the load-bearing simplifying step—dropping all O(b^{-1}) and z1=−z2 residues without proof—means the central claim is conditional on an unverified cancellation.

major comments (2)

The streamlined formulas are derived by discarding all O(b_i^{-1}) terms and all residues at z1 = −z2, based on the assertion that they must cancel in the full residue sum. The argument in §2.3 is that no combination of terms ∝ b^{-1} or ∝ b^{-1}e^{-bk/2} can be purely finite, and in §2.4 that any z1 = −z2 contribution has the structure (2.35) and must cancel entirely because its O(b_2^{-1}) part cannot cancel otherwise. This is not a proof for the cases that matter: a Laurent piece c_{-1}/b_i + c_0 would produce a finite c_0 if only the c_{-1}/b_i part cancels, and the paper provides no argument excluding such pieces at (g,n) = (1,2). Since v1,2, v1,1, v1,0 in Eqs. (2.50)-(2.52) and the SFF terms (3.9)-(3.11) are all obtained after these discards, a surviving finite term would change the claimed equality (1.26) = (1.30). Please either prove the cancellation for general residue sums or provide an independent verification of v1,0(b1,b2)—for example by direct numerical evaluation of the full residue sum at fixed b1,b2 or by a comparison with Stanford's recursion at selected values.
The central claim that the SFF agrees with universal RMT up to genus one includes the Airy τ^3 coefficient. Eqs. (1.27) and (1.31) show that the Airy τ^3 coefficients differ (log(2t/β) versus −γ − log(2βτ^2) − 1/3), and the reconciliation is deferred to prior work [19]. As written, the present paper does not establish the full agreement; it establishes agreement only for the non-Airy infinite-series terms. The reader should be told precisely which parts of (1.26) and (1.30) are proved here and which parts are imported from [19], and the imported result should be stated explicitly if the paper is to stand alone.

minor comments (6)

The displayed equation in §2.5 reads `−4z1z2F_1^{(p)}(z1,z2)e^{b1 z2}e^{b2 z2}/(2b1b2 y(z1))`; the first exponential should presumably be `e^{b1 z1}`, not `e^{b1 z2}`. The same typo appears in Eq. (2.49).
The equation contains a stray period inside the theta function: `θ(b2 − b1. )` should read `θ(b2 − b1)`.
The notation `+ O(b^{-1})` is used in a nonstandard sense: it is stated that the correct volumes are found by dropping all terms of this order. This usage should be defined explicitly, since in the standard meaning an O(b^{-1}) remainder is not a license to discard all O(b^{-1}) terms without further argument.
The convergence of the infinite residue sums is only discussed for b1 > b2; the analytic continuation in (2.39) and (2.49) is asserted without a detailed justification. A short remark on why the resulting function is analytic in (b1,b2) would improve rigor.
The display of v1,1(b1,b2) in (1.19) has a formatting issue: `b2 2b1 2` should be `(b2^2 b1)/2`, and the same expression in (2.54) has the factor 1/2 on the b1^3 and b2^3 terms, which is correct but not clearly reflected in the introduction's version.
The notation `O(t^{-1/2})` is used for subleading terms in t; since τ = t e^{-S0} is fixed, it would be helpful to explain explicitly which quantities are held fixed when t → ∞ in this expansion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the gravity-side volumes and the universal GOE SFF are computed by distinct routes, and the only self-citation ([19]) is independent Airy-model support.

full rationale

The central comparison is not circular. The unorientable JT gravity side of the SFF is computed from moduli-space volumes v_{g,k}(b1,b2), which are obtained by solving the loop equations of an orthogonal matrix model with the JT spectral curve y(z)=sin(2πz)/(4π) (Sections 2.2-2.5, Appendix A), then integrated against double-trumpet partition functions (Section 3.1, Appendix B). The universal RMT side is computed independently as the Laplace transform of the microcanonical GOE form factor, using only the GOE symmetry class and the JT leading density of states (Section 3.2, Appendix C). No parameter appearing in the GOE computation is fitted to the gravity-side result, and the two expressions are derived through different calculational chains before being compared in (1.26) versus (1.30). The streamlined volume formulas (2.57)-(2.58) rely on an asserted cancellation of all O(b_i^{-1}) contributions and of residues at z1=-z2; this is an unproved technical assumption that carries correctness risk, but it is not a circular reduction, because the discarded terms are not chosen so as to force the GOE answer. The only self-citation with real weight is reference [19], used to reconcile the Airy τ^3 coefficients; that is prior independent work on the unorientable Airy model, whose stated assumptions do not include the present full-JT target, so by the standard for external support it does not make the argument circular. The central claim therefore has independent content, and no step reduces by definition to its own inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The model has no fitted constants: the spectral curve is fixed by JT gravity, the regulator p is removed by cancellations, and τ and β are physical variables. The paper's central claim depends instead on a set of structural assumptions about the matrix-model duality and about unproved term cancellations, listed above.

assumptions (7)

domain assumption The orthogonal-matrix loop equations (2.1)-(2.2) correctly encode the double-scaled orthogonal matrix model correlation functions.
Invoked as the starting point in Section 2.1, following [9] and [19]; the contour deformation and double-cover coordinate are taken from [19].
domain assumption Unorientable JT gravity (time-reversal symmetric) is dual to the orthogonal matrix model with spectral curve y(z)=sin(2πz)/(4π).
Used throughout; proven in [12], conjectured in [9]; the paper builds on rather than re-derives this duality.
domain assumption The large-p limit of the (2,2p+1) minimal string spectral curve (1.10) regularizes the unorientable JT resolvents and volumes.
Introduced in Section 1.1, eq. (1.10), following Appendix F of [9] and [32,33]; the physical interpretation is a UV cutoff on the density of states.
standard math The GOE microcanonical form factor bGOE(x) (3.17) and the JT leading-order density ρ0(E)=sinh(2π√E)/(4π²) determine the canonical τ-scaled SFF.
Section 3.2, eqs. (3.15)-(3.18); the form factor is textbook [29] and the density follows from the JT spectral curve.
ad hoc to paper All O(b^{-1}) terms and all residues at z1=-z2 cancel and can be dropped from the volume formulas.
Sections 2.3 and 2.4, eqs. (2.27) and (2.35)-(2.36); this is an unproved structural assumption of the streamlined computation.
ad hoc to paper The polynomial part of unorientable volumes has the multiple-zeta structure of the conjecture (1.23)/(2.61).
Section 2.6; explicitly stated as a conjecture justified only by the computed low-genus examples.
domain assumption The τ-scaled limit τ=t e^{-S0} commutes with the genus expansion and the Airy τ³ term can be reconciled with GOE by the pseudo-renormalization of [19].
Section 1.1 and Section 3 comparison; the Airy reconciliation is imported from the authors' prior work [19] rather than shown here.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Chaos and moduli space volumes in unorientable JT gravity." pith.science (2026). https://pith.science/paper/4FEGIFSU

@misc{pith2026241108129,
  author       = {Pith},
  title        = {Pith review of: Chaos and moduli space volumes in unorientable JT gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4FEGIFSU}},
  note         = {Machine review of arXiv:2411.08129}
}

abstract

We show the late time, or $\tau-$scaled, limit of the canonical spectral form factor (SFF) in unorientable JT gravity agrees with universal random matrix theory (RMT) up to genus one in the topological expansion, establishing a key signature of quantum chaos for the time-reversal symmetric case. The loop equations for an orthogonal matrix model with spectral curve $y(z) \propto \sin(2\pi z)$ are used to compute the moduli space volumes of unorientable surfaces. The divergences of the unorientable volumes are regularized by first regularizing the resolvents of the orthogonal matrix model. To this end, we make use of the large $p$ limit of the $(2,2p+1)$ minimal string model. Using properties of the volumes and the loop equations, we derive streamlined formulas to compute the volumes for one and two boundaries, giving explicit results up to genus one. We find the general structure of the unorientable volumes to be written in terms of multiple polylogarithms and zeta values, with weight determined by the genus, number of boundaries, and number of crosscaps. In the $\tau-$scaled limit, contributions to the SFF from the divergent parts of the volume cancel, and the SFF becomes finite and independent of regularization. The SFF from universal RMT is a distinct computation, that depends on the leading order energy density of JT gravity, which we also derive up to genus one.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 25 canonical work pages

[12]

Stanford, A Mirzakhani recursion for non-orientable surfaces , arXiv:2303.04049

D. Stanford, A Mirzakhani recursion for non-orientable surfaces , arXiv:2303.04049

arXiv
[19]

Weber, J

T. Weber, J. Tall, F. Haneder, J.D. Urbina and K. Richter , Unorientable topological – 51 – gravity and orthogonal random matrix universality , Journal of High Energy Physics 2024 (2024) [2405.17177]

arXiv 2024
[1]

Jackiw, Lower dimensional gravity , Nuclear Physics, Section B 252 (1985) 343

R. Jackiw, Lower dimensional gravity , Nuclear Physics, Section B 252 (1985) 343 . – 50 –

work page 1985
[2]

Teitelboim, Gravitation and hamiltonian structure in two spacetime dim ensions, Physics Letters B 126 (1983) 41

C. Teitelboim, Gravitation and hamiltonian structure in two spacetime dim ensions, Physics Letters B 126 (1983) 41

work page 1983
[3]

Saad, S.H

P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral , 1903.11115

arXiv 1903
[4]

A simple model of quantum holography (part 1)

A. Kitaev, “A simple model of quantum holography (part 1) .” talk at KITP http://online.kitp.ucsb.edu/online/entangled15/kitaev/, April 7, 2015

work page 2015
[5]

A simple model of quantum holography (part 2)

A. Kitaev, “A simple model of quantum holography (part 2) .” talk at KITP http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, May 27, 2015

work page 2015
[6]

Maldacena and D

J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model , Physical Review D 94 (2016) [1604.07818]

arXiv 2016

Show all 51 references

[7]

Jensen, Chaos in AdS2 Holography, Physical Review Letters 117 (2016) 111601 [1605.06098]

K. Jensen, Chaos in AdS2 Holography, Physical Review Letters 117 (2016) 111601 [1605.06098]

2016 arXiv
[8]

Mertens and G.J

T.G. Mertens and G.J. Turiaci, Solvable models of quantum black holes: a review on jackiw–teitelboim gravity, Living Reviews in Relativity 26 (2023) 4

2023
[9]

Stanford and E

D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory , Advances in Theoretical and Mathematical Physics 24 1475
[10]

Dyson, Statistical Theory of the Energy Levels of Complex Systems

F.J. Dyson, Statistical Theory of the Energy Levels of Complex Systems. I , Journal of Mathematical Physics 3 (1962) 140

1962
[11]

Altland and M.R

A. Altland and M.R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures , Physical Review B 55 (1997) 1142

1997
[13]

Maldacena, D

J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two-dimensional nearly anti-de Sitter space , Progress of Theoretical and Experimental Physics 2016 (2016) [1606.01857]

2016 arXiv
[14]

Bohigas, M.J

O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws , Physical Review Letters 52 (1984) 1

1984
[15]

Saad, S.H

P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity , 1806.06840

arXiv
[16]

Cotler, G

J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Sa ad, S.H. Shenker et al., Black holes and random matrices , Journal of High Energy Physics 2017 (2017) [1611.04650]

2017 arXiv
[17]

Altland and J

A. Altland and J. Sonner, Late time physics of holographic quantum chaos , SciPost Physics 11 (2021) [2008.02271]

2021 arXiv
[18]

P. Saad, D. Stanford, Z. Yang and S. Yao, A convergent genus expansion for the plateau, Journal of High Energy Physics 2024 (2024) [2210.11565]

2024 arXiv
[20]

Norbury, Lengths of geodesics on non-orientable hyperbolic surface s, Geometriae Dedicata 134 (2008) 153

P. Norbury, Lengths of geodesics on non-orientable hyperbolic surface s, Geometriae Dedicata 134 (2008) 153

2008
[21]

Mirzakhani, Weil-Petersson volumes and intersection theory on the modu li space of curves , Journal of the American Mathematical Society 20 (2006) 1

M. Mirzakhani, Weil-Petersson volumes and intersection theory on the modu li space of curves , Journal of the American Mathematical Society 20 (2006) 1

2006
[22]

Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spac es of bordered Riemann surfaces, Inventiones Mathematicae 167 (2007) 179

M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spac es of bordered Riemann surfaces, Inventiones Mathematicae 167 (2007) 179

2007
[23]

Gendulphe, What’s wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces , 1706.08798

M. Gendulphe, What’s wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces , 1706.08798

arXiv
[24]

Eynard and N

B. Eynard and N. Orantin, Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models , 0705.3600

arXiv
[25]

Weber, F

T. Weber, F. Haneder, K. Richter and J.D. Urbina, Constraining Weil-Petersson volumes by universal random matrix correlations in low-dim ensional quantum gravity, Journal of Physics A: Mathematical and Theoretical 56 (2023) [2208.13802]

2023 arXiv
[26]

Blommaert, J

A. Blommaert, J. Kruthoﬀ and S. Yao, An integrable road to a perturbative plateau , Journal of High Energy Physics 2023 (2023) [2208.13795]

2023 arXiv
[27]

Okuyama and K

K. Okuyama and K. Sakai, ’t Hooft expansion of multi-boundary correlators in 2D topological gravity, Progress of Theoretical and Experimental Physics 2021 (2021) 083B03

2021
[28]

Okuyama and K

K. Okuyama and K. Sakai, Spectral form factor in the τ -scaling limit , Journal of High Energy Physics 2023 (2023) [2301.04773]

2023 arXiv
[29]

Mehta, Random Matrices, ISSN, Elsevier Science (2004)

M.L. Mehta, Random Matrices, ISSN, Elsevier Science (2004)

2004
[30]

Haake, Quantum signatures of chaos , Springer series in synergetics, Springer, Berlin [u.a.] (2010)

F. Haake, Quantum signatures of chaos , Springer series in synergetics, Springer, Berlin [u.a.] (2010)

2010
[31]

Eynard, Counting Surfaces , vol

B. Eynard, Counting Surfaces , vol. 70 of Progress in Mathematical Physics (2016)

2016
[32]

Artemev and I

A. Artemev and I. Chaban, (2 p + 1) minimal string and intersection theory 1 , 2403.02305

arXiv
[33]

Mertens and G.J

T.G. Mertens and G.J. Turiaci, Liouville quantum gravity – holography, JT and matrices, Journal of High Energy Physics 2021 (2021) [2006.07072]

2021 arXiv
[34]

Eynard, T

B. Eynard, T. Kimura and S. Ribault, Random matrices, 1510.04430

arXiv
[35]

Eynard, Topological expansion for the 1-hermitian matrix model cor relation functions, Journal of High Energy Physics 8 (2004) 845

B. Eynard, Topological expansion for the 1-hermitian matrix model cor relation functions, Journal of High Energy Physics 8 (2004) 845

2004
[36]

Waldschmidt, Multiple polylogarithms: An introduction , in Number Theory and Discrete Mathematics , (Basel), pp

M. Waldschmidt, Multiple polylogarithms: An introduction , in Number Theory and Discrete Mathematics , (Basel), pp. 1–12, Birkh¨ auser Basel, 2002

2002
[37]

Gil and J

J. Gil and J. Fresan, Multiple zeta values: from numbers to motives , in press (2017) . – 52 –

2017
[38]

Zhao, Multiple zeta functions, multiple polylogarithms and thei r special values , World Scientiﬁc (2016)

J. Zhao, Multiple zeta functions, multiple polylogarithms and thei r special values , World Scientiﬁc (2016)

2016
[39]

Witten, On quantum gauge theories in two dimensions , Communications in Mathematical Physics 141 (1991) 153

E. Witten, On quantum gauge theories in two dimensions , Communications in Mathematical Physics 141 (1991) 153

1991
[40]

Maximon, The dilogarithm function for complex argument , Proceedings of the Royal Society of London

L.C. Maximon, The dilogarithm function for complex argument , Proceedings of the Royal Society of London. Series A 459 (2003) 2807

2003
[41]

Duhr and F

C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses , JHEP 08 (2019) 135 [1904.07279]

2019 arXiv
[42]

Zagier, Values of zeta functions and their applications , in First European Congress of Mathematics Paris, July 6–10, 1992: Vol

D. Zagier, Values of zeta functions and their applications , in First European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2) , A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler, eds ., (Basel), pp. 497–512, Birkh¨ auser Basel (1994), DOI

1994
[43]

C. Duhr, Mathematical aspects of scattering amplitudes , in Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Thr ough the Precision Frontier: Amplitudes for Colliders , pp. 419–476, 2015, DOI [1411.7538]

2015 arXiv
[44]

Schlotterer and S

O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [1205.1516]

2013 arXiv
[45]

Mafra and O

C.R. Mafra and O. Schlotterer, Tree-level amplitudes from the pure spinor superstring, Phys. Rept. 1020 (2023) 1 [2210.14241]

2023 arXiv
[46]

Goncharov, Multiple polylogarithms, cyclotomy and modular complexes , Math

A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes , Math. Res. Lett. 5 (1998) 497 [1105.2076]

1998 arXiv
[47]

Goncharov, Multiple polylogarithms and mixed tate motives , math/0103059

A.B. Goncharov, Multiple polylogarithms and mixed tate motives , math/0103059

arXiv
[48]

Brown, Mixed tate motives over Z, Annals of Mathematics 172 (2012)

F. Brown, Mixed tate motives over Z, Annals of Mathematics 172 (2012)

2012
[49]

Gnutzmann and B

S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, an d Andreev star graphs. I. Construction and numerical results , Physical Review E 69 (2004) 16

2004
[50]

Griguolo, J

L. Griguolo, J. Papalini, L. Russo and D. Seminara, The resurgence of the plateau in supersymmetric N = 1 Jackiw-Teitelboim gravity, Journal of High Energy Physics 2024 (2024) 168 [2310.06768]

2024 arXiv
[51]

Turiaci and E

G.J. Turiaci and E. Witten, N = 2 JT supergravity and matrix models , JHEP 12 (2023) 003 [2305.19438]. – 53 –

2023 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [12]

Stanford, A Mirzakhani recursion for non-orientable surfaces , arXiv:2303.04049

D. Stanford, A Mirzakhani recursion for non-orientable surfaces , arXiv:2303.04049

arXiv

[2] [19]

Weber, J

T. Weber, J. Tall, F. Haneder, J.D. Urbina and K. Richter , Unorientable topological – 51 – gravity and orthogonal random matrix universality , Journal of High Energy Physics 2024 (2024) [2405.17177]

arXiv 2024

[3] [1]

Jackiw, Lower dimensional gravity , Nuclear Physics, Section B 252 (1985) 343

R. Jackiw, Lower dimensional gravity , Nuclear Physics, Section B 252 (1985) 343 . – 50 –

work page 1985

[4] [2]

Teitelboim, Gravitation and hamiltonian structure in two spacetime dim ensions, Physics Letters B 126 (1983) 41

C. Teitelboim, Gravitation and hamiltonian structure in two spacetime dim ensions, Physics Letters B 126 (1983) 41

work page 1983

[5] [3]

Saad, S.H

P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral , 1903.11115

arXiv 1903

[6] [4]

A simple model of quantum holography (part 1)

A. Kitaev, “A simple model of quantum holography (part 1) .” talk at KITP http://online.kitp.ucsb.edu/online/entangled15/kitaev/, April 7, 2015

work page 2015

[7] [5]

A simple model of quantum holography (part 2)

A. Kitaev, “A simple model of quantum holography (part 2) .” talk at KITP http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, May 27, 2015

work page 2015

[8] [6]

Maldacena and D

J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model , Physical Review D 94 (2016) [1604.07818]

arXiv 2016