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Weil-Petersson volumes for extended JT supergravity from ordinary differential equations

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Pith's one-line read This paper shows that one-boundary Weil-Petersson volumes for N=2 and N=4 JT supergravity are produced by two ordinary differential equations, confirming known results and adding genus-2 and genus-3 examples.

desk verdict Solid ODE-method extension: the new N=2 genus-2/3 volumes pass an exact independent cross-check, the N=4 results are new and provisional, and the real caveats are the unproved total-derivative property and a normalization choice that builds in the bosonic subsector. read the letter →

arxiv 2507.18715 v1 pith:B2YXQECQ submitted 2025-07-24 hep-th

Wasif Ahmed , Clifford V. Johnson , Krishan Saraswat This is my paper · ORCID
classification hep-th
keywords Weil-PeterssonvolumesJTsupergravitystringequationdiagonalresolventrandommatrixmodelsextendedsupersymmetrytopologicalrecursionmodulispaceofhyperbolicsurfaces
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

Two ordinary differential equations—the string equation that defines the double-scaled random matrix model and the equation satisfied by the diagonal resolvent of the auxiliary Hamiltonian—together generate the one-boundary Weil-Petersson volumes $V_{g,1}(b)$. This paper applies that ODE recipe to JT supergravity with extended supersymmetry. For $N=2$ it reproduces the known genus-1 volume and produces new genus-2 and genus-3 expressions, checked against the topological recursion derived from loop equations. For small and large $N=4$ it gives the first such volumes. In every extended case the bosonic JT volume appears as the highest-order term in the threshold energy $E_0$ (or in $J$ for $N=4$), with the lower orders carrying the supersymmetric corrections.

What carries the argument

The load-bearing objects are two equations: the string equation $uR^2-\frac{\hbar^2}{2}RR''+\frac{\hbar^2}{4}(R')^2=\tilde\Gamma^2$ with $R=\sum_k t_k R_k[u]+x$, and the resolvent equation $4(u-E)\widehat{R}^2-2\hbar^2\widehat{R}\widehat{R}''+\hbar^2(\widehat{R}')^2=1$. The argument is carried by the total-derivative identity $\widehat{R}_g=d\widehat{Q}_g/dx$ and by a recursion that constructs $\widehat{Q}_g$ from $\widehat{R}_g$ using only differentiations, starting from the highest power of $X=u_0-E$. The Fermi-surface data $u_0^{(p)}(\mu)$ are polynomials in the threshold energy (or in $J$), and their top coefficients are $(2\pi)^p$ or $4^{-p}$ times the bosonic JT values, which is why the normalization factors expose the bosonic subsector.

What would settle it

Run the paper's recursion at genus 4 and check whether $\widehat{R}_4$ can be written as $d\widehat{Q}_4/dx$ by the Appendix C procedure; if no such $\widehat{Q}_4$ exists, the definition of $V_{4,1}(b)$ collapses. A direct numerical cross-check would compare the resulting $V_{4,1}(b)$ with the value obtained from the $N=2$ topological recursion for the same model.

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Extended reading notes

Core claim

The paper's central claim is that once $u(x)$ solves the extended string equation, each genus-$g$ piece $\widehat{R}_g(x,E)$ of the resolvent is a total $x$-derivative, $\widehat{R}_g=d\widehat{Q}_g/dx$, so the volume is fixed solely by $u_0$ and its derivatives at the Fermi surface $x=\mu$. Applying this gives $V_{g,1}(b)$ for $N=2$ JT supergravity at $g=1,2,3$, matching the known genus-1 result and the recursion-based checks at higher genus. The same machinery produces new $V_{g,1}(b)$ for small $N=4$ (with $J\in\frac12\mathbb{Z}\setminus\{0\}$) and for large $N=4$, where the volumes reduce to rescaled $N=2$ volumes with the threshold energy shifted from $E_0$ to $E_0-E_\pm$. With the chosen normalizations, the highest power of $E_0$ or $J$ is exactly the bosonic JT volume, and the $E_0$-independent term is the $N=1$ volume.

Load-bearing premise

The load-bearing assumption is that the genus-$g$ resolvent piece is a total derivative at every genus once $u(x)$ solves the string equation; the paper verifies this for $g=1,2,3$ and says it has no closed-form proof. If a higher-genus counterexample existed, the boundary evaluation that defines the volumes would no longer be the whole answer.

Editorial extensions

If this is right

  • For $N=2$ JT supergravity the method confirms the known genus-1 volume and gives explicit new $V_{2,1}(b)$ and $V_{3,1}(b)$ polynomials that agree with the topological recursion where they overlap.
  • For small $N=4$ JT supergravity the new volumes are polynomials in $b^2$ and in $J$, with the bosonic JT volume sitting at the top power of $J$.
  • For large $N=4$ JT supergravity the one-boundary volumes are exactly the $N=2$ volumes rescaled by $(4\pi^2\omega_\alpha)^{1-2g}$ with $E_0$ replaced by $E_0-E_\pm$.
  • For one-boundary data, the ODE recursion needs only lower-genus one-boundary data rather than the full ladder of $V_{g,n}$, making it faster than standard volume recursions for this slice.
  • The same two-equation machinery works for any model whose tree-level profile $u_0$ obeys the string equation, so it is not specific to supersymmetric JT gravity.

Reading between the lines

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

  • A proof of the total-derivative property at all genera would turn the ODE recursion into a self-contained derivation of polynomiality and degree bounds for $V_{g,1}(b)$, bypassing the standard recursion; Appendix C looks like the right structure for such a proof.
  • The coefficient relations $(2\pi)^p$ and $4^{-p}$ suggest the bosonic subsector is controlled only by the first derivative $u'_0(\mu)$, so any string equation with the same structural form should exhibit the same top-order bosonic JT volume.
  • The large-$N=4$ rescaling relation suggests a sum rule: the full one-boundary volume is the sum of the two $N=2$ sector volumes with shifted thresholds; a mixed-boundary $V_{g,2}$ calculation would test whether the two sectors remain statistically independent beyond one boundary.
  • The threshold energy acts as a deformation parameter connecting the $N=1$ volume at $E_0=0$ to the bosonic volume at the top power; this suggests there may be a geometric interpretation of the intermediate powers as new moduli-space invariants.
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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

3 major / 6 minor

Summary. This paper develops and applies the "ODE method" of ref. [1] to compute one-boundary Weil-Petersson volumes for extended JT supergravity. The method uses the string equation to determine the perturbative solution u(x) and the Gel'fand-Dikii equation for the diagonal resolvent bR(x,E). The central technical step is the assertion, verified by explicit construction for g=1,2,3, that bR_g(x,E) is a total x-derivative when u(x) solves the string equation. With this input, the authors define V_{g,1}(b) from the boundary term at x=mu via Eqs. (4.1)-(4.2). They compute N=2 volumes through genus 3, reproduce V_{1,1} of Turiaci-Witten, and confirm their genus-2 and genus-3 results by an independent evaluation of the Turiaci-Witten recursion in Section 5.2. They then present new predictions for small and large N=4 JT supergravity, including explicit V_{g,1} for g=1,2,3 and a rescaling relation for large N=4. Appendices provide recursion relations for the Gel'fand-Dikii polynomials and a Mathematica implementation.

Significance. If the unproved total-derivative property holds to all orders, the paper provides an efficient and apparently universal ODE-based route to one-boundary Weil-Petersson volumes. The N=2 cross-checks against the independent Turiaci-Witten recursion are a genuine strength: the recursive derivations in Sections 2 and 3 are explicit, the construction of bQ_g in Appendix C is algorithmic, and the genus-2 and genus-3 agreement in Section 5.2 is quantitative. The paper also ships a reproducible Mathematica implementation in Appendix A for Gel'fand-Dikii polynomials. The main significance is therefore a confirmation and extension of known N=2 results plus a set of new N=4 predictions. The significance is tempered by two facts: the general total-derivative property is not proved, and the normalization used for N=4 is chosen so that the bosonic JT volume appears at the highest order in J, so that particular feature is partly conventional rather than an independent prediction.

major comments (3)
  1. [Section 3, Eq. (3.9); Section 1.3] The central structural assumption is that bR_g(x,E)=d_x bQ_g(x,E) for every genus when u(x) solves the string equation. The paper verifies this only for g=1,2,3 by explicit construction and admits in Section 1.3 that no closed-form proof is known. This property is load-bearing because the volume definition in Eqs. (4.1)-(4.2) evaluates only the boundary term at x=mu; without it, the integral over x would depend on the full profile of u_0 and would not be guaranteed to produce the polynomial Weil-Petersson volumes. The supporting argument that polynomiality 'must' force the total-derivative form cannot serve as independent support in the new N=4 cases, where polynomiality is not independently known. The explicit g=1-3 results, including the N=2 check against Turiaci-Witten, are protected by the explicit bQ_g constructed in Eqs. (3.14)-(3.18), so this gap does not invalidate those numbers; however, the statement in Section 7 that the method has 'correctly defined these whole new classes' of volumes goes beyond what is proved. Please either supply a proof of the total-derivative property or clearly present the general property and the N=4 predictions as conditional on a conjecture.
  2. [Section 5.1, after Eq. (5.13); Eq. (4.2)] The normalization K_{g,1}=4^{1-2g} for N=4 is chosen, as the text states, precisely so that the bosonic JT volume appears at the highest order in J. Consequently the observation that the bosonic JT volume is a subsector of the N=4 volumes is, at least in part, a consequence of this normalization convention rather than an independent prediction. This does not affect the numerical content of the volumes, but the paper should distinguish this convention-dependent feature from genuinely derived results, especially in the summary of 'novel properties' in Section 7.
  3. [Section 6.2, Eq. (6.21)] The relation V^{N=4,pm}_{g,1}(b)=V^{N=2}_{g,1}(b)/(4pi^2 omega_alpha)^{2g-1}|_{E0->E0-E_pm} is quoted as exact, but the footnote to this equation states that the denominator X of bQ_g is replaced by X_pm=u0-E_pm-E and that the uniformizing variable must be redefined as E0-E_pm-E=z^2. Since the volume is obtained by an inverse Laplace transform in z, the relation (6.21) requires an explicit demonstration that the substitution E0->E0-E_pm commutes with the Laplace-transform step and with the normalization. Please provide that derivation or state the precise conditions under which (6.21) holds.
minor comments (6)
  1. [Section 1.2] The phrase 'Meanwhile in N = JT supergravity' is missing the subscript 1; it should read 'N=1 JT supergravity'.
  2. [Section 3 and Appendix C] The word 'Anzatz' should be 'ansatz'.
  3. [Section 3] The word 'perturabtively' is a typo for 'perturbatively'.
  4. [Table 1] The caption appears as 'T able 1' in the text; this formatting should be corrected.
  5. [Section 1.1] The word 'tesslation' should be 'tessellation'.
  6. [Section 7] In the sentence about n-point energy (or loop) correlators, 'correlates' should be 'correlators'.

Circularity Check

1 steps flagged · score 2.0 of 10

Mostly self-contained: N=2 results are checked against an independent recursion, and the N=4 volumes follow from explicit bQ_g constructions; the only by-construction element is the N=4 normalization that makes the bosonic leading term exact.

  1. fitted input called prediction [Section 5.1 (after eq. (5.13)); Section 4 eq. (4.2); Section 6.1 and Section 7]
    "Therefore, to ensure we get exactly the bosonic volume at highest order in J we must multiply the bQg by 4^{2g−1} which is precisely what Kg,1 does in equation (4.2) in the N = 4 volume definition."

    The paper later presents the bosonic-JT subsector as an emergent, observed property: 'we see it naturally emerge here for N=4' (Section 7). For N=4, however, the overall normalization K_{g,1}=4^{1−2g} was chosen in eq. (4.2) explicitly so that the highest-order-in-J coefficient equals the bosonic volume. That leading coefficient is therefore not an independent prediction of the method; it is enforced by the normalization convention, while the lower-order terms in J and b are still genuinely computed from the string-equation derivatives and the explicit bQ_g. The circularity is partial and does not affect the central N=2 verification or the polynomial form of the new volumes.

full rationale

The paper's central derivation is self-contained at the orders displayed. The genus-1, genus-2, and genus-3 volumes come from explicit total-derivative expressions bQ_1, bQ_2, bQ_3 (eqs. (3.14)-(3.18)) evaluated at the Fermi surface, with u_0 derivatives fixed by the string equation. The N=2 results are checked twice: V_{1,1}, V_{2,1}, V_{3,1} from the ODE method agree with the independent Turiaci-Witten topological recursion (Section 5.2), including intermediate V_{1,2}, V_{0,4}, V_{1,3}, V_{2,2}; the bosonic JT volumes appear at the expected highest powers and match refs. [27,48,53]. Citations to refs. [1,37,38,52] are inputs to the method, not substitutes for the derivation; no uniqueness theorem is imported to forbid alternatives. The unproved total-derivative conjecture (Section 3, eq. (3.9)) is a genuine gap in generality, but it is not circular for the computed genera, where bQ_g is exhibited by construction. The only by-construction element is the N=4 normalization K_{g,1}=4^{1−2g}, which is chosen to make the bosonic JT volume appear exactly at highest order in J; this is a convention, not a fitted parameter, and it does not contaminate the N=2 agreement or the computation of the remaining polynomial coefficients. Overall, no load-bearing circularity; score 2 reflects the minor normalization convention and the self-referential nature of the new N=4 definitions.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard resolvent theory, the string-equation definitions of the models taken from prior work, the identification of loop correlators with volumes, and the unproven total-derivative property. No new physical entities are introduced.

free parameters (1)
  • Volume normalization K_{g,1} = (2π)^{2g-1} for N=2; 4^{1-2g} for small N=4; (4π^2ω_α)^{2g-1} for large N=4
    Chosen by hand in equation (4.2) and equation (6.21) to match Turiaci-Witten conventions for N=2 and to make the bosonic JT volume appear at highest order in E_0 or J. It rescales the definition of V_{g,1}.
assumptions (5)
  • standard math The Gel'fand-Dikii equation (3.1) and its asymptotic series (1.13) correctly describe the diagonal resolvent of the auxiliary Hamiltonian (1.2).
    Standard result from Gel'fand and Dikii [10], used in ref. [1].
  • domain assumption The string equation (2.1) with the t_k and \\Gamma from refs. [36,37,52,38] defines the matrix models for N=2 and N=4 JT supergravity.
    The paper takes these models as input; it does not re-derive them.
  • domain assumption The boundary term \Q_g(μ, z^2 + E_0) after inverse Laplace transform defines the Weil-Petersson volume V_{g,1}(b) in equations (1.25) and (4.1).
    Identification of the matrix model loop correlator with moduli space volumes, following refs. [1,28,36].
  • ad hoc to paper \R_g(x,E) is a total x-derivative at every genus when u(x) solves the string equation, equation (3.9).
    Unproven; checked at g=1,2,3 via the recursion. Load-bearing: without it, W_{g,1} would not be determined by Fermi-surface data alone and V_{g,1} would not be a polynomial in b. The paper states 'we do not have a closed-form proof' in Section 1.3.
  • domain assumption For small N=4, the parameter J is a non-zero half-integer and the sign choice w = sgn(J)(-1)^{2J+1} is required, as stated in Section 6.1.
    Needed for the Bessel resummation and spectral density match in refs. [37,38].

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Pith. "Pith review of Weil-Petersson volumes for extended JT supergravity from ordinary differential equations." pith.science (2026). https://pith.science/paper/B2YXQECQ

@misc{pith2026250718715,
  author       = {Pith},
  title        = {Pith review of: Weil-Petersson volumes for extended JT supergravity from ordinary differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2YXQECQ}},
  note         = {Machine review of arXiv:2507.18715}
}
abstract

Recent work [1] produced an efficient method for computing Weil-Petersson volumes using two ordinary differential equations (ODEs) that appear naturally in double scaled random matrix models. One is the defining string equation of the model and the other is the Gel'fand-Dikii equation satisfied by the diagonal resolvent of an auxiliary Hamiltonian used to compute correlators of macroscopic loops. In concert, when applied to Jackiw-Teitelboim gravity, the recursive expansion of these two ODEs efficiently define, order by order in genus, the Weil-Petersson volumes $V_{g,1}(b)$ for bordered hyperbolic Riemann surfaces with one geodesic boundary (length $b$) and genus $g$. The method works equally well for both ordinary and ${ N}{=}1$ supersymmetric JT gravity cases. This paper explores the method to higher genus, verifying some conjectures of ref.[1], and deriving several useful recursive formulae for general use. The method is then applied to the new examples furnished by recent matrix model definitions of JT supergravity with extended supersymmetry, and several example expressions for the volumes are derived, confirming and extending ${ N}{=}2$ examples of Turiaci and Witten, and furnishing new formulae for the cases with small and large ${ N}{=}4$ supersymmetry. The prospects for extending the ODE method to the full set of $V_{g,n}(\{ b_i\})$, ($i=1,\ldots ,n$), are discussed.

Figures

Figures reproduced from arXiv: 2507.18715 by the authors.

Figure 1
Figure 1. The connected correlator of multiple loops, ⟨Z(β1)· · ·Z(βn)⟩, can be written in terms of Vg,n({bi}), the volume of the moduli space of a prototype surface of genus g with n boundaries. Riemann surfaces with geodesic boundaries of lengths {bi}. Much has been learned about these volumes 5 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Schematic depiction of gluing a trumpet to a bordered higher genus Riemann surface. In fact, for genus g = 0, where n-point connected correlators of Z(β) are given by the simple formula [9, 41]: ⟨Z(β1)· · ·Z(βn)⟩ = √ β1 · · · βn 2π n 2 βT h ∂ n−2 x e −βT u0(x) i x=µ , (1.29) where βT ≡ P i βi , it is straightforward [12] to use the trumpet to convert this into expressions 9 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. (a) The typical N =1 behaviour of a density. (b) The possible N >1 situation with some BPS states at E=0 and a non-BPS sector beginning at some threshold energy E0. its derivatives, which solves the leading equation (obtained by setting ℏ=0 in (2.1)): u0R2 0 = Γe2 , (2.4) where: R0 ≡ G0 + x , with G0 ≡ X∞ k=1 tku k 0 . (2.5) Working to order O(ℏ 2 ) to begin with, we have: R2 = u2G˙ 0 − u ′′ 0 6 G¨ 0 − u ′2 0 12 ...… view at source ↗

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Works this paper leans on

56 extracted references · 26 canonical work pages

  1. [1]

    C. V. Johnson,Random matrix model of the Virasoro minimal string, Phys. Rev. D110 (2024) 066015 [2401.06220]

    arXiv 2024

  2. [2]

    Brezin and V

    E. Brezin and V. A. Kazakov,Exactly Solvable Field Theories Of Closed Strings, Phys. Lett. B236 (1990) 144

    work page 1990

  3. [3]

    M. R. Douglas and S. H. Shenker,Strings In Less Than One-Dimension, Nucl. Phys.B335 (1990) 635

    work page 1990

  4. [4]

    D. J. Gross and A. A. Migdal,Nonperturbative Two-Dimensional Quantum Gravity, Phys. Rev. Lett.64 (1990) 127

    work page 1990

  5. [5]

    ’t Hooft,A planar diagram theory for strong interactions, Nucl

    G. ’t Hooft,A planar diagram theory for strong interactions, Nucl. Phys.B72 (1974) 461

    work page 1974

  6. [6]

    Brezin, C

    E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber,Planar diagrams, Commun. Math. Phys.59 (1978) 35

    work page 1978

  7. [7]

    Bessis, C

    D. Bessis, C. Itzykson and J. B. Zuber,Quantum field theory techniques in graphical enumeration, Adv. Appl. Math.1 (1980) 109

    work page 1980

  8. [8]

    E. P. Wigner,Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math.62 (1955) 548

    work page 1955

Show all 56 references
  1. [9]

    Banks, M

    T. Banks, M. R. Douglas, N. Seiberg and S. H. Shenker,Microscopic And Macroscopic Loops In Nonperturbative Two- Dimensional Gravity, Phys. Lett.B238 (1990) 279. 54

    1990

  2. [10]

    I. M. Gel’fand and L. A. Dikii,Asymptotic behavior of the resolvent of sturm-liouville equations and the algebra of the korteweg-de vries equations, Russ. Math. Surveys30 (1975) 77

    1975

  3. [11]

    V. A. Kazakov,The appearance of matter fields from quantum fluctuations of 2-d gravity, Mod. Phys. Lett.A4 (1989) 2125

    1989

  4. [12]

    T. G. Mertens and G. J. Turiaci,Liouville quantum gravity – holography, JT and matrices, JHEP 01 (2021) 073 [2006.07072]

    2021 arXiv

  5. [13]

    Jackiw,Lower Dimensional Gravity, Nucl

    R. Jackiw,Lower Dimensional Gravity, Nucl. Phys.B252 (1985) 343

    1985

  6. [14]

    Teitelboim,Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys

    C. Teitelboim,Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett.126B (1983) 41

    1983

  7. [15]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten,Developments in Topological Gravity, Int. J. Mod. Phys. A33 (2018) 1830029 [1804.03275]

    2018 arXiv

  8. [16]

    Okuyama and K

    K. Okuyama and K. Sakai,JT gravity, KdV equations and macroscopic loop operators, JHEP 01 (2020) 156 [1911.01659]

    2020 arXiv

  9. [17]

    C. V. Johnson,Nonperturbative Jackiw-Teitelboim gravity, Phys. Rev. D101 (2020) 106023 [1912.03637]

    2020 arXiv

  10. [18]

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

    1903 arXiv

  11. [19]

    C. V. Johnson,Jackiw-Teitelboim supergravity, minimal strings, and matrix models, Phys. Rev. D 103 (2021) 046012 [2005.01893]

    2021 arXiv

  12. [20]

    T. R. Morris,2-d quantum gravity, multicritical matter and complex matrices,

  13. [21]

    Dalley, C

    S. Dalley, C. V. Johnson and T. Morris,Multicritical complex matrix models and nonperturbative 2-d quantum gravity, Nucl. Phys.B368 (1992) 625

    1992

  14. [22]

    Dalley, C

    S. Dalley, C. V. Johnson, T. R. Morris and A. Watterstam,Unitary matrix models and 2-d quantum gravity, Mod. Phys. Lett.A7 (1992) 2753 [hep-th/9206060]

    1992 arXiv

  15. [23]

    C. V. Johnson,Explorations of nonperturbative Jackiw-Teitelboim gravity and supergravity, Phys. Rev. D103 (2021) 046013 [2006.10959]

    2021 arXiv

  16. [24]

    Witten,On the Structure of the Topological Phase of Two-dimensional Gravity, Nucl

    E. Witten,On the Structure of the Topological Phase of Two-dimensional Gravity, Nucl. Phys. B 340 (1990) 281

    1990

  17. [25]

    Distler,2-D Quantum Gravity, Topological Field Theory and the Multicritical Matrix Models, Nucl

    J. Distler,2-D Quantum Gravity, Topological Field Theory and the Multicritical Matrix Models, Nucl. Phys. B342 (1990) 523. 55

    1990

  18. [26]

    Witten,Two-dimensional gravity and intersection theory on moduli space, Surveys Diff

    E. Witten,Two-dimensional gravity and intersection theory on moduli space, Surveys Diff. Geom. 1 (1991) 243

    1991

  19. [27]

    Mirzakhani,Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent

    M. Mirzakhani,Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math.167 (2006) 179

    2006

  20. [28]

    Eynard and N

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

    arXiv

  21. [29]

    Stanford and E

    D. Stanford and E. Witten,JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys.24 (2020) 1475 [1907.03363]

    2020 arXiv

  22. [30]

    C. V. Johnson,Supersymmetric Virasoro minimal strings, Phys. Rev. D110 (2024) 066016 [2401.08786]

    2024 arXiv

  23. [31]

    Förste, J

    S. Förste, J. Kames-King and M. Wiesner,Towards the Holographic Dual of N = 2 SYK, JHEP 03 (2018) 028 [1712.07398]

    2018 arXiv

  24. [32]

    Stanford and E

    D. Stanford and E. Witten,Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [1703.04612]

    2017 arXiv

  25. [33]

    T. G. Mertens, G. J. Turiaci and H. L. Verlinde,Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [1705.08408]

    2017 arXiv

  26. [34]

    Heydeman, L

    M. Heydeman, L. V. Iliesiu, G. J. Turiaci and W. Zhao,The statistical mechanics of near-BPS black holes, J. Phys. A55 (2022) 014004 [2011.01953]

    2022 arXiv

  27. [35]

    Heydeman, X

    M. Heydeman, X. Shi and G. J. Turiaci,Can black holes preserveN >4 supersymmetry?, 2504.20146

  28. [36]

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

    2023 arXiv

  29. [37]

    C. V. Johnson and M. Usatyuk,God of the Gaps: Random matrix models and the black hole spectral gap, 2407.17583

    arXiv

  30. [38]

    C. V. Johnson and M. Kolanowski,Extended JT supergravity and random matrix models: The power of the string equation, 2507.07185

    arXiv

  31. [39]

    Lowenstein,Open-closed string duality, branes, and topological recursion, JHEP 07 (2024) 056 [2404.13175]

    A. Lowenstein,Open-closed string duality, branes, and topological recursion, JHEP 07 (2024) 056 [2404.13175]

    2024 arXiv

  32. [40]

    Wolpert,On the kähler form of the moduli space of once punctured tori, Commentarii Mathematici Helvetici 58 (1983) 246

    S. Wolpert,On the kähler form of the moduli space of once punctured tori, Commentarii Mathematici Helvetici 58 (1983) 246. 56

    1983

  33. [41]

    P. H. Ginsparg and G. W. Moore,Lectures on 2-D gravity and 2-D string theory, inTheoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, pp. 277–469, 10, 1993, hep-th/9304011

    1993 arXiv

  34. [42]

    Brezin and A

    E. Brezin and A. Zee,Universality of the correlations between eigenvalues of large random matrices, Nucl. Phys. B402 (1993) 613

    1993

  35. [43]

    C. W. J. Beenakker,Universality in the random-matrix theory of quantum transport, Phys. Rev. Lett.70 (1993) 1155

    1993

  36. [44]

    Beenakker,Universality of Brézin and Zee’s spectral correlator, Nuclear Physics B422 (1994) 515 [cond-mat/9310010]

    C. Beenakker,Universality of Brézin and Zee’s spectral correlator, Nuclear Physics B422 (1994) 515 [cond-mat/9310010]

    1994 arXiv

  37. [45]

    Jancovici and P

    B. Jancovici and P. J. Forrester,Derivation of an asymptotic expression in Beenakker’s general fluctuation formula for random-matrix correlations near an edge, Phys. Rev. B50 (1994) 14599

    1994

  38. [46]

    Nakanishi and M

    T. Nakanishi and M. Näätänen,Areas of two-dimensional moduli spaces, Proceedings of the American Mathematical Society129 (2001) 3241

    2001

  39. [47]

    P. G. Zograf,The weil-petersson volume of the moduli space of punctured spheres, inMapping class groups and moduli spaces of Riemann surfaces: proceedings of workshops, vol. 150 of Contemporary Mathematics, pp. 367–372. American Mathematical Society, Providence, RI, 1993

    1993

  40. [48]

    R. C. Penner,Weil-petersson volumes, Journal of Differential Geometry35 (1992) 559

    1992

  41. [49]

    Collier, L

    S. Collier, L. Eberhardt, B. Mühlmann and V. A. Rodriguez,The Virasoro Minimal String, 2309.10846

    arXiv

  42. [50]

    C. V. Johnson,Further aspects of Supersymmetric Virasoro Minimal Strings, 2506.19000

    arXiv

  43. [51]

    C. V. Johnson,A differential equation for a class of correlation kernels, 2505.10622

    arXiv

  44. [52]

    C. V. Johnson,Nonperturbative random matrix model of N=2 JT supergravity, Phys. Rev. D110 (2024) 106019 [2306.10139]

    2024 arXiv

  45. [53]

    Do,Moduli spaces of hyperbolic surfaces and their weil-petersson volumes, 1103.4674

    N. Do,Moduli spaces of hyperbolic surfaces and their weil-petersson volumes, 1103.4674

    arXiv

  46. [54]

    Eynard,A short overview of the ”Topological recursion”, 1412.3286

    B. Eynard,A short overview of the ”Topological recursion”, 1412.3286

    arXiv

  47. [55]

    Lowenstein,Exact Expressions For Infinitely Many Weil-Petersson Volumes, 2407.16039

    A. Lowenstein,Exact Expressions For Infinitely Many Weil-Petersson Volumes, 2407.16039

    arXiv

  48. [56]

    C. V. Johnson, F. Rosso and A. Svesko,Jackiw-Teitelboim supergravity as a double-cut matrix model, Phys. Rev. D104 (2021) 086019 [2102.02227]. 57

    2021 arXiv

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Reviewed August 15, 2026 · model on record in the stance chip above.