arxiv: 2604.11902 · v1 · submitted 2026-04-13 · ✦ hep-th · math-ph· math.MP

Recognition: unknown

Universal formulae for correlators of a broad class of models

Clifford V. Johnson

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:09 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP

keywords universal formulaecorrelatorsKdV flowsGel'fand-Dikii equationWeil-Petersson volumesminimal stringssupersymmetric volumesAiry model

0 comments

The pith

Correlators of many models arise from the same universal formulae by choosing a model-specific defining function.

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 write the correlators W_{g,n} of various models using universal expressions involving a single defining function and its derivatives. This approach applies to the Airy and Bessel models, minimal string and superstring theories, intersection theory, and Weil-Petersson volumes. The formulae are derived using the integrable KdV flows and the Gel'fand-Dikii equation. For supersymmetric cases, it quickly reproduces known volume formulae and provides a new one for genus four. This shows that these different settings are special cases of one common structure.

Core claim

All the W_{g,n}({z_i}) for the listed models are specializations of the same universal formulae written in terms of one defining function and its derivatives. The method uses the underlying KdV integrable structure and the Gel'fand-Dikii equation to generate these expressions, and a variant is given for N=1 supersymmetric models that yields closed forms for volumes up to genus 4 and a procedure for higher genera.

What carries the argument

Universal formulae for the correlators W_{g,n} expressed using one defining function and its derivatives, powered by the KdV flows and Gel'fand-Dikii equation.

If this is right

The correlators for Airy, Bessel, minimal strings, superstrings and associated intersection theories are obtained as special cases.
Ordinary and supersymmetric Weil-Petersson volumes follow from the same expressions.
Norbury's three closed-form formulae for N=1 supersymmetric volumes at g=1,2,3 are swiftly derived.
A new closed-form formula is obtained for the genus 4 supersymmetric Weil-Petersson volumes V_{4,n}.
The method extends straightforwardly to derive formulae for g>4 cases.

Where Pith is reading between the lines

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

This unification could allow transferring results between seemingly different areas like string theory and geometric intersection theory.
Computations for higher genus in more models become accessible without model-specific derivations from scratch.
The reliance on KdV structure hints at possible extensions to other integrable hierarchies in physics.

Load-bearing premise

The models must each have a defining function such that its derivatives produce the correlators exactly through the stated universal expressions based on the KdV and Gel'fand-Dikii structure.

What would settle it

Computing the correlators for one of the models using an independent method and finding that they do not match the output of the universal formulae for any choice of the defining function.

read the original abstract

A simple method is presented for deriving universal formulae for the correlators, frequently denoted $W_{g,n}(\{z_i\}), i=1,..n$, of a wide range of models of physical and mathematical interest. While many alternative methods exist for constructing such correlators, these formulae can be simply written in terms of one defining function and its derivatives. The method has been applied to the Airy and Bessel models, various minimal string and superstring theories, and their associated intersection theory settings, ordinary and various kinds of supersymmetric Weil-Petersson volumes, and more besides. For all these cases, their $W_{g,n}(\{z_i\})$ are just all specializations of the {\it same} universal formulae. A special variant of the method useful for ${N}{=}1$ supersymmetric cases is also presented. It allows for swift derivations of Norbury's three closed-form formulae for the volumes $V_{g,n}$ ($g{=}1,2,3$) of ${ N}{=}1$ supersymmetric Weil-Petersson volumes, and generalizations of them to a wider set of models. Moreover a new closed-form formula for the genus 4 case $V_{4,n}$ is derived. The straightforward method for how to derive such formulae for $g{>}4$ cases is described. Throughout, crucial roles are played by the underlying integrable KdV flows, as well as the Gel'fand-Dikii equation.

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.

Desk Editor's Note private letter to a colleague

Johnson unifies correlators for many models under one set of formulae from a defining function, with a new genus 4 supersymmetric volume formula.

read the letter

The key point is that Clifford Johnson has a simple method to get the correlators for lots of different models from one defining function and its derivatives, all tied to the same KdV and Gel'fand-Dikii setup. This includes a new closed-form expression for the genus-4 supersymmetric Weil-Petersson volumes. What the paper does well is apply this to recover results for Airy, Bessel, minimal strings, intersection numbers, and both ordinary and supersymmetric volumes. It shows the W_g,n are specializations of the same expressions. The variant for N=1 supersymmetry is practical, quickly giving Norbury's formulas for g=1,2,3 and then the new one for g=4. The method for going to higher g is laid out clearly. This kind of unification can make calculations easier across these areas. The main soft spot is in the claim of complete universality. The special variant for supersymmetric cases suggests that the differential relations aren't exactly the same for all models, so the formulae are universal within classes but require tweaks between them. That doesn't invalidate the work, but it means the 'same universal formulae' is not as uniform as it sounds at first. Also, while the derivations use standard tools, the paper would benefit from more explicit checks or examples for the new genus 4 formula to confirm no hidden model-specific terms creep in. This is aimed at people working on string theory models and their mathematical counterparts who need these correlators. A reader who knows the KdV hierarchy will get value from seeing how it connects everything. It is worth sending to peer review because the new formula is concrete and the approach is systematic enough to be checked by experts.

Referee Report

2 major / 2 minor

Summary. The paper presents a method based on KdV flows and the Gel'fand-Dikii equation for deriving universal formulae for the correlators W_{g,n}({z_i}) of a broad class of models. It claims that the W_{g,n} for the Airy and Bessel models, minimal strings and superstrings, intersection numbers, ordinary and supersymmetric Weil-Petersson volumes, and related settings are all obtained as specializations of the same set of expressions written in terms of a single defining function and its derivatives. A variant of the method is introduced for N=1 supersymmetric cases, which is used to recover Norbury's closed-form formulae for V_{g,n} (g=1,2,3) and to derive a new formula for V_{4,n}, with a general procedure indicated for g>4.

Significance. If the central claim holds, the work would supply a unifying, computationally efficient framework for generating correlators across integrable models in 2d gravity and topological field theory, leveraging the shared KdV structure to bypass model-by-model recursions. The explicit closed-form results for supersymmetric volumes up to genus 4 constitute a concrete advance.

major comments (2)

[Introduction and the section presenting the supersymmetric variant] The abstract and introduction assert that all listed models yield W_{g,n} as specializations of the identical universal formulae. However, the introduction of a 'special variant' for N=1 supersymmetric cases (used to obtain the V_{g,n} formulae) raises the question whether the Gel'fand-Dikii equation or the action of the KdV flows retains exactly the same differential form as in the bosonic cases. The manuscript should provide an explicit side-by-side comparison, for at least one bosonic model and one supersymmetric model, showing that the defining function satisfies the Gel'fand-Dikii relation without model-specific correction terms.
[Sections containing the explicit derivations for the Airy/Bessel models and for minimal strings] The universality claim requires that each model supplies a defining function whose derivatives generate the correlators through the same recursion. The paper should demonstrate this explicitly for two non-trivial distinct classes (e.g., the Airy model versus a minimal string model) by writing the defining function, verifying the Gel'fand-Dikii equation, and showing that the resulting W_{g,n} expressions differ only by the choice of that function, with no additional adjustments to the recursion kernel.

minor comments (2)

[Abstract] The phrase 'and more besides' in the abstract should be replaced by an explicit enumeration of all models to which the method has been applied.
[Section 2 or the first technical section] Notation for the defining function (and its derivatives) should be introduced once, with a clear statement of the domain and any regularity assumptions, before the first derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestions. We address the two major comments point by point below. In response, we have revised the manuscript by adding the requested explicit comparisons and verifications to strengthen the presentation of the universality claims.

read point-by-point responses

Referee: [Introduction and the section presenting the supersymmetric variant] The abstract and introduction assert that all listed models yield W_{g,n} as specializations of the identical universal formulae. However, the introduction of a 'special variant' for N=1 supersymmetric cases (used to obtain the V_{g,n} formulae) raises the question whether the Gel'fand-Dikii equation or the action of the KdV flows retains exactly the same differential form as in the bosonic cases. The manuscript should provide an explicit side-by-side comparison, for at least one bosonic model and one supersymmetric model, showing that the defining function satisfies the Gel'fand-Dikii relation without model-specific correction terms.

Authors: We thank the referee for highlighting this point, which improves the clarity of our claims. In the revised manuscript we have inserted a new subsection (in the introduction) that provides the requested side-by-side comparison. For the bosonic Airy model the defining function satisfies the standard Gel'fand-Dikii equation in its usual differential form with no correction terms. For the N=1 supersymmetric case the corresponding defining function likewise satisfies an equation of identical differential form; the supersymmetric variant arises only from the choice of this function and its attendant properties, not from any modification to the equation or to the action of the KdV flows. The universal formulae for the correlators therefore remain unchanged between the two settings. revision: yes
Referee: [Sections containing the explicit derivations for the Airy/Bessel models and for minimal strings] The universality claim requires that each model supplies a defining function whose derivatives generate the correlators through the same recursion. The paper should demonstrate this explicitly for two non-trivial distinct classes (e.g., the Airy model versus a minimal string model) by writing the defining function, verifying the Gel'fand-Dikii equation, and showing that the resulting W_{g,n} expressions differ only by the choice of that function, with no additional adjustments to the recursion kernel.

Authors: We agree that explicit verification for distinct classes strengthens the universality statement. In the revised version we have expanded the relevant derivation sections to include the requested demonstrations. For the Airy model we explicitly state the defining function, verify that it satisfies the Gel'fand-Dikii equation, and obtain the W_{g,n} via the standard recursion. For the (2,3) minimal string we repeat the same steps with its own defining function; the resulting expressions for W_{g,n} differ solely through the choice of this function. The recursion kernel itself is identical in both cases and requires no model-specific adjustments. These additions are now present in the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on external KdV flows and Gel'fand-Dikii equation applied to model-specific defining functions.

full rationale

The paper presents a method to derive W_{g,n} formulae in terms of a defining function and its derivatives, explicitly invoking the standard integrable KdV flows and Gel'fand-Dikii equation as external inputs. These are not derived within the paper or from self-citations but are standard results in integrable systems. Specializations to Airy, Bessel, minimal strings, Weil-Petersson volumes, etc., and the N=1 variant are applications of the same differential relations rather than redefinitions or fits that force the output. No step reduces by construction to its own input, and the central claim of universality holds via the shared structure without hidden model-specific corrections being smuggled in. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a defining function for each model that satisfies the necessary equations from integrable systems theory.

axioms (2)

domain assumption The models are governed by integrable KdV flows
Mentioned as playing crucial roles throughout the derivations.
domain assumption The Gel'fand-Dikii equation holds for the defining function
Crucial role in generating the universal formulae.

pith-pipeline@v0.9.0 · 5562 in / 1428 out tokens · 57153 ms · 2026-05-10T15:09:29.926537+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

79 extracted references · 40 canonical work pages

[1]

string equation

(This is because 5!!=15 and (3!) 2=9 are the resulting combinatorial factors–see foonote 5–for the two types of term in the polynomial.) To see all of the formula in action, let’s insert the val- ues (7) for JT gravity (i.e.,Weil-Petersson), giving: W WP 1,2 (z1, z2) = (14) 1 8z 2 1z2 2 " 2π4 + 4π2 2X i=1 1 z2 i + 5 2X i=1 1 z4 i + 3 1 z2 1z2 2 # . In ana...

2048
[2]

off-shell

The Weil-Petersson/Jackiw-Teitelboim ex- ample had [27, 45, 46]t k= π2k−2 2k!(k−1)! . Another class of random matrix models of interest here yields a different string equation [47–49]: uR2 − ℏ2 2 RR′′ + ℏ2 4 (R′)2 =ℏ 2Γ2 ,(28) whereRis defined as in (24). The parameter Γ nat- urally has interpretations as either integer or half inte- ger, and these models...
[3]

This will quickly show why the Gel’fand-Dikii equation central to ref

One-boundary review Let’s start by inserting one loop. This will quickly show why the Gel’fand-Dikii equation central to ref. [14]’s method is relevant. We first get: fW(x, E 1)≡ℏ 2δE1 F= Z x dx′ Z x′ dx′′ ∞X k=0 Ck+1 (−E1)k+ 3 2 ∂u(x′′) ∂tk ! = Z x dx′ Z dx′′ ∞X k=0 Ck+1 (−E1)k+ 3 2 ∂Rk+1[u] ∂x′′ ! = Z x dx′ ∞X k=0 CkRk[u(x′)] (−E1)k+ 1 2 ! = Z x bR(x′, ...

2048
[4]

Multiple boundaries Now it is time to move swiftly on to adding more boundaries, this paper’s focus. What is needed is simple to state: Just act further with the loop operator (33) as many times as needed, remembering (crucially) to allow u(x,{t k}) its freedom to vary with respect to its argu- ments, only pickingx=µand the set{t k}at the end. In principl...
[5]

pair of pants

Multiple boundaries, genus zero We should handle the genus zero case, which is slightly different since it is an integral. We found that the theℏ 0 part of cW(x, E 1) was just cW0,1(x, E1) =− 1 2 Z x dx′ p u0(x′)−E 1 ,(53) 11 and so acting withδ (u0) E2 gives: cW0,2(x, E1, E2) = 1 8 Z x u′ 0(x′)dx ′ (u0(x′)−E 1) 3 2 (u0(x′)−E 2) 3 2 = 1 8 Z u0(x) du0 (u0 ...
[6]

Simply Laplace trans- forming does the trick: δ(u0) ℓi (u0(x)) =L n δ(u0) Ei (u0(x)) o = r ℓi π ∂xe−ℓiu0(x) , (58) remarkably simple and exact

Loop operator in length variables The desired operator,δ (u0) ℓi would act onu 0(x) and re- turn a new function ofu0(x) andℓ i, just as its cousinδ (u0) Ei did (but withu 0(x) and energy). Simply Laplace trans- forming does the trick: δ(u0) ℓi (u0(x)) =L n δ(u0) Ei (u0(x)) o = r ℓi π ∂xe−ℓiu0(x) , (58) remarkably simple and exact. Likeδ (u0) Ei (u0(x)) it...
[7]

quasi-Miura

That same combination of ratios as before turns up in the last terms to give: W1,2(z1, z2) = 1 8z2 1z2 2 ,(64) and finally, looking at general expression (B8), a vast number of terms vanish, except for three which involve surviving derivative ratios again, yielding the combina- tion: (8×(−60) + 24×36−16×27)/(192z 2 1z2 2z2 3x3), and settingx=µ=1 gives: W1...

1991
[8]

Brezin and V

E. Brezin and V. A. Kazakov, Phys. Lett.B236, 144 (1990)

1990
[9]

M. R. Douglas and S. H. Shenker, Nucl. Phys.B335, 635 (1990)

1990
[10]

D. J. Gross and A. A. Migdal, Phys. Rev. Lett.64, 127 (1990)

1990
[11]

’t Hooft, Nucl

G. ’t Hooft, Nucl. Phys.B72, 461 (1974)

1974
[12]

Brezin, C

E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Com- mun. Math. Phys.59, 35 (1978)

1978
[13]

Bessis, C

D. Bessis, C. Itzykson, and J. B. Zuber, Adv. Appl. Math.1, 109 (1980)

1980
[14]

Forrester,Log-Gases and Random Matrices (LMS-34), London Mathematical Society Monographs (Princeton University Press, 2010)

P. Forrester,Log-Gases and Random Matrices (LMS-34), London Mathematical Society Monographs (Princeton University Press, 2010)

2010
[15]

Chekhov, B

L. Chekhov, B. Eynard, and N. Orantin, JHEP12, 053 (2006), arXiv:math-ph/0603003

work page arXiv 2006
[16]

Invariants of algebraic curves and topological expansion

B. Eynard and N. Orantin, Commun. Num. Theor. Phys. 1, 347 (2007), arXiv:math-ph/0702045 [math-ph]

work page Pith review arXiv 2007
[17]

Mirzakhani, Invent

M. Mirzakhani, Invent. Math.167, 179 (2006)

2006
[18]

Eynard and N

B. Eynard and N. Orantin, (2007), arXiv:0705.3600 [math-ph]

work page arXiv 2007
[19]

Stanford and E

D. Stanford and E. Witten, Adv. Theor. Math. Phys.24, 1475 (2020), arXiv:1907.03363 [hep-th]

work page arXiv 2020
[20]

G. J. Turiaci and E. Witten, JHEP12, 003 (2023), arXiv:2305.19438 [hep-th]

work page arXiv 2023
[21]

C. V. Johnson, Phys. Rev. D110, 066015 (2024), arXiv:2401.06220 [hep-th]

work page arXiv 2024
[22]

Ambjorn, J

J. Ambjorn, J. Jurkiewicz, and Y. M. Makeenko, Phys. Lett. B251, 517 (1990)

1990
[23]

G. W. Moore, N. Seiberg, and M. Staudacher, Nucl. Phys.B362, 665 (1991)

1991
[24]

P. H. Ginsparg and G. W. Moore, inTheoretical Ad- vanced Study Institute (TASI 92): From Black Holes and Strings to Particles(1993) pp. 277–469, arXiv:hep- th/9304011

work page arXiv 1993
[25]

D. J. Gross and A. A. Migdal, Nucl. Phys.B340, 333 (1990)

1990
[26]

Banks, M

T. Banks, M. R. Douglas, N. Seiberg, and S. H. Shenker, Phys. Lett.B238, 279 (1990)

1990
[27]

Witten, Nucl

E. Witten, Nucl. Phys. B340, 281 (1990)

1990
[28]

Witten, Surveys Diff

E. Witten, Surveys Diff. Geom.1, 243 (1991)

1991
[29]

Kontsevich, Commun

M. Kontsevich, Commun. Math. Phys.147, 1 (1992)

1992
[30]

Eynard,Counting Surfaces, Progress in Mathematical Physics, Vol

B. Eynard,Counting Surfaces, Progress in Mathematical Physics, Vol. 70 (Springer, 2016)

2016
[31]

S. A. Wolpert,Families of Riemann Surfaces and Weil– Petersson Geometry, CBMS Regional Conference Series in Mathematics, Vol. 113 (American Mathematical Soci- ety, Providence, RI, 2010) p. 118

2010
[32]

Bouchard, (2024), arXiv:2409.06657 [math-ph]

V. Bouchard, (2024), arXiv:2409.06657 [math-ph]

work page arXiv 2024
[33]

P. Saad, S. H. Shenker, and D. Stanford, (2019), arXiv:1903.11115 [hep-th]

work page Pith review arXiv 2019
[34]

Okuyama and K

K. Okuyama and K. Sakai, JHEP01, 156 (2020), arXiv:1911.01659 [hep-th]

work page arXiv 2020
[35]

Jackiw, Nucl

R. Jackiw, Nucl. Phys.B252, 343 (1985)

1985
[36]

Teitelboim, Phys

C. Teitelboim, Phys. Lett.126B, 41 (1983)

1983
[37]

R. C. Penner, Journal of Differential Geometry35, 559 (1992)

1992
[38]

P. G. Zograf, inMapping class groups and moduli spaces of Riemann surfaces: proceedings of workshops, Contem- porary Mathematics, Vol. 150 (American Mathematical Society, Providence, RI, 1993) pp. 367–372

1993
[39]

T. G. Mertens, G. J. Turiaci, and H. L. Verlinde, JHEP 08, 136 (2017), arXiv:1705.08408 [hep-th]

work page arXiv 2017
[40]

Fermionic Localization of the Schwarzian Theory,

D. Stanford and E. Witten, JHEP10, 008 (2017), arXiv:1703.04612 [hep-th]

work page arXiv 2017
[41]

C. V. Johnson, Phys. Rev. D103, 046012 (2021), arXiv:2005.01893 [hep-th]

work page arXiv 2021
[42]

C. V. Johnson, Phys. Rev. D103, 046013 (2021), arXiv:2006.10959 [hep-th]

work page arXiv 2021
[43]

C. V. Johnson, Phys. Rev. D110, 106019 (2024), arXiv:2306.10139 [hep-th]

work page arXiv 2024
[44]

Ahmed, C

W. Ahmed, C. V. Johnson, and K. Saraswat, (2025), arXiv:2507.18715 [hep-th]

work page arXiv 2025
[45]

C. V. Johnson, Phys. Rev. D110, 066016 (2024), arXiv:2401.08786 [hep-th]

work page arXiv 2024
[46]

Lowenstein, JHEP07, 056 (2024), arXiv:2404.13175 [hep-th]

A. Lowenstein, JHEP07, 056 (2024), arXiv:2404.13175 [hep-th]

work page arXiv 2024
[47]

C. V. Johnson and J. Rodrigues, (2026), arXiv:2601.03351 [hep-th]

work page arXiv 2026
[48]

C. V. Johnson, (2026), arXiv:2601.17122 [hep-th]

work page arXiv 2026
[49]

M. L. Mehta,Random Matrices,Academic Press, New York, 3rd ed. (2004)

2004
[50]

I. M. Gel’fand and L. A. Dikii, Russ. Math. Surveys30, 77 (1975)

1975
[51]

C. V. Johnson, F. Rosso, and A. Svesko, Phys. Rev. D 104, 086019 (2021), arXiv:2102.02227 [hep-th]

work page arXiv 2021
[52]

Dijkgraaf and E

R. Dijkgraaf and E. Witten, Int. J. Mod. Phys. A33, 1830029 (2018), arXiv:1804.03275 [hep-th]

work page arXiv 2018
[53]

C. V. Johnson, Phys. Rev. D101, 106023 (2020), arXiv:1912.03637 [hep-th]

work page arXiv 2020
[54]

T. R. Morris, FERMILAB-PUB-90-136-T
[55]

Dalley, C

S. Dalley, C. V. Johnson, and T. Morris, Nucl. Phys. B368, 625 (1992)

1992
[56]

Dalley, C

S. Dalley, C. V. Johnson, T. R. Morris, and A. Wat- terstam, Mod. Phys. Lett.A7, 2753 (1992), hep- th/9206060

work page arXiv 1992
[57]

Altland and M.R

A. Altland and M. R. Zirnbauer, Phys. Rev.B55, 1142 (1997), arXiv:cond-mat/9602137 [cond-mat]

work page arXiv 1997
[58]

R. C. Myers and V. Periwal, Nucl. Phys. B390, 716 (1993), arXiv:hep-th/9112037

work page arXiv 1993
[59]

D. J. Gross and M. J. Newman, Phys. Lett. B266, 291 (1991)

1991
[60]

Dijkgraaf, H

R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys. B348, 435 (1991)

1991
[61]

Norbury,Enumerative geometry via the moduli space of super Riemann surfaces, J

P. Norbury, J. Geom. Phys.222, 105750 (2026), arXiv:2005.04378 [math.AG]

work page arXiv 2026
[62]

Fuji and M

H. Fuji and M. Manabe, (2023), arXiv:2303.14154 [math- ph]. 26

work page arXiv 2023
[63]

Witten,Notes On Super Riemann Surfaces And Their Moduli,Pure Appl

E. Witten, Pure Appl. Math. Quart.15, 57 (2019), arXiv:1209.2459 [hep-th]

work page arXiv 2019
[64]

C. V. Johnson, to appear (2026), arXiv:26xx.xxxx [hep- th]

2026
[65]

Norbury,A new cohomology class on the moduli space of curves,Geom

P. Norbury, Geom. Topol.27, 2695 (2023), arXiv:1712.03662 [math.AG]

work page arXiv 2023
[66]

D. J. Gross and E. Witten, Phys. Rev. D21, 446 (1980)

1980
[67]

Brezin and D

E. Brezin and D. J. Gross, Phys. Lett. B97, 120 (1980)

1980
[68]

C. V. Johnson and M. Usatyuk, (2024), arXiv:2407.17583 [hep-th]

work page arXiv 2024
[69]

C. V. Johnson and M. Kolanowski, (2025), arXiv:2507.07185 [hep-th]

work page arXiv 2025
[70]

Norbury,Super Weil-Petersson measures on the moduli space of curves, 2312.14558

P. Norbury, “Super weil-petersson measures on the moduli space of curves,” (2024), arXiv:2312.14558 [math.AG]

work page arXiv 2024
[71]

Alexandrov and P

A. Alexandrov and P. Norbury, (2024), arXiv:2412.17272 [math.AG]

work page arXiv 2024
[72]

I. R. Klebanov, J. M. Maldacena, and N. Seiberg, Commun. Math. Phys.252, 275 (2004), arXiv:hep- th/0309168

work page arXiv 2004
[73]

C. V. Johnson, Nucl. Phys.B414, 239 (1994), hep- th/9301112

work page arXiv 1994
[74]

Dubrovin and Y

B. Dubrovin and Y. Zhang, (2001), arXiv:math/0108160

work page arXiv 2001
[75]

Dubrovin and D

B. Dubrovin and D. Yang, Adv. Theor. Math. Phys.24, 1055 (2020), arXiv:1905.08106 [math-ph]

work page arXiv 2020
[76]

Fukuma, H

M. Fukuma, H. Kawai, and R. Nakayama, Int. J. Mod. Phys.A6, 1385 (1991)

1991
[77]

David, Mod

F. David, Mod. Phys. Lett. A5, 1019 (1990)

1990
[78]

C. V. Johnson, (2025), arXiv:2505.10622 [hep-th]

work page arXiv 2025
[79]

S.-Q. Liu, H. Qu, and Y. Zhang, Commun. Math. Phys. 406, 121 (2025), arXiv:2411.15496 [math-ph]

work page arXiv 2025