Extended JT supergravity and random matrix models: The power of the string equation

Clifford V. Johnson; Maciej Kolanowski

REVIEW 2 major objections 4 minor 58 references

Extended JT supergravity and random matrix models: The power of the string equation

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that the leading string equation of a double-scaled random matrix model, together with the demand that the spectral density have only a simple pole at E=0, forces the BPS sector of extended JT supergravity to be…

desk verdict A checkable, honest extension of the string-equation toolkit to N=3 and large N=4 JT supergravity; the central BPS-from-non-BPS claim is real but rests on an unproven analyticity assumption that the authors flag. read the letter →

arxiv 2507.07185 v1 pith:25BKQDY3 submitted 2025-07-09 hep-th gr-qc

Clifford V. Johnson , Maciej Kolanowski This is my paper · ORCID

classification hep-thgr-qc

keywords extendedJTsupergravityrandommatrixmodelstringequationBPSspectrummulticriticalspectraldensitynon-perturbativecompletionN=4

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

Extended JT supergravity, a two-dimensional model of near-extremal black hole quantum mechanics, is believed to be equivalent to double-scaled random matrix models, and this paper tries to sharpen exactly what the matrix model's string equation knows about the supergravity spectrum. The authors show that the leading-order string equation, plus the analytic requirement that the spectral density have only a single simple pole at zero energy, fixes the BPS sector entirely from the non-BPS continuum. In their refined procedure, the residue of the non-BPS density at E=0 directly gives the number of BPS states, and matching the remaining expansion determines the multicritical couplings. Re-running the method on N=2 and small N=4 reproduces the known spectra, while new N=3 and large N=4 spectra are shown to fit when split into separate one-pole sectors. The payoff is a sharp, testable claim: each of these extended supergravity theories should admit a multicritical random matrix description, with BPS and non-BPS sectors bound together by the string equation.

What carries the argument

The load-bearing object is the leading-order string equation $u_0(\sum_k t_k u_0^k+x)^2=\tilde\Gamma^2$ (with $\tilde\Gamma=\hbar\Gamma$ counting BPS states), together with the integral representation of the leading spectral density, $\rho_0(E)=\frac{1}{2\pi\hbar}\int_{E_0}^E f(u_0)/\sqrt{E-u_0}\,du_0$, where $f(u_0)=\sum_k t_k k u_0^{k-1}\pm|\tilde\Gamma|/(2u_0^{3/2})$. By changing to the variable $\lambda=(E-E_0)/E_0$, the pole structure at $E=0$ becomes visible; the coefficient of the simple pole is what fixes $\tilde\Gamma(E_0)$, and the polynomial pieces $w_k(\lambda)$ determine the multicritical couplings $t_k$. The whole method is a two-step algorithm: extract $\tilde\Gamma$ from the residue, then match the remaining expansion order by order to sum the $t_k$.

What would settle it

A direct check would be to compute the two-boundary cylinder amplitude in large N=4 JT supergravity with one boundary in the plus sector and one in the minus sector of the proposed split; if that amplitude does not vanish, the two sectors are not statistically independent and the central split of the density fails. A second check: find an extended JT spectrum whose non-BPS density has two simple poles at E=0 or a pole of order two that does not vanish after charge quantization; the residue selection rule would then give a BPS count inconsistent with an independent gravity computation.

Watch

Extended reading notes

Core claim

The central claim is that the BPS sector of extended JT supergravity is not an independent input but a forced output of the string equation. Starting from the leading density of states $\rho_0(E)$ of the non-BPS continuum, the paper rewrites the integral representation in the variable $u_0$ and expands around the threshold $E=E_0(1+\lambda)$. Analyticity, specifically the fact that the matrix model can tolerate only a simple pole at $E=0$, gives a residue formula $\pm|\tilde\Gamma(E_0)|=-\hbar E_0\oint_{\lambda=-1}\rho_0(\lambda,E_0)$, and the residue is exactly the BPS degeneracy. For N=2 this reproduces $\tilde\Gamma=\sin(2\pi\sqrt{E_0})/4\pi^2$; for small N=4 it forces angular momentum quantization; for the new large N=4 and N=3 spectra, the same logic works after splitting the density into components that each have a single pole. The paper therefore establishes the string equation as a predictive constraint on extended JT supergravity spectra, not merely a bookkeeping device.

Load-bearing premise

The argument presumes that summing the entire infinite multicritical series in $f(u_0)=\sum_k t_k k u_0^{k-1}$ does not introduce new singularities, so the only admissible pole in the leading spectral density remains the single simple pole at $E=0$; if a supergravity density had additional poles, the residue method and the forced BPS/non-BPS link would need new justification.

Editorial extensions

If this is right

For N=2 JT supergravity, the BPS degeneracy formula $\tilde\Gamma=\sin(2\pi\sqrt{E_0})/4\pi^2$ follows from the non-BPS density alone, with no separate input.
For small N=4, the requirement of a single simple pole forces $J\in\frac12\mathbb{Z}$; at physical angular momenta it yields $\tilde\Gamma=2$ and a resummed set of couplings $t_k=J^{2k+4}\pi^{2k+2}J_{k+1}(2\pi J)/(k!(2k+1)(2k+3)(2\pi J)^{k+1})$.
For large N=4, the total density must be decomposed into two one-pole components $\rho^{\pm}$; treating them as statistically independent matrix models recovers all BPS degeneracies, BPS energies, and the spin condition that selects which short multiplets have BPS states.
For the anomalous SO(3) N=3 model, a multicritical 0A matrix model exists with couplings $t_k=2\pi^{k+1}J_{k+1}(2\pi\sqrt{E_0})/((2k+1)E_0^{k/2}k!)$, and a 0B realization may exist; the other two N=3 realizations do not fit the present framework.
In all models, sectors without BPS states generically produce a multi-valued $u_0(x)$ and therefore lack a non-perturbative completion of this matrix model class, even though the perturbative genus expansion is well-defined.

Reading between the lines

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

Editorial inference: the residue method reads like a general spectral-to-matrix-model dictionary: given any candidate positive density with a single simple pole at E=0 and a branch cut above E0, the same two-step algorithm would produce tks and a predicted BPS count, so it could be used to screen future proposed N>4 spectra before a matrix model is constructed.
Editorial inference: the non-perturbative instability of gap sectors without BPS states suggests a selection rule beyond perturbation theory: only spectra with a nonzero protected BPS ground-state sector can be completed as this class of matrix ensemble. A numerical search over string-equation solutions near finite $\hbar$ could test whether the phase diagram (regular at large $\hbar$, ill-defined
Editorial inference: because the split of large N=4 into plus and minus sectors is motivated by a partial-fraction identity, one might test the same statistical-independence hypothesis at the level of higher spectral correlators: if independent, the mixed-sector contribution to the spectral form factor should be absent, giving a cleaner signal than the cylinder amplitude alone.

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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

Here is my take. The genuinely new content is the residue-based method in Section 2 and its application to two recently derived spectra: large N=4 and one N=3 variant. The shift to λ=E/E0−1 and the formula (2.11) for Γ(E0) are clean and easy to check; Appendix A's generating function for the t_k is a useful addition. The authors recover the BPS multiplicities for these new cases without putting them in by hand, and they isolate the condition that selects which short multiplet carries BPS states. That is a real step beyond refs. [13,14]. The N=2 and small N=4 re-derivations also check out as consistency tests. I also think the nonperturbative section (7) is careful and valuable.

The soft spots are real but mostly flagged by the authors. The biggest one is the assumption after (2.9d) that the infinite sum over multicritical sectors does not change the analytic structure at λ=−1. Everything downstream, including (2.11), depends on that. The contour in (2.11) is only formal because λ=−1 sits on the square-root branch cut; the first term in (2.10) needs to be regular there for the residue to isolate Γ. This is asserted, not proved. It may be true for all admissible densities, but the paper does not give an actual admissibility criterion. The authors explicitly call it a presumption, which is honest, but it is still load-bearing.

Two smaller concerns. For small N=4, the subtraction that defines eρ0 builds in the cancellation of the double pole; the subsequent recovery of J ∈ Z/2 is less astonishing than it looks, though it agrees with the earlier work. For large N=4, the split into two statistically independent matrix models is a conjecture, as the authors say; the negative sign in ρ_- is unusual and the prediction that wormholes between sectors vanish is testable, so this is fine, but it is not evidence yet. And the N=3 section covers only the anomalous SO(3) variant; the other two are left open, which the authors acknowledge.

Who should read it: people working on JT supergravity, matrix models, and near-BPS black hole spectra. It is a serious within-field contribution, not a revolution. The math is mostly checkable, the limitations are honestly stated, and the new cases are genuinely new. My recommendation: send it to referees, with a request that they push on the analyticity assumption. If the authors can prove or sharpen the regularity condition on ρ0(E), the method will be on much firmer ground. As it stands, conditional acceptance feels right.

Referee Report

2 major / 4 minor

Summary. This paper develops a refined method for extracting the spectrum of extended JT supergravity from the leading-order string equation of double-scaled multicritical random matrix models. The authors change variables to E=E0(1+lambda), rewriting the leading spectral density in the form (2.10), where one term is regular and the other has a simple pole at lambda=-1 whose residue is identified with the BPS degeneracy parameter Gamma(E0). Matching the pole residue and then expanding in E0 fixes the multicritical coefficients tk(E0). The method is applied to N=2 and small N=4 JT supergravity (reproducing known results), to large N=4 JT supergravity, and to one of the N=3 variants, yielding a conjecture that these theories admit matrix model descriptions. The paper also discusses perturbative and non-perturbative issues, including single-valuedness of u0(x) as a criterion for non-perturbative completability.

Significance. If valid, this is a significant conceptual step: it turns the earlier 'miracle' of the BPS/non-BPS linkage into a systematic residue-extraction procedure, and it generates new, testable conjectures about large N=4 and N=3 JT supergravity. The paper is explicit and checkable, with concrete coefficient matchings and a universal formula for the tks in Appendix A. The authors are also transparent about the assumptions they make. However, the central claim is conditional on an explicitly stated but unproven analytic-structure assumption, and the large N=4 result depends on a conjectured split into statistically independent sectors; these points prevent the result from being a fully closed theorem at this stage.

major comments (2)

[Section 2, Eq. (2.10)-(2.11)] The residue formula (2.11) is the load-bearing step of the paper: it identifies the coefficient of the simple pole at lambda=-1 with +/-|Gamma(E0)| and hence with the BPS sector. This identification presupposes that the first term of (2.10), built from f0(u0)=sum_k t_k k u0^{k-1}, is regular at lambda=-1 and that only the explicit sqrt(lambda)/(1+lambda) term carries the pole. The paper states this as a presumption immediately after (2.9d): 'We presume that this analytic structure is not changed upon performing the sum in f=sum_k t_k k u0^{k-1}.' This is not proven, and it is not a technicality: the infinite sum over multicritical sectors could in principle develop additional poles or branch points at lambda=-1, or move the branch cut, in which case (2.11) would not isolate the BPS degeneracy. Moreover, lambda=-1 lies on the branch cut of sqrt(lambda), so the contour integral in (2.11) is formal unless the f0 contribution is regular across that cut, which is exactly the unproven assertion. The examples provide consistency checks, but the paper does not derive a general admissibility criterion for rho0(E) beyond the asserted presumption. I ask the authors to either prove or justify the analytic-structure assumption for the class of tk generated by these models, or to explicitly reframe the central claim as a conjecture and state precisely which parts are established only modulo this assumption.
[Section 5, Eqs. (5.2)-(5.4)] The analysis of large N=4 JT supergravity rests on the proposal that the total density (5.1a) can be written as rho_+ - rho_-, with rho_+ and rho_- given by (5.3) and each generated by a statistically independent matrix model. This is a postulate, not a derivation: the text says 'we decided to keep terms ... separate' and 'we postulate that these are two statistically-independent sectors.' The subsequent recovery of the BPS data from rho_+ (Eqs. (5.5)-(5.7)) is then used to argue that the split is justified, which is somewhat circular: the BPS output is predicated on the split, so it cannot independently confirm the split. The paper correctly notes that a cylinder computation could test the proposal, but as written the central claim for large N=4 is conditional on an untested conjecture. I recommend that the authors either provide additional evidence for statistical independence (for example, a cylinder-amplitude computation or a more detailed argument from the derivation in ref. [19]), or clearly mark the large-N=4 results as conditional on this conjecture in the abstract and in the summary of results.

minor comments (4)

[Section 5, after Eq. (5.10)] The displayed expression 'alpha j+ + j+ / 1 + alpha' in the sentence after (5.10) appears to contain a typo; the intended quantity is probably (alpha j_+ + j_-)/(1 + alpha).
[Section 6, Eq. (6.14)] The claimed proportionality t_k^{N=4} = 2/(k+3/2) t_k^{N=3} does not obviously follow by direct substitution from the formulas in (4.13) and (6.13); a direct substitution appears to produce a ratio depending on E0. Please clarify the definitions used here or correct the formula.
[Section 1.3.1] There are several typos, for example 'detertminant' and the duplicated 'systems' in the sentence following (1.3); a careful proofread is recommended.
[Section 2, footnote 12] The reuse of the symbol lambda for the unscaled eigenvalues in Section 1 and for the expansion variable in Section 2 is a potential source of confusion, although the footnote helps; a different symbol for the new variable would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the BPS sector is derived as the residue of the non-BPS density under the string-equation ansatz, not fitted; self-citations to prior multicritical constructions are re-derived and are not load-bearing.

full rationale

The paper's central derivation is self-contained: Section 2 derives f(u0)=f0±|Γ|/(2u0^{3/2}) directly from the leading string equation without needing the full ansatz of Ref. [13], and equation (2.11) defines Γ(E0) as the residue of the input non-BPS density at the pole forced by the string equation. In the N=2 example the BPS coefficient sin(2π√E0)/(4π^2) is an output of the residue computation, not an input to the fit of the tk, so the claimed 'prediction' is not a fitted parameter renamed as a prediction. The same holds for small and large N=4 and N=3, where the BPS degeneracies (or their absence) emerge from residue/regularity conditions. The paper does rely on the unproven analytic-structure assumption after Eq. (2.9d), but that is an assumption about convergence/singularity structure and a possible correctness risk, not circular reasoning. Citations to the authors' earlier work [13,14] supply motivation and prior examples, but the key steps re-derived here give them independent support; no load-bearing argument reduces to a self-citation alone.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claim rests on the standard JT/matrix model dictionary and two case-specific ad hoc assumptions: the preservation of analytic structure (single pole) and the statistical independence of the large N=4 sectors. No new particle-like entities are introduced.

assumptions (6)

domain assumption The double-scaled random matrix model is described by the string equation u R^2 - (ℏ^2/2) R R'' + (ℏ^2/4) (R')^2 = ℏ^2 Γ^2, with R = Σ_k t_k R_k[u] + x.
This is the established framework for JT gravity as a matrix integral, reviewed in Section 1.3 and used throughout; the paper does not re-derive it from the matrix integral.
standard math The leading spectral density equals ρ0(E) = (1/2πℏ) ∫_μ^{-∞} Θ(E-u0(x)) / √(E-u0(x)) dx, and after a change of variables to u0, f(u0) = -dx/du0.
This integral representation is a known result from the orthogonal polynomial formalism (refs. [38,39,13]); the paper uses it as the bridge between the string equation and the spectrum.
ad hoc to paper The analytic structure of ρ0(E) is unchanged by the sum over k in f(u0) = Σ_k t_k k u0^{k-1}; i.e., the only possible pole is a simple pole at E=0.
Stated as a presumption in Section 2 immediately after (2.9d). This assumption is load-bearing: it allows the residue formula (2.11) for Γ(E0) and rules out densities with double poles, which then forces the angular momentum quantization in small N=4 and the sector split in large N=4.
ad hoc to paper The large N=4 density can be split into two statistically independent sectors ρ+ and ρ-, each described by its own matrix model, with the total density being their difference.
Proposed in Section 5 as a conjecture, motivated by the Poisson-resummed sector structure of ref. [19]. It is not derived from a gravitational computation and is presented as a testable prediction (vanishing cylinder amplitude across sectors).
domain assumption For N=3, the shifted-wall string equation (u-σ)R^2 - (ℏ^2/2)RR'' + (ℏ^2/4)(R')^2 = ℏ^2 Γ^2 with σ=E0 describes the model.
The paper uses the generalized string equation from refs. [40,41] to accommodate the hard edge at E0, in addition to the standard string equation.
domain assumption The parameter Γ counts the BPS degeneracy (number of zero-energy states) and ℏ = e^{-S0} sets the extremal entropy.
This is the dictionary of the JT supergravity/matrix model correspondence established in refs. [13,14] and assumed here.

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Pith. "Pith review of Extended JT supergravity and random matrix models: The power of the string equation." pith.science (2026). https://pith.science/paper/25BKQDY3

@misc{pith2026250707185,
  author       = {Pith},
  title        = {Pith review of: Extended JT supergravity and random matrix models: The power of the string equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25BKQDY3}},
  note         = {Machine review of arXiv:2507.07185}
}

abstract

A number of supersymmetric Jackiw-Teitelboim (JT) gravity theories are known to be described (in the Euclidean path integral formulation) by double-scaled random matrix models. Such matrix models can be characterized using a certain ``string equation''. It was shown recently that in extended supergravity, when the number of BPS states scales as ${\rm e}^{S_0}$, where $S_0$ is the extremal entropy, a special ansatz for the leading order solution of the string equation yields the supergravity spectrum. Somewhat miraculously, the construction showed that the functional form of the non-BPS (continuum) sector predicts the precise form of the BPS sector, showing the robustness of the supergravity/matrix-model correspondence. In this paper, we refine the analysis and show that the string equation, combined with some simple requirements on solutions, are powerful tools for constraining the spectrum of extended JT supergravity theories. We re-explore the cases of ${N}{=}2$ and (small) ${N}{=}4$ JT supergravity, and then explore the new cases of spectra from ${N}{=}3$ and large ${N}{=}4$ JT supergravity (recently derived by Heydeman, Shi, and Turiaci) showing that our approach also works naturally for (nearly) all the models. Based on this success, we conjecture that these new supergravity models also have matrix model descriptions.

Figures

Figures reproduced from arXiv: 2507.07185 by the authors.

**Figure 1.** Left: A generic density with ends at λ±. Magnifying the infinitessimal region near λ=λ− recovers universal physics in the double scaling limit. The universal physics to be found in the neighbourhood of one or other endpoint, can be written in terms of the orthogonal polynomial quantities that survive the limit. All that is really needed for the purposes of this paper is that the universal physics of the scaled endpo… view at source ↗

**Figure 2.** (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. 1.4.1 N =1 JT supergravity The simplest natural solution in the positive x > 0 regime is simply u0(x) = 0, and indeed this readily emerged in studies of N =1 JT supergravity [46, 47]. The leading string equation is simply: u0R2 0 = 0 , with R0[u0] ≡ X∞ k… view at source ↗

**Figure 3.** The solid curve is a plot of u0(x) for (a) E0 = 0.2, and (b) E0 = 0.25, the values indicated by the horizontal dashed line. The sign of the derivative (and hence of the Jacobian f(u0)) remains positive as long as E0≤ 1 4 . Contrast with the cases presented in [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The solid curve is a plot of u0(x) for (a) E0 = 0.5, and (b) E0 = 1.1, the values indicated by the horizontal dashed line. The sign of the derivative (and hence of the Jacobian f(u0)) changes sign when E0> 1 4 , as can be seen in both cases. However, it remains positiv…

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Loop equations and the topological phase of multi-cut matrix models.,

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Jackiw-Teitelboim supergravity as a double-cut matrix model,

C. V. Johnson, F. Rosso, and A. Svesko, “Jackiw-Teitelboim supergravity as a double-cut matrix model,” Phys. Rev. D104 no. 8, (2021) 086019, arXiv:2102.02227 [hep-th]

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Factorization properties of critical matrix models,

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Quantum field theory techniques in graphical enumeration,

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Nonperturbative two-dimensional quantum gravity, again,

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The Boundary cosmological constant in stable 2-D quantum gravity,

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On loop equations in KdV exactly solvable string theory,

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On integrable c < 1 open string theory,

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Microscopic And Macroscopic Loops In Nonperturbative Two- Dimensional Gravity,

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Backlund transformations, D-branes, and fluxes in minimal type 0 strings,

J. E. Carlisle, C. V. Johnson, and J. S. Pennington, “Backlund transformations, D-branes, and fluxes in minimal type 0 strings,” J. Phys. A40 (2007) 12451–12462, arXiv:hep-th/0501006 [hep-th]

2007 arXiv
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Jackiw-Teitelboim supergravity, minimal strings, and matrix models,

C. V. Johnson, “Jackiw-Teitelboim supergravity, minimal strings, and matrix models,” Phys. Rev. D103 no. 4, (2021) 046012, arXiv:2005.01893 [hep-th]

2021 arXiv
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Explorations of nonperturbative Jackiw-Teitelboim gravity and supergravity,

C. V. Johnson, “Explorations of nonperturbative Jackiw-Teitelboim gravity and supergravity,” Phys. Rev. D103 no. 4, (2021) 046013, arXiv:2006.10959 [hep-th]

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Fermionic Localization of the Schwarzian Theory,

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Solving the Schwarzian via the Conformal Bootstrap,

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D-branes and fluxes in supersymmetric quantum mechanics,

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2008 arXiv
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Supersymmetric Sachdev-Ye-Kitaev models,

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Consistency Conditions for Non-Perturbative Completions of JT Gravity,

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arXiv
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Volumes for Extended JT Supergravity from the Gelfand-Dikii Equation,

W. Ahmed, C. V. Johnson, and K. Saraswat, “Volumes for Extended JT Supergravity from the Gelfand-Dikii Equation,” to appear(07, 2025) , arXiv:25xx.xxxx [hep-th]

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Unitary and Hermitian matrices in an external field,

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Random matrix model with external source and a constrained vector equilibrium problem,

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Gaussian matrix model in an external field and non-intersecting brownian motions,

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Solving Puzzles in Deformed JT Gravity: Phase Transitions and Non-Perturbative Effects,

C. V. Johnson and F. Rosso, “Solving Puzzles in Deformed JT Gravity: Phase Transitions and Non-Perturbative Effects,” JHEP 04 (2021) 030, arXiv:2011.06026 [hep-th]

2021 arXiv
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The Microstate Physics of JT Gravity and Supergravity,

C. V. Johnson, “The Microstate Physics of JT Gravity and Supergravity,” arXiv:2201.11942 [hep-th]. 41

arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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