$q$-deformed Weil-Petersson volumes from the spectral edge of DSSYK and supersymmetric extensions

Joydeep Naskar; Krishan Saraswat; Wasif Ahmed

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$q$-deformed Weil-Petersson volumes from the spectral edge of DSSYK and supersymmetric extensions

Wasif Ahmed, Joydeep Naskar, Krishan Saraswat

High Energy Physics - Theory · Submitted 9 Oct 2026

Pith's verdict: The N=1 candidate construction is the real content; the bosonic edge-zoom story is a reformulation with an unproven all-genus step.

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arXiv ID2610.13899 (version 2)

Pith IDOYG7XS3P, Pith's permanent record of this paper and its review

CitationJoydeep Naskar, Krishan Saraswat, Wasif Ahmed. "$q$-deformed Weil-Petersson volumes from the spectral edge of DSSYK and supersymmetric extensions." arXiv:2610.13899 (2026). https://pith.science/paper/OYG7XS3P

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Editor's note

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Here's my take on arXiv:2610.13899. The paper has two parts and they should be graded differently. The N=1 half is the genuinely new part: a hard-edge construction of candidate q-deformed super Weil-Petersson volumes, the new series xi_q, the replacement rule xi_q(2m) -> (-1)^m zeta^odd_q(2m), and an honest discussion of why a discrete N=1 analogue fails. The bosonic half is mostly a reformulation: density (1.12) is exactly what you get by translating the Do-Norbury spectral curve into ODE language, so the Appendix B reproductions matching known low-genus volumes is not a surprise.

What the paper does well: the algebraic rewritings of the DSSYK density in terms of zeta_q are clean; the sample ODE computations are internally consistent and match the published low-genus volumes; Appendix C's asymptotic analysis of xi_q is real work; and the Section 3.4 obstruction is reported openly, not hidden.

The soft spot is the load-bearing edge step, and the stress-test note lands. Eq. (2.9) is called a leading truncation of (2.8), but gamma=-4 and C' are fixed by requiring the q->1 limit to be JT gravity, not by taking the finite-q edge expansion of the DSSYK density. The actual leading coefficient in the exponent of (2.8) near E=-a is 8/a, and 4/a for the N=1 hard edge; both are q-dependent and vanish in the scaling used in Eq. (2.10). So at fixed q, density (1.12) is not the local limit of the DSSYK density—it is a nearby ansatz with the same functional form and the correct continuum limit. Since the ODE method reads all KdV times from the full Taylor data at E=0, the omitted subleading terms can shift higher t_k and hence higher-genus W_{g,n}. The all-genus claim therefore rests on low-genus samples plus the known equivalence with the Do-Norbury curve, not on the edge-zoom derivation. This is a real gap, not a nitpick.

Minor: Eq. (3.10) has typos—xi_q(2m) should be xi_q(2), and the last line uses zeta_q where xi_q is meant. Fixable. The authors already label the N=1 objects as candidates, so I won't hold the Section 3.4 obstruction against them.

Who gets value: people working on DSSYK/JT/matrix-model connections to Weil-Petersson volumes. The N=1 candidate volumes and the xi/zeta^odd relation are worth knowing; the bosonic edge-origin story should be presented as a suggestive prescription, not a derivation. I would send it to a referee, with the request that the authors either prove the ODE/topological-recursion matching for these densities at finite q or explicitly restrict the all-genus claim to the checked genera, and fix the typos.

Referee report

Significance

If the all-genus statement is accepted, the paper gives a clean physical reinterpretation of q-deformed Weil-Petersson volumes as local soft-edge data of DSSYK, and it constructs a new family of candidate N=1 q-deformed super Weil-Petersson volumes with the correct continuum limit. The algebraic rewriting of the DSSYK densities is explicit and checkable, the sample ODE computations in Appendix B are detailed, and Section 3.4 openly reports the obstruction to a discrete N=1 analogue rather than hiding it. The main weakness is that the bosonic all-genus reproduction claim is supported only by low-genus checks plus an assumed ODE/topological-recursion equivalence, and the finite-q edge truncation is a prescription rather than a controlled limit; the N=1 results are correspondingly 'candidate' statements.

Major objections (2)

[Section 2, Eqs. (2.8)-(2.11), with Appendix B] The all-genus claim that the density (1.12) reproduces the q-deformed Weil-Petersson volumes is supported only by the low-genus sample computations in Appendix B. The step from (2.8) to (2.9) is a leading small-epsilon truncation: the omitted prefactor corrections and the next-order terms inside the exponent of (2.8) are not negligible at finite q, and Eq. (2.10) fixes gamma and C' by matching the q->1 JT limit rather than by matching the finite-q coefficient read off from (2.8). Since the ODE method reads every KdV time from the full Taylor expansion of the density through Eqs. (A.22)-(A.24), these omitted terms can in principle shift all higher KdV times and hence all higher-genus volumes. Please either supply a proof or a precise cited equivalence showing that the Appendix A.3 matching applies to (1.12) and (1.11) for all stable (g,n), or explicitly state that the reproduction is verified only up to the cases listed in Appendix B.
[Section 3.2, Eqs. (3.6)-(3.13), with Appendix C] The candidate N=1 volumes are presented as having fixed degree 2g-2 and as satisfying the scaled q->1 limit (1.15), and the map to the Do-Norbury volumes in Eq. (3.13) is stated as an identity. However, only the cases displayed in Eq. (3.10) are computed, and the relation (3.13) is verified on those cases. The asymptotic result (C.6) makes the general pattern plausible, but a general argument for all stable (g,n) is not given. The manuscript should either prove that the degree and limit statements follow structurally from the N=1 ODE formulas for every stable (g,n), or mark Eqs. (1.15) and (3.13) explicitly as conjectures verified in low-genus examples. The word 'candidate' is appropriate, but it should be applied consistently to the all-genus statements and not only to the individual volumes.

Minor objections (5)

[Eq. (3.10)] In the expression for V^q_{2,1}, the argument 'xi_q(2m)' should be 'xi_q(2)'; as written it is dimensionally inconsistent with the grading deg(xi_q(2m)) = 2m.
[Appendix B, Eq. (B.9)] Two coefficients in the formula for W_{2,1} are both labelled C_2; the coefficient of z^{-4} should be renumbered, for example as C_4, to match the sum over k.
[Appendix B, after Eq. (B.14)] There is a duplicated sentence fragment: 'which will give the following result: which will give:' should be reduced to a single clause.
[Section 3.3 and Eq. (3.17)] The term 'irrelevant' is defined by vanishing in the scaled q->1 limit, not by numerical smallness at finite q. A phrase such as 'continuum-irrelevant' would prevent the reader from misreading the even-sector contributions as negligible in the finite-q candidate volumes.
[References] Reference [83] is listed only as 'arXiv, 2026' with no title or arXiv number; this should be completed before publication.

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The paper

The paper claims

The paper shows that the q-deformed Weil-Petersson volumes of double-scaled SYK arise from local data at the spectral edge: the soft edge reproduces the known volumes, and the hard edge of $\mathcal{N}=1$ DSSYK yields candidate…

The reading

The paper tries to establish that q-deformed Weil-Petersson volumes — the objects that form the top-degree part of the discrete volumes of double-scaled SYK and reduce to ordinary Weil-Petersson volumes as $q\to 1$ — are encoded in purely local data at the spectral edges of the DSSYK matrix model. It shows that zooming into the soft edge of the DSSYK density, keeping only the leading square-root profile dressed by an exponential built from the q-zeta series $\zeta_q(2m)$, yields a disk density whose ODE-method volumes match the known q-deformed Weil-Petersson volumes in the sample checks. Applying the same prescription to the hard edge of $\mathcal{N}=1$ DSSYK produces candidate q-deformed $\mathcal{N}=1$ super Weil-Petersson volumes: fixed-degree polynomials in the boundary lengths and the $\xi_q(2m)$ that reduce to the standard super volumes in a scaled $q\to 1$ limit. The paper further shows that these candidates differ from the previously defined $\mathcal{N}=1$ q-deformed volumes only by 'irrelevant' even-sector terms that vanish in that limit, so both constructions share a common hard-edge origin. If the picture is right, soft edges generate bosonic q-deformed volumes, hard edges generate supersymmetric ones, and the same recipe can be turned on other supersymmetric SYK models such as $\mathcal{N}=2$.

What carries the argument

The engine of the argument is the edge-zoomed disk density: rewrite the full DSSYK density of states so that its exponential contains the q-zeta series, then keep only the leading near-edge form $\rho_0(\epsilon) = C'\sqrt{\epsilon}\,\exp\big[-\sum_{m\ge 1}\frac{\zeta_q(2m)}{m}(\gamma\epsilon)^m\big]$ at the soft edge, and the analogous $\rho_0(\epsilon) = C'\epsilon^{-1/2}\exp\big[-\sum_{m\ge 1}\frac{\xi_q(2m)}{m}(\gamma\epsilon)^m\big]$ at the $\mathcal{N}=1$ hard edge. These densities are converted into volumes by the ODE method: the string equation built from Gel'fand-Dikii polynomials, whose KdV times $t_k$ are fixed by expanding the density in canonical form, determines the derivatives of the potential $u_0$ at the Fermi surface, and those derivatives enter universal formulas for the correlators $W_{g,n}$ whose inverse Laplace transforms are the volumes. The comparison with the earlier $\mathcal{N}=1$ construction runs through the identity $\xi_q(2m) = \xi^{\mathrm{odd}}_q(2m) + \xi^{\mathrm{even}}_q(2m)$, with $\xi^{\mathrm{odd}}_q(2m) = (-1)^m\zeta^{\mathrm{odd}}_q(2m)$ and $\lim_{q\to 1}(1-q)^{2m}\xi^{\mathrm{even}}_q(2m) = 0$, so the even sector is irrelevant in the continuum limit. The scaling limits $\lim_{q\to1}(1-q)^{2m}\zeta_q(2m)=\zeta(2m)$ and $\lim_{q\to1}(1-q)^{2m}\xi_q(2m)=(-1)^m(1-2^{-2m})\zeta(2m)$ turn the edge densities into the JT and $\mathcal{N}=1$ super JT disk densities, which fixes the normalizations $C'$ and $\gamma$.

Extended reading notes

Core claim

The paper's central claim is that the q-deformed Weil-Petersson volumes do not require the global spectral curve of the DSSYK model: they are reproduced, to the orders checked, by a double-scaled matrix model whose disk density $\rho_0(E) = \frac{1}{2\pi\hbar\sqrt{E}}\exp\big[-\sum_{m\ge 1}\frac{(-4)^m\zeta_q(2m)}{m} E^m\big]$ is obtained by rewriting the DSSYK density in terms of the q-zeta series and zooming into the soft edge at $E=-a$. The same edge-zooming prescription applied to the hard edge at $E=0$ of $\mathcal{N}=1$ DSSYK gives the density $\rho_0(E) = \frac{1}{2\pi\hbar\sqrt{E}}\exp\big[-\sum_{m\ge 1}\frac{4^m\xi_q(2m)}{m} E^m\big]$ with a new q-series $\xi_q(2m)$, and feeding it through the ODE method produces candidate q-deformed $\mathcal{N}=1$ super Weil-Petersson volumes $V^{q\mathrm{SWP}}_{g,n}$: polynomials of degree $2g-2$ satisfying $\lim_{q\to 1}(1-q)^{2g-2}V^{q\mathrm{SWP}}_{g,n}(b_i/(1-q)) = V^{\mathrm{SWP}}_{g,n}(b_1,\dots,b_n)$. These candidates are not identical to the previously defined $\mathcal{N}=1$ q-deformed volumes, but the difference is carried by an 'irrelevant' piece: splitting $\xi_q(2m)$ into odd and even index sectors, the even sector vanishes in the scaled $q\to 1$ limit, the odd sector equals $(-1)^m\zeta^{\mathrm{odd}}_q(2m)$, and discarding the even sector recovers the earlier volumes up to an overall sign $(-1)^n$. The paper also records an obstruction it cannot resolve: a discrete analogue of $\mathcal{N}=1$ super volumes built from the full $\mathcal{N}=1$ DSSYK spectral curve fails, because the residue at the hard-edge ramification point introduces an exponential factor of ill-defined degree.

Load-bearing premise

The load-bearing premise is the edge zoom-and-truncate step: the leading near-edge form of the DSSYK density is assumed to fix all KdV times and therefore all higher-genus volumes, and only low-genus sample checks are supplied, so the claim fails if subleading terms of the full density contribute at higher genus.

Editorial extensions

If this is right

The known q-deformed Weil-Petersson volumes are fixed by the near-edge profile of the DSSYK density alone, so any matrix model with the same soft-edge shape produces the same volumes without any reference to a discrete volume construction.
The string-equation ODE method becomes a working bridge from a DSSYK edge density to volumes, an alternative route to topological recursion on the global spectral curve.
The previously defined $\mathcal{N}=1$ q-deformed super volumes acquire a matrix-model home: up to the discarded even-sector terms, they are the volumes of the double-scaled model living at the hard edge of $\mathcal{N}=1$ DSSYK.
The candidate $\mathcal{N}=1$ volumes are q-deformations by the paper's own criteria — fixed degree $2g-2$ and the scaled $q\to1$ limit reproducing $V^{\mathrm{SWP}}_{g,n}$ — even though they are not the same polynomials as the earlier construction.
A discrete analogue of $\mathcal{N}=1$ super volumes, in the spirit of the bosonic discrete volumes, is obstructed: the hard-edge residue of the full $\mathcal{N}=1$ spectral curve generates terms of ill-defined degree, a limitation the authors explicitly acknowledge.

Reading between the lines

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

Read as a recipe, the edge-zooming prescription makes a concrete prediction: zooming into the BPS-related edge of an $\mathcal{N}=2$ DSSYK model should produce q-deformed $\mathcal{N}=2$ super Weil-Petersson volumes that reduce to the known continuum $\mathcal{N}=2$ volumes, and computing a genus-one sample volume would test the picture immediately.
The even sector $\xi^{\mathrm{even}}_q(2m)$ is only 'irrelevant' in the $q\to1$ scaling; at finite $q$ it genuinely changes the volumes, and one might ask whether it encodes physical content such as Ramond-sector or BPS data instead of being a term to discard.
The paper does not elevate its own observation that the hard edge of $\mathcal{N}=1$ carries $\zeta^{\mathrm{odd}}_q$ while the soft edge carries the complementary $\zeta^{\mathrm{even}}_q=\zeta_{q^2}$ into a general claim; I would conjecture that the pair of edges of any DSSYK variant organises the available q-deformation functions in exactly this way.

Original abstract

The $q$-deformed Weil-Petersson volumes of Do and Norbury are the top-degree parts of Okuyama's discrete volumes, which are defined by topological recursion on the spectral curve of the double-scaled SYK (DSSYK) model. We show that the same $q$-deformed volumes can also be obtained from a double-scaled matrix model whose disk density is found by zooming in on the soft edges of the DSSYK density. Applying the same procedure to the hard edge of $\mathcal{N}=1$ DSSYK, we obtain candidate $q$-deformed $\mathcal{N}=1$ super Weil-Petersson volumes. They are polynomials of fixed degree, and they reduce to $\mathcal{N}=1$ super Weil-Petersson volumes in a scaled $q\to1$ limit. These candidate volumes differ from the $\mathcal{N}=1$ volumes of Do and Norbury in (arXiv:2510.12431) up to what we call ``irrelevant'' contributions which exactly vanish in the scaled $q\to 1$ limit. Together, these results demonstrate how $q$-deformed volumes can be naturally associated to the spectral edges of various DSSYK models and provides an intuitive way to construct them in supersymmetric contexts.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-12: 1 request, 5,247 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.79 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The $q$-deformed Weil-Petersson volumes of Do and Norbury are the top-degree parts of Okuyama's discrete volumes, which are defined by topological recursion on the spectral curve of the double-scaled SYK (DSSYK) model. We show that the same q-deformed volumes can be obtained from a double-scaled matrix model whose disk density is found by zooming in on the soft edges of the DSSYK density. Applying the same procedure to the hard edge of $\mathcal{N}=1$ DSSYK, we obtain candidate $q$-deformed $\mathcal{N}=1$ super Weil-Petersson volumes. They are polynomials of fixed degree, and they reduce to the $\mathcal{N}=1$ super Weil-Petersson volumes in a scaled $q\to1$ limit. These candidate volumes differ from the $\mathcal{N}=1$ volumes of Do and Norbury in (arXiv:2510.12431) up to what we call "irrelevant" contributions which exactly vanish in the scaled $q\to 1$ limit. Together, these results demonstrate how q-deformed volumes can be naturally associated to the spectral edges of various DSSYK models and provides an intuitive way to construct them in supersymmetric contexts.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The $q$-deformed Weil-Petersson volumes of Do and Norbury are the top-degree parts of Okuyama's discrete volumes, which are defined by topological recursion on the spectral curve of the double-scaled SYK (DSSYK) model.	background 1.00	0.08	0.08	asserted 1.00
s1	We show that the same q-deformed volumes can be obtained from a double-scaled matrix model whose disk density is found by zooming in on the soft edges of the DSSYK density.	result 0.76	0.93	0.30	asserted 1.00
s2	Applying the same procedure to the hard edge of $\mathcal{N}=1$ DSSYK, we obtain candidate $q$-deformed $\mathcal{N}=1$ super Weil-Petersson volumes.	contribution 0.56	0.92	0.14	asserted 0.91
s3	They are polynomials of fixed degree, and they reduce to the $\mathcal{N}=1$ super Weil-Petersson volumes in a scaled $q\to1$ limit.	result 1.00	0.83	0.15	asserted 1.00
s4	These candidate volumes differ from the $\mathcal{N}=1$ volumes of Do and Norbury in (arXiv:2510.12431) up to what we call "irrelevant" contributions which exactly vanish in the scaled $q\to 1$ limit.	result 0.90	0.81	0.18	asserted 1.00
s5	Together, these results demonstrate how q-deformed volumes can be naturally associated to the spectral edges of various DSSYK models and provides an intuitive way to construct them in supersymmetric contexts.	interpretation 1.00	0.80	0.78	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

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