Half-Wormholes in a Supersymmetric SYK Model

Saurabh Natu; Stefan Forste

REVIEW 3 major objections 4 minor 38 references

Half-Wormholes in a Supersymmetric SYK Model

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Half-wormholes restore factorization in the supersymmetric SYK model at one time point.

desk verdict First half-wormhole computation for N=1 SUSY SYK, with a solid ⟨Z²⟩ calculation and a neat SUSY-breaking observation, but the factorization claim depends on an unproven counting step in Appendix C. read the letter →

arxiv 2411.10155 v1 pith:BDGLK3WV submitted 2024-11-15 hep-th cond-mat.str-elgr-qc

Stefan Forste , Saurabh Natu This is my paper · ORCID

classification hep-thcond-mat.str-elgr-qc

keywords half-wormholessupersymmetricSYKmodelfactorisationproblemwormholesaddleslargeNlimitsupersymmetrybreakingrandomcouplingscollectivefields

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

This paper tries to establish that the factorization puzzle in the $N=1$ supersymmetric SYK model is cured by half-wormholes, exactly as it is in the ordinary SYK model. In the simplified setting where time is collapsed to a single instant, the authors show that the non-averaged square of the partition function is reproduced at large $N$ by adding a half-wormhole contribution to the averaged wormhole result: $Z^2\approx \langle Z^2\rangle+\Phi(0)$, with the variance of the error canceling. The same saddle-point analysis yields $\langle Z^4\rangle=3\langle Z^2\rangle^2$, so the partition function is effectively Gaussian. A second, independent finding is that wormhole and half-wormhole saddles each force all supersymmetry parameters to zero, breaking the $N=1$ supersymmetry completely.

What carries the argument

The engine of the argument is the half-wormhole saddle: a collective-field configuration that contributes to the fourth moment $\langle Z^4\rangle$ through one of the three pairings of four boundaries into two wormholes, but contributes nothing to the averaged second moment $\langle Z^2\rangle$, so that adding it to the non-averaged expression restores factorization. Technically, the paper rewrites $Z^2$ as an integral over a single auxiliary field $\Sigma$ and studies the integrand $\Phi(\Sigma)$; at $\Sigma=0$, $\langle\Phi(0)\rangle=0$ while $\langle\Phi(0)^2\rangle=2\langle Z^2\rangle^2$, which identifies the half-wormhole region. Wormhole saddles are the non-trivial solutions of the saddle-point equations after contour deformation, solved in Section 2.2.

What would settle it

Evaluate the unresolved triple sum below Eq. (90) exactly or numerically to large $N$; if the configurations with $k_{LR}=k_{L'R'}=k_{LR'}=k_{RL'}=N/4$ are not exponentially suppressed relative to the two-pair saddle, then $\langle Z^4\rangle=3\langle Z^2\rangle^2$ fails and the error cancellation in (55) collapses.

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Extended reading notes

Core claim

For the $\mathcal{N}=1$ supersymmetric SYK model reduced to a single time point, the paper's central claim is that factorization at fixed couplings is restored in the large $N$ limit by including half-wormhole saddle points. Concretely, $Z^2\approx \langle Z^2\rangle+\Phi(0)$, where $\Phi(0)$ is the half-wormhole contribution: it has zero average but nonzero second moment, and the mean squared error vanishes because the wormhole-pairing contributions to $\langle Z^4\rangle$, $\langle Z^2\rangle^2$, $\langle \Phi(0)^2\rangle$, and $\langle Z^2\Phi(0)\rangle$ count as $3, 1, 2,$ and $4$, respectively, giving $(3-1+2-4)\langle Z^2\rangle^2=0$. Along the way the paper establishes $\langle Z^4\rangle=3\langle Z^2\rangle^2$, consistent with a Gaussian distribution for $Z$. It also shows that the nonzero fermion-bilinear saddle value $G^{LR}_{\psi\psi}$ forces, through the composite-field transformation rules, every supersymmetry parameter $\epsilon_\alpha$ to vanish; hence wormholes and half-wormholes both break supersymmetry completely.

Load-bearing premise

The argument leans on the assumption, stated in footnote 4, that in the large-$N$ limit only the three wormhole-pairing saddles contribute to the fourth moment and that all other contributions, including mixed derivative terms, are exponentially suppressed; the decisive triple sum is plotted, not evaluated.

Editorial extensions

If this is right

In the one-time-point $N=1$ supersymmetric SYK model, the non-averaged $Z^2$ factorizes at large $N$ once half-wormholes are included: $Z^2\approx \langle Z^2\rangle+\Phi(0)$ and the mean squared error vanishes by the pairing count $(3-1+2-4)\langle Z^2\rangle^2=0$.
The fourth moment satisfies $\langle Z^4\rangle=3\langle Z^2\rangle^2$, so at leading order $Z$ is Gaussian-distributed even though the couplings are fixed.
Both wormhole and half-wormhole configurations force all supersymmetry parameters to zero, so these saddles break $N=1$ supersymmetry completely.
The averaged moments are nonzero only when $N$ is a multiple of four, tying the wormhole story to that divisibility condition.

Reading between the lines

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

Beyond the paper, the same half-wormhole mechanism is testable at fixed couplings: a direct evaluation of $Z^2$ in the one-time-point $N=1$ model, analogous to the ordinary SYK computation, should reproduce $Z^2\approx \langle Z^2\rangle+\Phi(0)$.
Beyond the paper, the fact that both wormhole and half-wormhole saddles kill every supersymmetry parameter suggests that a bulk dual such as supersymmetric JT gravity should contain SUSY-breaking saddles with the same pairing structure; checking the $N=2$ or complex SYK variants would show how generic this is.
Beyond the paper, the one-time-point setting leaves open whether the restoration persists once time dependence and the Schwarzian mode are included; if the half-wormhole contribution is purely a saddle-counting effect of the instant reduction, the factorization story would look different in the full model.

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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

Short version: this is the first half-wormhole analysis for the N=1 supersymmetric SYK model at one time point, and the SUSY-breaking observation is clean and correct. The main factorization claim, though, is only as solid as an unproven large-N counting assumption in Appendix C, so treat it as conditional rather than established.

What the paper does well: the exact computation of ⟨Z²⟩ in Section 2 is careful, and the saddle-point match including one-loop determinants is genuine evidence that the wormhole interpretation is right. The SUSY-breaking argument in Section 4 is simple and convincing: a nonzero G_ψψ saddle forces all ϵ_α to zero, so wormholes and half-wormholes both completely break the N=1 supersymmetry. That observation will be useful to people working on supersymmetric SYK and JT supergravity. The citation pattern is fine; the paper builds directly on Saad-Shenker-Stanford-Yao and cites the surrounding literature.

The soft spot is the α=4 counting in Appendix C. Equation (91) leaves α undetermined, and the argument that only two-derivative terms contribute at leading order is heuristic. The triple sum in (90) is not evaluated; Fig. 1 only shows one symmetric mixed configuration, and footnote 4 refers to a computation that isn't shown. If an unsuppressed mixed configuration exists, then ⟨Z⁴⟩≠3⟨Z²⟩² and the cancellation in (55) fails. This is a real gap, not a nitpick. The authors are honest about it — they say they 'argue' and 'assume' — but honesty doesn't close it. The relation ⟨Z⁴⟩=3⟨Z²⟩² is consistent with a Gaussian Z, and the plotted suppression is suggestive, so I wouldn't bet against the result, but it is not yet demonstrated.

A minor limitation: factorization restoration is only claimed at saddle-point values of σ_ψψ, not for the full self-averaging region; the authors acknowledge this. That's acceptable for what they set out to do.

Who this is for: people working on wormholes without averaging, SYK variants, and supersymmetric JT gravity. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to close the α=4 gap — either by evaluating the triple sum in a large-N saddle approximation or by providing a direct fixed-coupling check along the lines of Mukhametzhanov. If they close it, this is a solid paper. As it stands, the main result is conditional.

Referee Report

3 major / 4 minor

Summary. The paper studies the N=1 supersymmetric SYK model with time reduced to a point. The ensemble average of the partition function squared is computed exactly and reproduced by a saddle-point analysis after a contour deformation; the saddle points are interpreted as wormholes. The fourth moment is then analyzed to identify half-wormhole contributions to the non-averaged Z^2, with the claim that including half-wormholes restores factorization. It is also argued that both wormhole and half-wormhole saddle configurations break supersymmetry completely.

Significance. If the central claim holds, the paper extends the half-wormhole/factorization program to a supersymmetric version of SYK, which is a natural and nontrivial generalization of [21]. The exact ⟨Z^2⟩ computation with the one-loop determinant matching is a concrete technical achievement. The supersymmetry-breaking observation is interesting and potentially relevant for holographic applications. However, the factorization-restoring argument depends on an unproved large-N counting assumption, so the central claim is conditional.

major comments (3)

[Appendix C, Eq. (91)] The derivation of ⟨Z^4⟩ = 3⟨Z^2⟩^2 (Eq. (92)) rests on the unproven assumption that α=4. The triple sum in Eq. (90) is not evaluated; Fig. 1 plots only the symmetric mixed configuration with kLR=kL'R'=kLR'=kRL'=N/4 and does not establish suppression of all other mixed configurations. Footnote 4 asserts that mixed products did not contribute in a previous computation, but that computation is not shown and does not cover all terms in (77). Since Eq. (55) cancels the error only when ⟨Z^4⟩ = 3⟨Z^2⟩^2, the factorization restoration claim is conditional on this counting. Please provide a closed-form evaluation of the sum (90) or a rigorous large-N argument that all contributions beyond the three pairing saddles (42) are subleading.
[Section 3.2, Eq. (55)] The values ⟨Φ(0)^2⟩ = 2⟨Z^2⟩^2 and ⟨Z^2Φ(0)⟩ = 2⟨Z^2⟩^2 are asserted through the pairing-counting statement 'for the last two terms only two remain'. No explicit computation of these averaged products is presented. If the counting is affected by the same unsettled issue as the α=4 assumption, the cancellation in (55) fails. Please provide the explicit expressions or a controlled derivation for these two correlation functions.
[Section 2.2, Eq. (29)] The constant term in L1 appears to be incorrect. Evaluating (22) at the saddle (26) with m=0 gives L1 = -N + (3N/4) log 3 + (N/2) log J - N log 2, whereas (29) gives -N + (N/2) log(3^{3/2}/2) + N log sqrt(J) = -N + (3N/4) log 3 + (N/2) log J - (N/2) log 2. The displayed L1 therefore yields e^{L1} a factor 2^{N/2} too large, and the sum over the four saddle points would not reproduce Eq. (16). Please correct Eq. (29) and verify that the saddle-point sum, including one-loop determinants, reproduces Eq. (16) with the corrected constant.

minor comments (4)

[Figure 1] The caption does not define the base of the logarithm or the exact ratio plotted (including N-dependent prefactors); please make the plot reproducible.
[Footnote 4] The phrase 'the previous computation' is vague; please give a reference or an appendix where that computation is shown.
[Section 2.3, after Eq. (41)] The notation 'the ψψ label is suppressed on the rhs' is confusing; please define σαβ and gαβ explicitly as (σψψ)αβ and (gψψ)αβ.
[Section 5] The concluding remark that a self-averaging region could not be depicted is an honest limitation, but the main text would benefit from a one-sentence recap of how this limitation affects the comparison with [21].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the half-wormhole factorization check is a self-contained moment calculation, with the only weak point being a heuristic large-N counting assumption rather than a circular reduction.

full rationale

I walked the claimed derivation chain and found no step in which a prediction is equivalent to its inputs by construction. The averaged second moment ⟨Z^2⟩ is first computed by direct integration in Appendix A and then independently reproduced by a sum over four saddle points in Section 2.2, so the wormhole identification is checked against an exact calculation. The fourth moment is a separate collective-field computation: ⟨Z^4⟩=3⟨Z^2⟩^2 in Eq. (92) is not assumed to prove factorization, but is obtained from a saddle-point count with an explicit, if heuristic, argument that α=4. The half-wormhole object Φ(0) is defined through the collective-field integral in Eq. (46), and its second moment ⟨Φ(0)^2⟩=2⟨Z^2⟩^2 in Eq. (50) is traced to two saddle configurations rather than fitted to force the error to vanish. The mean-squared-error check in Eqs. (52)-(55) is a consistency condition using these independently counted degeneracies, and no free parameter is adjusted to produce the cancellation. The paper relies methodologically on Refs. [21] and [32], but those are external prior works, not self-citations, and they are not used to forbid alternatives. The authors’ own background citations [36,37] are not load-bearing for the factorization claim. The genuine weakness is that the large-N counting behind α=4 in Appendix C, especially the unevaluated triple sum (90) and the assertion that no mixed products contribute, is heuristic; however, an unproven or plausibly incomplete counting argument is a correctness risk, not circularity. The derivation is therefore self-contained with respect to the circularity criteria.

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

The paper introduces no new free parameters or invented entities; it applies the half-wormhole concept from [21] to a new model. The main assumptions are the standard large-N saddle point approximation, the one-time-point restriction, the validity of the contour deformation, and the leading-order counting of ⟨Z^4⟩ contributions.

assumptions (5)

domain assumption The large N limit of the integrals is dominated by saddle points with one-loop determinants.
Used throughout (Sec. 2.2, 3.2) to equate the exact integral for ⟨Z^2⟩ with the sum over saddle points; standard in SYK but not proven here.
ad hoc to paper Contributions to ⟨Z^4⟩ from configurations other than the three pairing saddles (42) are exponentially suppressed or subleading in N.
Assumed to set α=4 and to obtain ⟨Z^4⟩=3⟨Z^2⟩^2 (Eqs. 91-92); partially supported by Fig. 1, but not computed in closed form.
domain assumption Contour deformation to real g_b, σ_b, g_ψ, σ_ψ is valid and the saddle point sum over m∈{0,1,2,3} gives the full result.
Inherited from Ref. [21]; needed to identify wormhole and half-wormhole saddles.
domain assumption The supersymmetric SYK model with time reduced to a point is representative of the factorization phenomenon.
The paper restricts to time-independent fields following [21]; stated in Sec. 2.
domain assumption The bulk dual of the N=1 supersymmetric SYK model is supersymmetric JT gravity.
Cited from Refs. [36,37] by the same first author; used to motivate the model and the wormhole interpretation.

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Cite this review

Pith. "Pith review of Half-Wormholes in a Supersymmetric SYK Model." pith.science (2026). https://pith.science/paper/BDGLK3WV

@misc{pith2026241110155,
  author       = {Pith},
  title        = {Pith review of: Half-Wormholes in a Supersymmetric SYK Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDGLK3WV}},
  note         = {Machine review of arXiv:2411.10155}
}

abstract

We identify half-wormhole contributions to the non averaged $\mathscr{N} =1$ supersymmetric SYK model in which time has been reduced to a point. As in previously studied examples, the inclusion of half-wormholes restores factorisation in the large $N$ limit. Wormholes as well as half-wormholes break supersymmetry.

Figures

Figures reproduced from arXiv: 2411.10155 by the authors.

**Figure 1.** Log K1| kLR=kL′R′=kLR′=kRL′= N 4 / K1| kLR=kL′R′=N/2 as a function of N/8 where α is the number of contributions from the sum in (77). Finally, we are going to argue that α = 4. Also for all other terms in (77) one can perform first the G αβ bb integrations such that one is left only with G αβ ψψ and Σαβ ψψ integrals. The Σαβ ψψ integrals are performed using (61). Now we assume that all terms contributing at lea… view at source ↗

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Pith tools

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

[1] [21]

Wormholes without averaging,

P. Saad, S. H. Shenker, D. Stanford, and S. Yao, “Wormholes without averaging,” JHEP 09 (2024) 133, arXiv:2103.16754 [hep-th]

arXiv 2024

[2] [1]

The Large N limit of superconformal field theories and supergravity,

J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.2 (1998) 231–252, arXiv:hep-th/9711200. 4In the previous computation we observed that from the power in (83) no mixed products contributed at leading order. 18

arXiv 1998

[3] [2]

Wormholes in AdS,

J. M. Maldacena and L. Maoz, “Wormholes in AdS,” JHEP 02 (2004) 053, arXiv:hep-th/0401024

arXiv 2004

[4] [5]

Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information,

D. Marolf and H. Maxfield, “Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information,” JHEP 08 (2020) 044, arXiv:2002.08950 [hep-th]

arXiv 2020

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Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,

C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,” Phys. Lett. B126 (1983) 41–45

1983

[6] [7]

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R. Jackiw, “Lower dimensional gravity,” Nuclear Physics B252 (1985) 343–356. https://www.sciencedirect.com/science/article/pii/0550321385904481

arXiv 1985

[7] [8]

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P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th]

arXiv 1903

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JT gravity and the ensembles of random matrix theory,

D. Stanford and E. Witten, “JT gravity and the ensembles of random matrix theory,” Adv. Theor. Math. Phys.24 no. 6, (2020) 1475–1680, arXiv:1907.03363 [hep-th]

arXiv 2020