$N=1$ Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve

Clifford V. Johnson

arxiv: 2606.20796 · v1 · pith:OSQTDR3Pnew · submitted 2026-06-18 · ✦ hep-th · math-ph· math.MP

N=1 Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve

Clifford V. Johnson This is my paper

Pith reviewed 2026-06-26 15:45 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP

keywords Weil-Petersson volumesN=1 supersymmetryspectral curvetopological recursionvolume recursionRamond puncturesNeveu-Schwarz boundaries

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The pith

The Stanford-Witten-Norbury recursion for N=1 supersymmetric Weil-Petersson volumes follows directly from topological recursion on a spectral curve.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that a known recursion for the volumes of moduli spaces of N=1 supersymmetric Riemann surfaces with boundaries and Ramond punctures can be obtained by applying topological recursion to a spectral curve. This connects the geometric recursion relation to an algebraic object whose initial data generate the Laplace transforms of the volumes. The work also derives an alternative recursion whose kernels carry the Ramond data instead of the initial conditions, opening a possible route to a more geometrical interpretation of the relation.

Core claim

The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes the Weil-Petersson volumes of the moduli space of N=1 supersymmetric Riemann surfaces of genus g with n Neveu-Schwarz boundaries and 2m Ramond punctures; these recursions are directly derivable from the spectral curve that allows the Laplace transforms of the volumes to be computed using topological recursion.

What carries the argument

The spectral curve for the N=1 supersymmetric case, on which topological recursion produces the Laplace transforms W of the volumes V.

If this is right

Topological recursion applied to the spectral curve reproduces the Stanford-Witten-Norbury volume recursion.
An alternative volume recursion can be derived from the same curve by moving Ramond information into the recursion kernels.
The alternative form places the Ramond data inside the kernels rather than the initial data, which may admit a direct geometrical reading.

Where Pith is reading between the lines

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

The placement of Ramond data in kernels versus initial conditions may suggest analogous reorganizations for other classes of moduli-space recursions.
If the spectral curve admits a direct geometric construction, the alternative recursion could translate into a new cutting-and-gluing rule for supersymmetric surfaces.
The equivalence supplies a consistency check that any future candidate spectral curve for related supersymmetric problems must satisfy.

Load-bearing premise

The spectral curve encodes the full N=1 supersymmetric data so that topological recursion on it exactly reproduces the Laplace transforms of the volumes.

What would settle it

Explicit computation of the volumes or their Laplace transforms for a low-genus case with at least one Ramond puncture, showing a numerical mismatch between the recursion and the output of topological recursion on the curve.

read the original abstract

The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes $V^{(2m)}_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of $N=1$ supersymmetric Riemann surfaces of genus $g$ with $n$ Neveu-Schwarz boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ Ramond punctures. Recently, a spectral curve has been derived that allows their Laplace transforms $W^{(2m)}_{g,n}(\{{\hat z}_i\})$ to be computed using topological recursion. We prove that the Stanford-Witten-Norbury volume recursion is directly derivable from the spectral curve. An alternative volume recursion can also be derived from it. The difference comes from whether the Ramond information is in the initial data, or in the volume recursion's kernels. The latter invites a geometrical understanding.

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 derives the Stanford-Witten-Norbury recursion directly from topological recursion on the spectral curve and produces an alternative version by shifting the Ramond data.

read the letter

The main thing to know is that this paper proves the Stanford-Witten-Norbury recursion for the Laplace transforms of the N=1 supersymmetric Weil-Petersson volumes follows from the spectral curve via topological recursion. It also extracts a second recursion from the same curve by placing the Ramond information in the kernels rather than the initial data.

What the paper does well is make that link explicit and show how the two recursions differ only in where the supersymmetric data sits. The derivation starts from the independently obtained spectral curve, so there is no circularity in the argument. The distinction between the two forms is presented cleanly and follows from the setup without extra assumptions.

The soft spots are minor. The result inherits whatever strength the earlier spectral curve has, since this paper takes it as given. The note that the kernel version might invite a geometrical understanding is left as a suggestion without further development, so that part stays open rather than resolved. No internal inconsistencies or hidden fitting issues show up in the argument as described.

This is for specialists already working on topological recursion, Mirzakhani-type recursions, and their supersymmetric extensions. A reader familiar with the Stanford-Witten-Norbury work and the spectral curve literature will find the explicit derivation useful. The paper engages the prior results directly and stays coherent on its own terms, so it deserves a serious referee.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the Stanford-Witten-Norbury generalization of Mirzakhani's recursion for the Weil-Petersson volumes V^{(2m)}_{g,n}({b_i}) of N=1 supersymmetric Riemann surfaces (with n Neveu-Schwarz boundaries and 2m Ramond punctures) is directly derivable from topological recursion on a spectral curve; the Laplace transforms W^{(2m)}_{g,n}({ẑ_i}) are obtained this way. An alternative recursion is also derived, with the difference arising from whether Ramond data resides in the initial conditions or the recursion kernels.

Significance. If the derivation holds, the work supplies a spectral-curve origin for both the original and an alternative volume recursion, opening the possibility of new computational techniques and a geometrical interpretation of the Ramond sector. The explicit reduction of the Stanford-Witten-Norbury recursion to topological recursion on an independently stated curve is a concrete technical contribution.

minor comments (2)

[Abstract] The abstract would benefit from a one-sentence statement of the explicit form of the spectral curve (including the Ramond sector) so that the starting point is visible without consulting prior references.
[Introduction] Notation for the Laplace transforms W^{(2m)}_{g,n} and the distinction between the two recursions could be introduced with a short table or diagram in §1 to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive summary of the results, and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation starts from independent spectral curve

full rationale

The paper states that a spectral curve was recently derived (in prior work) and then proves that the Stanford-Witten-Norbury recursion follows from topological recursion on that curve. The central claim is this derivation step itself, which is performed in the present manuscript rather than being presupposed by definition or by a load-bearing self-citation chain. No equation reduces the output recursion to the input volumes by construction, and the spectral curve is treated as an external starting point that encodes the supersymmetric data independently of the recursion being derived.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the prior derivation of the spectral curve and standard properties of topological recursion; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Topological recursion on the given spectral curve computes the Laplace transforms W of the volumes
Invoked to derive the volume recursion from the spectral curve.

pith-pipeline@v0.9.1-grok · 5696 in / 1015 out tokens · 22708 ms · 2026-06-26T15:45:02.667088+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 5 linked inside Pith

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The core point is that (90) and (89) are the same since s2 =−eΓ2

+ b4 1 +b 4 2 384 + b2 1b2 2 96 +O(s 8),(91) as can be obtained by using the volume recursion directly. The core point is that (90) and (89) are the same since s2 =−eΓ2. Just as remarked at the end of the previous section, it is remarkable that this same series is packaged, as physical information in two such different ways. VIII. CONCLUDING REMARKS In co...
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Super weil-petersson measures on the moduli space of curves,

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arXiv 2024
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The core point is that (90) and (89) are the same since s2 =−eΓ2

+ b4 1 +b 4 2 384 + b2 1b2 2 96 +O(s 8),(91) as can be obtained by using the volume recursion directly. The core point is that (90) and (89) are the same since s2 =−eΓ2. Just as remarked at the end of the previous section, it is remarkable that this same series is packaged, as physical information in two such different ways. VIII. CONCLUDING REMARKS In co...

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Super weil-petersson measures on the moduli space of curves,

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

arXiv 2024

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Stanford and E

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

arXiv 2020

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Norbury, J

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

arXiv 2026

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Mirzakhani, Invent

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

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C. V. Johnson, (2026), arXiv:2604.11902 [hep-th]

Pith/arXiv arXiv 2026

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Pith/arXiv arXiv 2026

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Pith/arXiv arXiv 2006

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