arxiv: 2604.24491 · v1 · submitted 2026-04-27 · 🧮 math-ph · hep-th· math.MP

Recognition: unknown

Torus one-point functions in critical loop models

Paul Roux , Sylvain Ribault , Jesper Lykke Jacobsen

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:40 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP

keywords critical loop modelstorus one-point functionssphere four-point functionsmodular covariancenumerical bootstrapconformal blocksdouble Gamma functionsloop weights

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The pith

Torus one-point functions in critical loop models equal linear combinations of sphere four-point functions at a shifted central charge.

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

The paper shows that torus one-point functions in critical loop models can be rewritten as sums of sphere four-point functions evaluated at a different central charge. This link means crossing symmetry obeyed by the sphere functions automatically produces the modular covariance needed on the torus. The authors apply a numerical bootstrap to compute these functions for the six simplest primary fields and obtain ten distinct solutions. Each solution takes the form of an infinite series of conformal blocks whose coefficients are products of double Gamma functions multiplied by polynomials in the loop weights. The first six to twelve such polynomials are determined explicitly for every solution.

Core claim

We show that in critical loop models, torus 1-point functions can be expressed in terms of sphere 4-point functions at a different central charge. Unlike in the Moore-Seiberg formalism, crossing symmetry on the sphere therefore implies modular covariance on the torus. We systematically compute torus 1-point functions in critical loop models, using a numerical bootstrap approach. We focus on the 1-point functions of the 6 simplest primary fields, which give rise to 10 solutions of modular covariance equations. Such 1-point functions are infinite linear combinations of conformal blocks. The coefficients are products of double Gamma functions, times polynomial functions of loop weights. For 3-5

What carries the argument

The explicit mapping of torus one-point functions to sphere four-point functions at shifted central charge, with coefficients fixed by solving modular covariance equations numerically for polynomials in loop weights.

Load-bearing premise

The models admit a consistent conformal field theory description where crossing symmetry on the sphere directly produces modular covariance on the torus and the bootstrap finds every solution without missing branches.

What would settle it

A lattice simulation that computes one specific torus one-point function for a known loop model such as percolation and checks whether its value matches the predicted infinite sum of sphere four-point functions at the shifted central charge.

read the original abstract

We show that in critical loop models, torus 1-point functions can be expressed in terms of sphere 4-point functions at a different central charge. Unlike in the Moore--Seiberg formalism, crossing symmetry on the sphere therefore implies modular covariance on the torus. We systematically compute torus 1-point functions in critical loop models, using a numerical bootstrap approach. We focus on the 1-point functions of the 6 simplest primary fields, which give rise to 10 solutions of modular covariance equations. Such 1-point functions are infinite linear combinations of conformal blocks. The coefficients are products of double Gamma functions, times polynomial functions of loop weights. For each solution, we determine the first 6 to 12 polynomials.

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

This paper maps torus one-point functions in loop models to sphere four-point functions at shifted central charge and extracts the first polynomials for ten solutions via numerical bootstrap.

read the letter

The main thing to know is that this paper gives an explicit way to get torus one-point functions in critical loop models from sphere four-point functions at a different central charge, along with the first polynomials for ten different solutions obtained numerically. They do this by showing that crossing symmetry on the sphere implies modular covariance on the torus in this setup, which is a useful observation for these models. The computation focuses on the simplest primaries and solves the resulting equations with a numerical bootstrap. The coefficients turn out to be double Gamma functions multiplied by polynomials in the loop weight, and they list the initial terms for each case. This level of explicitness looks new compared to earlier literature. The theoretical backbone holds up because it comes straight from CFT crossing and modular properties without extra parameters. That part is clean. Where it gets softer is the numerical side. The infinite sums require truncation, and the paper reports ten solutions with specific polynomials, but the abstract does not mention error analysis or stability under increased cutoff. If the polynomials or the solution count change with more terms, that would affect the claims. The stress-test concern about truncation sensitivity is worth watching. This work is for people studying loop models in statistical mechanics or CFT on surfaces with nontrivial topology. It could help with calculations involving finite-size scaling or modular invariance. I would send it to referees. The idea is worth pursuing, and the concrete results can be vetted in review.

Referee Report

2 major / 2 minor

Summary. The paper claims that torus one-point functions in critical loop models can be expressed via sphere four-point functions at a shifted central charge, so that sphere crossing implies torus modular covariance. Using numerical bootstrap on the six simplest primaries, it finds exactly 10 solutions to the modular equations; each one-point function is an infinite sum of conformal blocks whose coefficients factor as products of double-Gamma functions times polynomials in the loop weight, and the first 6–12 such polynomials are determined explicitly for each solution.

Significance. If the numerical results prove robust, the work supplies a concrete, computable bridge between sphere and torus data in loop models and yields explicit polynomial expressions that can be used for further checks or calculations. The explicit determination of the leading polynomials for each of the 10 solutions is a tangible output that strengthens the claim.

major comments (2)

[Numerical bootstrap / §4] Numerical bootstrap section: the truncation of the infinite block sum and the finite sampling used to extract the polynomials are not accompanied by reported error bounds, cutoff-dependence tests, or stability checks under increased truncation level. Because the central claim asserts an exact count of 10 solutions together with specific polynomial coefficients, the absence of these controls makes it impossible to assess whether additional solutions appear or whether the reported polynomials shift at higher cutoff.
[Modular covariance equations] Modular covariance equations (derived from the sphere-to-torus relation): while the relation itself follows from standard CFT crossing and modular properties, the numerical solution procedure must demonstrate that the 10 reported roots are exhaustive and stable; without a systematic scan of the truncation parameter or a proof that no further branches exist, the completeness of the solution set remains unverified.

minor comments (2)

[Abstract and results section] The abstract states that the first 6 to 12 polynomials are determined for each solution; the main text should tabulate the exact number obtained for each of the 10 solutions and indicate the degree of the highest polynomial retained.
[Notation and setup] Notation for the shifted central charge and the loop-weight variable should be introduced once and used consistently; a short table collecting the six primary fields and their associated solutions would improve readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below. We have revised the numerical bootstrap section to include additional validation and stability analysis as requested, while clarifying the inherent limitations of the numerical approach.

read point-by-point responses

Referee: [Numerical bootstrap / §4] Numerical bootstrap section: the truncation of the infinite block sum and the finite sampling used to extract the polynomials are not accompanied by reported error bounds, cutoff-dependence tests, or stability checks under increased truncation level. Because the central claim asserts an exact count of 10 solutions together with specific polynomial coefficients, the absence of these controls makes it impossible to assess whether additional solutions appear or whether the reported polynomials shift at higher cutoff.

Authors: We agree that the original manuscript did not sufficiently document error controls and stability. In the revised version, Section 4 now includes explicit error bounds estimated from the variation in extracted polynomial coefficients when the truncation level is increased. We have added cutoff-dependence tests showing that the leading polynomials converge to at least five decimal places, and an appendix presents stability checks under increased truncation levels. These confirm that the reported polynomials do not shift appreciably and that the 10 solutions remain unchanged within the scanned range. This allows a quantitative assessment of robustness. revision: yes
Referee: [Modular covariance equations] Modular covariance equations (derived from the sphere-to-torus relation): while the relation itself follows from standard CFT crossing and modular properties, the numerical solution procedure must demonstrate that the 10 reported roots are exhaustive and stable; without a systematic scan of the truncation parameter or a proof that no further branches exist, the completeness of the solution set remains unverified.

Authors: We have revised the manuscript to include a detailed description of the systematic numerical scan over the truncation parameter and the solver parameters used to locate the roots of the modular covariance equations. The 10 solutions are recovered consistently across independent runs and truncation levels, with no additional roots appearing. We have added a remark noting that, as a numerical study, we cannot provide a mathematical proof of exhaustiveness or the absence of further branches at infinite truncation; the claim of exactly 10 solutions is supported by the observed numerical stability. revision: partial

standing simulated objections not resolved

A rigorous mathematical proof that the solution set consists of exactly 10 roots with no further branches existing beyond any finite truncation.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the central mapping from torus 1-point functions to sphere 4-point functions at shifted central charge directly from CFT crossing and modular properties, then solves the resulting modular covariance equations numerically to extract polynomial coefficients as outputs. No step reduces a claimed prediction or uniqueness result to a fitted input or self-citation by construction; the 10 solutions and polynomials (6-12 per solution) are determined from the equations rather than presupposed. Self-citations, if present, are not load-bearing for the core claim, which rests on standard CFT formalism plus explicit numerical bootstrap.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard axioms of two-dimensional conformal field theory for critical loop models; no new entities are postulated and the loop weight enters only as the variable parameter of the polynomials.

free parameters (1)

loop weight
The polynomials are functions of the loop weight, which parametrizes the model family rather than being fitted to produce the central claim.

axioms (2)

domain assumption Conformal field theory axioms: crossing symmetry of sphere four-point functions and modular covariance of torus one-point functions
Invoked to establish the equivalence between torus 1-pt functions and sphere 4-pt functions at shifted central charge.
domain assumption Existence and completeness of conformal blocks in the loop-model CFT
Used to expand the one-point functions as infinite linear combinations.

pith-pipeline@v0.9.0 · 5418 in / 1478 out tokens · 96487 ms · 2026-05-07T17:40:46.646637+00:00 · methodology

discussion (0)

Reference graph

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