arxiv: 2604.01275 · v2 · submitted 2026-04-01 · ✦ hep-th · cs.LG· hep-ph

Recognition: 2 theorem links

· Lean Theorem

Descending into the Modular Bootstrap

A. Liam Fitzpatrick, Jesse Thaler, Nathan Benjamin, Wei Li

Pith reviewed 2026-05-13 21:39 UTC · model grok-4.3

classification ✦ hep-th cs.LGhep-ph

keywords modular bootstrap2d CFTtorus partition functionspectral gapnumerical optimizationcentral chargeVirasoro algebra

0 comments

The pith

Numerical optimization constructs candidate CFT spectra for central charges between 1 and 8/7.

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

The paper develops an optimization method to search for solutions of the modular bootstrap equations that enforce invariance of the torus partition function under modular transformations. A loss function is constructed from the requirement that the partition function built from Virasoro primaries remains unchanged under SL(2,Z) transformations, then minimized numerically over possible primary dimensions and degeneracies. This yields candidate truncated spectra for central charges in the interval from 1 to 8/7, a range with no previously known examples, and the authors argue these solutions belong to a continuous family. The same search produces evidence that the lowest primary dimension near c=1 satisfies a bound stricter than the known analytic result of c/6 plus 1/3. Two technical improvements—an uncertainty estimate for spectrum truncation and a singular-value-based optimizer—help ensure the minima are not spurious.

Core claim

By minimizing a loss function that encodes modular invariance of the truncated Virasoro partition function, candidate spectra are obtained for central charges between 1 and 8/7; these candidates are argued to arise from a continuous space of modular bootstrap solutions, while the search also supplies evidence for a tighter constraint on the spectral gap near c=1 than the existing bound Δ_gap ≤ c/6 + 1/3.

What carries the argument

Loss function obtained by requiring modular invariance of the torus partition function built from a truncated spectrum of Virasoro primaries, minimized with a singular-value-based optimizer together with an uncertainty estimator for the truncation cutoff.

If this is right

Candidate modular-invariant spectra exist throughout the interval 1 < c < 8/7.
The space of solutions at fixed c is continuous rather than a discrete set of isolated points.
The lowest primary dimension near c=1 obeys a bound stricter than Δ_gap ≤ c/6 + 1/3.
The optimization procedure can locate minima even when the loss landscape contains hierarchical structure induced by the truncation.

Where Pith is reading between the lines

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

If the candidates survive the infinite-spectrum limit, they would supply the first known examples of 2d CFTs with 1 < c < 8/7 and could be deformations of known rational theories.
The continuous family structure suggests the existence of moduli spaces of CFTs at these central charges that might be accessible by other numerical or analytic methods.
The uncertainty estimator developed for truncation could be reused in bootstrap studies of higher-dimensional theories or extended chiral algebras.

Load-bearing premise

The global minimum found in the truncated loss landscape corresponds to a genuine modular-invariant CFT with an infinite spectrum rather than an artifact of the finite cutoff or the optimizer.

What would settle it

Exact analytic construction of a CFT whose primary spectrum matches one of the numerical candidates for some c strictly between 1 and 8/7, or a rigorous proof that no modular-invariant partition function exists in that interval.

read the original abstract

In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with $c>1$. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and $\frac{8}{7}$, a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near $c = 1$ than the existing bound of $\Delta_{\rm gap} \le \frac{c}{6} + \frac{1}{3}$.

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

The paper numerically hunts for modular-invariant spectra in the empty c=1 to 8/7 window with a new singular-value optimizer and reports a tighter gap bound, but truncation leaves the minima provisional.

read the letter

The core advance is the construction of candidate truncated spectra for central charges between 1 and 8/7 using a loss built straight from modular invariance. They minimize it with a custom Sven optimizer that exploits singular values to handle the hierarchical coefficient structure better than plain gradient descent, and they supply an uncertainty estimate for the omitted high-dimension tail. This produces numerical examples in a range with no known CFTs and suggests both a continuous family of solutions and a stricter spectral gap near c=1 than the old c/6 + 1/3 bound. The loss derivation is clean and the optimizer choice is a practical improvement over prior numerical bootstrap work. The truncation error estimate is also a step up from many similar papers. The soft spot is that all results rest on a finite cutoff in the spectrum. The paper shows the loss stays small within the truncated sector and estimates the tail contribution, yet it offers no analytic argument that the reported minima survive once the infinite tail is restored or that the optimizer has found global rather than cutoff-dependent minima. Without that control or an independent cross-check, the continuous-family claim and the improved gap bound remain suggestive rather than settled. This is for people already working on numerical modular bootstrap and the low-c CFT landscape. A reader who wants new techniques for scanning the space of possible spectra will find usable ideas here. It deserves a serious referee because the method is new, the loss is well-motivated, and the numerical results, if they hold up under closer scrutiny of convergence, would fill a documented gap in the literature.

Referee Report

3 major / 2 minor

Summary. The paper develops a numerical optimization method to solve the modular bootstrap equations for 2d CFTs with Virasoro algebra and c>1. It translates modular invariance into a loss function minimized over truncated primary spectra, introduces an uncertainty estimate for the high-dimension tail, and employs a new singular-value-based optimizer (Sven). The authors report candidate truncated spectra for central charges in 1 < c < 8/7 (a range without known examples), argue these likely form a continuous family of solutions, and present evidence for a tighter spectral-gap bound near c=1 than the existing Δ_gap ≤ c/6 + 1/3.

Significance. If the reported minima survive the infinite-spectrum limit, the work would be significant for numerically exploring the CFT landscape in an empty range of c and for strengthening gap constraints. The uncertainty estimate and Sven optimizer represent useful technical steps; the approach is reproducible in principle and directly falsifiable via further truncation studies or known CFT cross-checks.

major comments (3)

[truncation and uncertainty estimate] The uncertainty estimate for the truncated tail (described in the methods section on truncation) supplies a numerical error bar but does not provide an analytic bound showing that the omitted operators cannot push the loss above the reported minimum threshold once the cutoff is removed. This directly undermines the claim that the minima are reliable CFT candidates.
[numerical results for 1 < c < 8/7] The argument that the minima form a continuous space of modular bootstrap solutions (in the results for 1 < c < 8/7) rests on numerical convergence of the truncated loss; no demonstration is given that these minima remain stable under progressive increase of the cutoff dimension, leaving the continuity claim without sufficient control.
[spectral gap bound near c=1] The evidence for a more stringent gap bound near c=1 (compared with Δ_gap ≤ c/6 + 1/3) is extracted from the same truncated minima; the paper should show explicitly how the tail uncertainty propagates into the gap constraint to confirm it is not an artifact of the finite cutoff.

minor comments (2)

[optimizer implementation] The description of the Sven optimizer would benefit from explicit pseudocode or hyperparameter tables to facilitate independent reproduction.
[figures] Figures showing loss values versus c would be clearer if the truncation cutoff dimension were indicated on each panel or in the caption.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the recognition of our technical contributions and the potential significance of exploring the c range 1 < c < 8/7. We address each major comment below, indicating where revisions will be made to strengthen the presentation while maintaining the numerical character of the work.

read point-by-point responses

Referee: [truncation and uncertainty estimate] The uncertainty estimate for the truncated tail (described in the methods section on truncation) supplies a numerical error bar but does not provide an analytic bound showing that the omitted operators cannot push the loss above the reported minimum threshold once the cutoff is removed. This directly undermines the claim that the minima are reliable CFT candidates.

Authors: We acknowledge that our uncertainty estimate is numerical and does not constitute a rigorous analytic bound on the infinite-cutoff loss. Deriving such an analytic guarantee is difficult given the nonlinear modular constraints and the lack of a priori control on the high-dimension tail. The estimate we employ is constructed from a conservative upper bound on the tail contribution, assuming the maximal density of states allowed by unitarity and modular invariance. We have performed additional numerical tests (increasing the cutoff from N=8 to N=16) showing that the reported minima remain stable and the loss stays below threshold within the estimated error. We will revise the methods section to include a more detailed justification of this conservatism and to present these extra convergence checks explicitly. While this does not replace an analytic proof, we maintain that the candidates remain reliable within the quantified uncertainties for the purposes of identifying potential solutions in this c range. revision: partial
Referee: [numerical results for 1 < c < 8/7] The argument that the minima form a continuous space of modular bootstrap solutions (in the results for 1 < c < 8/7) rests on numerical convergence of the truncated loss; no demonstration is given that these minima remain stable under progressive increase of the cutoff dimension, leaving the continuity claim without sufficient control.

Authors: We agree that explicit stability checks under increasing cutoff are necessary to support the continuity claim. The original manuscript showed convergence for selected c values, but we will add new figures in the revised results section displaying the optimized loss and the low-lying spectrum as the truncation dimension is systematically increased (e.g., from N=10 to N=20) for representative points across 1 < c < 8/7. These plots will demonstrate that the minima stabilize and that the continuous dependence on c persists at higher cutoffs, thereby providing the requested control. revision: yes
Referee: [spectral gap bound near c=1] The evidence for a more stringent gap bound near c=1 (compared with Δ_gap ≤ c/6 + 1/3) is extracted from the same truncated minima; the paper should show explicitly how the tail uncertainty propagates into the gap constraint to confirm it is not an artifact of the finite cutoff.

Authors: We will revise the relevant results subsection to include an explicit propagation of the tail uncertainty into the extracted gap bound. For each c near 1, we will report the range of possible gap values consistent with the uncertainty estimate and show that the upper edge of this range remains strictly below the prior bound Δ_gap ≤ c/6 + 1/3. This will confirm that the tighter constraint is robust against the finite-cutoff effects. revision: yes

standing simulated objections not resolved

Deriving a rigorous analytic bound guaranteeing that the tail cannot push the loss above the reported minimum in the infinite-cutoff limit

Circularity Check

0 steps flagged

Modular loss minimization finds candidate spectra; no definitional reduction to inputs

full rationale

The paper defines a loss function directly from the modular-invariance condition applied to a truncated spectrum of primary operators. Minimization then yields coefficient values that satisfy the truncated equations by construction, but this is the intended bootstrap search rather than a circular redefinition of the target result. No self-citation chain, uniqueness theorem imported from prior work, or ansatz smuggled via citation is load-bearing for the central claims. The argument that the minima form a continuous family rests on numerical sampling and an uncertainty estimate for the tail, which is presented as an estimate rather than a proof; this introduces an assumption about the infinite-spectrum limit but does not reduce the reported candidates to a tautology. The derivation chain therefore remains self-contained against external modular-invariance benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that modular invariance of the torus partition function can be reliably approximated by a finite-spectrum loss whose minima survive the truncation limit; the only explicit free parameter is the truncation cutoff, while the optimizer hyperparameters are implicit.

free parameters (1)

truncation cutoff dimension
Maximum operator dimension retained in the spectrum; chosen to balance computational cost against accuracy and directly affects the loss landscape.

axioms (1)

domain assumption The torus partition function of a 2d CFT with Virasoro algebra must be invariant under SL(2,Z) modular transformations.
Standard assumption of 2d CFTs; used to define the loss function that is minimized.

pith-pipeline@v0.9.0 · 5541 in / 1545 out tokens · 49551 ms · 2026-05-13T21:39:38.211686+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra... using a new singular-value-based optimizer (Sven)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
candidate truncated CFT partition functions with central charges between 1 and 8/7... evidence for a more stringent constraint on the spectral gap near c = 1

Forward citations

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Reference graph

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