REVIEW
What crossing symmetry forbids just above $c=1$
High Energy Physics - Theory · Submitted 2 Nov 2026 · 54 references
Pith's verdict: Serious numerical bootstrap result: new excluded region just above c=1 and Δ_gap < 0.17, with the interval claim resting on grids rather than proof.
T0 · AI review
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Referee report
The paper
The paper claims
Crossing symmetry bans operator dimensions $0.17$ to $1$ just above $c=1$; with the modular bootstrap this forces a compact unitary CFT's lightest primary below $0.17$.
The reading
What carries the argument
The machine is the numerical conformal bootstrap run on Virasoro blocks, the holomorphic-times-antiholomorphic contributions of Virasoro-symmetry multiplets. At each candidate point $(c,\Delta_\phi)$ the paper asks whether a linear functional $\alpha$ on the derivatives $\partial_z^m\partial_{\bar z}^n$ (with $m<n$, $m+n$ odd, $m+n\le\Lambda$) of the crossing combination $F_h(z)F_{\bar h}(\bar z)-F_h(1-z)F_{\bar h}(1-\bar z)$ at $z=\bar z=1/2$ can satisfy $\alpha[F_{0,0}]>0$ while staying non-negative on every block with $\Delta_O\ge\max(\Delta_\phi,\ell_O)$; such an $\alpha$ is a certificate that no unitary CFT has a primary $\phi$ of that dimension. Zamolodchikov's recursion in the elliptic nome writes each Taylor coefficient as a polynomial in the conformal weight divided by a Kac-determinant denominator, and for $1<c<25$ that denominator is positive on $h\ge0$, so the semidefinite program solver can search for $\alpha$ directly. A certificate found at order $\Lambda$ remains valid at every higher order, so the excluded region can only grow with $\Lambda$, and the boundary $c_{\rm min}(\Delta_\phi)$ is located by bisection. At the boundary the zeros of the extremal functional reproduce the dimensions of the known theories (twist $10/7$ and $16/7$ families for the $N=1$ model; $8/5$ and $12/7$ for the two $c=8/7$ theories), which is how the paper identifies the sharp minima with the $\mathbb{Z}_5$ parafermion and its orbifold. The compact-boson fact that $\cos(X/R)\times\cos(X/R)$ contains only $\cos(2X/R)$ below the continuum is what lets every $\Delta_\phi\le2$ satisfy the no-lighter-primary assumption at $c=1$.
Extended reading notes
Core claim
On its own terms, the paper's claim is a two-part statement about the $(c,\Delta_\phi)$ plane for a real scalar Virasoro primary $\phi$ whose self-OPE contains no lighter primary. First, just above $c=1$ the crossing equation of $\langle\phi(0)\phi(z,\bar z)\phi(1)\phi(\infty)\rangle$ has no solution with non-negative squared OPE coefficients in the interval $0.17\lesssim\Delta_\phi\lesssim1$: a positive linear functional on the identity cancels the crossing of every allowed Virasoro block. At $c=1$ the same claim is false, because the compact boson realizes every $\Delta_\phi\le2$. Second, combining the excluded interval with the modular-bootstrap bound that the lightest primary near $c=1$ is a scalar below about $1/2$ forces $\Delta_{\rm gap}<0.167$ at $c=1.0005$ and $\Delta_{\rm gap}<0.169$ at $c=1.001$ (inequality (5)), with the same argument excluding gaps in $0.3<\Delta_{\rm gap}<0.5$ up to $c=1.14$. Away from $c=1$, the boundary $c_{\rm min}(\Delta_\phi)$ of the excluded region has two sharp minima near $\Delta_\phi=1/2$ and $4/7$ that approach $c=8/7$ within $2\times10^{-7}$ at order $\Lambda=31$; these are exactly the dimensions of the twist field of $\mathrm{WB}_2(5,7)$ and the energy operator $\epsilon_1$ of the $\mathbb{Z}_5$ parafermion, the two known theories at that central charge.
Load-bearing premise
The load-bearing premise is that the crossing exclusions are valid everywhere, although positivity of the functional is checked only on a sample: for spins above 40 on 28 fixed-twist lines plus large-spin and large-dimension limits, near the poles of the block recursion on finite grids, and across the interval between the lowest excluded dimension and the modular bound on grids of spacing at most 0.07 — a single negative value between sampled points would invalidate the exclusion and with it the $\Delta_{\rm gap}<0.17$ bound.
Editorial extensions
If this is right
- At $c=1.0005$ and $c=1.001$, every compact unitary bosonic CFT must contain a scalar lighter than about $0.17$; no such theory can have a lightest primary anywhere in the range $0.17\lesssim\Delta\lesssim1$.
- Up to $c=1.14$, gaps in $0.3<\Delta_{\rm gap}<0.5$ are impossible, filling a window in which modular bootstrap alone had found no candidate spectra below $c=1.06$.
- The bounds are monotone in the numerical order: a functional that excludes a point at order $\Lambda$ works at every higher order, so the excluded interval grows and the $\Delta_{\rm gap}<0.17$ bound can only tighten as $\Lambda$ increases.
- If the excluded interval reaches $c=1$ as $\Lambda\to\infty$, the space of allowed theories is discontinuous at $c=1$, since the compact boson realizes every $\Delta_\phi\le2$ there.
- The two theories at $c=8/7$ sit exactly at the sharp minima of the boundary and saturate the upper bound on the lightest scalar in $\phi\times\phi$ heavier than $\phi$ ($8/5$ for the twist field, $12/7$ for the parafermion energy operator), so crossing symmetry singles out their spectra at those points.
Reading between the lines
- My reading: the lower endpoint of the excluded interval at $c=1.0005$ falls from $0.229$ at $\Lambda=11$ to $0.159$ at $\Lambda=27$, so the $0.167$ and $0.169$ certificates are conservative; the true bound is plausibly smaller, and an analytic functional realizing it would be the natural limit of this construction.
- My inference: the argument compresses the open existence question into the band $\Delta_{\rm gap}<0.17$ — at $c=1.001$ a compact unitary CFT either contains a very light scalar or does not exist, so constructing or excluding such a theory is now a sharply focused problem rather than a search over all gaps below $1/2$.
- My testable suggestion: run the same crossing scan with positivity imposed on a dense twist grid (spacing $10^{-4}$) near the poles of the block recursion at $c\approx1.156$; if the plateau just below the $N=1$ minimal model's central charge persists and its extremal zeros keep matching the twist-$10/7$ and twist-$16/7$ families, it would indicate a genuine family of crossing solutions there rathe
Original abstract
What does a unitary two-dimensional conformal field theory look like just above $c=1$? Using the numerical conformal bootstrap, we study the four-point function of a scalar Virasoro primary $\phi$ such that no primary lighter than $\phi$ appears in its operator product expansion with itself. Slightly above $c=1$, crossing symmetry excludes the interval $0.17\lesssim\Delta_\phi\lesssim1$, although the compact boson realizes every such $\Delta_\phi$ at $c=1$. Combining this exclusion with results from the modular bootstrap, we show that the lightest primary of a compact unitary conformal field theory at $c=1.0005$ or $1.001$ has dimension $\Delta_{\rm gap}<0.17$. Further from $c=1$, the largest excluded central charge at fixed $\Delta_\phi$ has two sharp minima near $\Delta_\phi=1/2$ and $4/7$. At these values it lies within $2\times10^{-7}$ of $c=8/7$, where the $\mathbb{Z}_5$ parafermion and its charge-conjugation orbifold have scalars of exactly these dimensions.
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.
The abstract, sentence by sentence
What does a unitary two-dimensional conformal field theory look like just above $c=1$? Using the numerical conformal bootstrap, we study the four-point function of a scalar Virasoro primary $\phi$ such that no primary lighter than $\phi$ appears in its operator product expansion with itself. Slightly above $c=1$, crossing symmetry excludes the interval $0.17\lesssim\Delta_\phi\lesssim1$, although the compact boson realizes every such $\Delta_\phi$ at $c=1$. Combining this exclusion with results from the modular bootstrap, we show that the lightest primary of a compact unitary conformal field theory at $c=1.0005$ or $1.001$ has dimension $\Delta_{\rm gap}<0.17$. Further from $c=1$, the largest excluded central charge at fixed $\Delta_\phi$ has two sharp minima near $\Delta_\phi=1/2$ and $4/7$. At these values it lies within $2\times10^{-7}$ of $c=8/7$, where the $\mathbb{Z}_5$ parafermion and its charge-conjugation orbifold have scalars of exactly these dimensions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | What does a unitary two-dimensional conformal field theory look like just above $c=1$? | background 0.87 | 0.04 | 0.02 | unclear 1.00 |
| s1 | Using the numerical conformal bootstrap, we study the four-point function of a scalar Virasoro primary $\phi$ such that no primary lighter than $\phi$ appears in its operator product expansion with itself. | contribution 0.64 | 0.88 | 0.03 | asserted 0.76 |
| s2 | Slightly above $c=1$, crossing symmetry excludes the interval $0.17\lesssim\Delta_\phi\lesssim1$, although the compact boson realizes every such $\Delta_\phi$ at $c=1$. | result 1.00 | 0.90 | 0.54 | asserted 0.97 |
| s3 | Combining this exclusion with results from the modular bootstrap, we show that the lightest primary of a compact unitary conformal field theory at $c=1.0005$ or $1.001$ has dimension $\Delta_{\rm gap}<0.17$. | result 0.99 | 0.94 | 0.49 | asserted 1.00 |
| s4 | Further from $c=1$, the largest excluded central charge at fixed $\Delta_\phi$ has two sharp minima near $\Delta_\phi=1/2$ and $4/7$. | result 1.00 | 0.87 | 0.09 | asserted 1.00 |
| s5 | At these values it lies within $2\times10^{-7}$ of $c=8/7$, where the $\mathbb{Z}_5$ parafermion and its charge-conjugation orbifold have scalars of exactly these dimensions. | result 0.88 | 0.71 | 0.08 | asserted 0.99 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2611.03217 by the authors.
References
Discussion
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