Cubic Redundancy Channel in Conformal Bootstrap

Updated 6 July 2026

The cubic redundancy channel is a forbidden region in bootstrap parameter space that arises by imposing a spectral gap in the cubic B-channel, preventing accidental O(3) symmetry enhancement.
It distinguishes the cubic fixed point from the O(3) model by exploiting representation-theoretic mechanisms and redundant operators to exclude low-lying scalar primaries.
Numerical bootstrap analyses using gaps in the B, X, and Z channels reveal an exclusion strip along ΔX = ΔZ, which serves as a diagnostic tool for symmetry differentiation.

Searching arXiv for the exact topic and closely related bootstrap work to ground the article in the current literature. Searching arXiv for "Cubic Redundancy Channel" and related conformal bootstrap papers. The cubic redundancy channel is a numerical-conformal-bootstrap obstruction to accidental symmetry enhancement in the three-dimensional cubic model. In the formulation introduced for cubic versus $O(3)$ criticality, it is the strip of disallowed parameter space that appears when one imposes a gap on the leading scalar in the cubic $B$ -channel, a gap justified by the presence of redundant operators in the cubic theory but incompatible with the $O(3)$ -symmetric spectrum. The resulting exclusion band lies along the locus $\Delta_X=\Delta_Z$ , precisely where cubic data would recombine into the $O(3)$ tensor sector, so the channel functions as a diagnostic for and obstruction to effective $O(3)$ enhancement (Kousvos et al., 7 Jul 2025).

1. Definition within the redundancy-channel framework

A redundancy channel is a forbidden region in bootstrap parameter space that emerges after imposing spectral gaps motivated by redundant operators. The setup begins with crossing equations written for a symmetry $H$ , while the same equations may also admit theories with a larger symmetry $G\supset H$ . In that situation, operators that are redundant in the $H$ -theory need not appear as physical primaries in the $G$ -theory, so one can impose a gap that is justified for $B$ 0 but incompatible with $B$ 1; numerically, that gap carves out the region where the solution would otherwise exhibit the larger symmetry (Kousvos et al., 7 Jul 2025).

For the cubic model, $B$ 2 and $B$ 3. The cubic redundancy channel is the strip of disallowed parameter space in the $B$ 4 plane, and likewise in the full $B$ 5 space, produced by imposing a gap

$B$ 6

in the cubic $B$ 7-channel. The strip lies along $B$ 8, and that alignment is not incidental: in the cubic bootstrap, the equality $B$ 9 is the signature that the assumed cubic symmetry is effectively enhancing to $O(3)$ 0 (Kousvos et al., 7 Jul 2025).

2. Cubic versus $O(3)$ 1 criticality

The field-theoretic starting point is the hypercubic action in $O(3)$ 2,

$O(3)$ 3

with $O(3)$ 4 if all indices are equal and $O(3)$ 5 otherwise. For $O(3)$ 6, nonzero $O(3)$ 7 gives the cubic model with global symmetry $O(3)$ 8, while setting $O(3)$ 9 yields the $\Delta_X=\Delta_Z$ 0 model (Kousvos et al., 7 Jul 2025).

The relevant operator content is organized by how $\Delta_X=\Delta_Z$ 1 representations decompose under the cubic group. The leading rank-2 traceless symmetric tensor

$\Delta_X=\Delta_Z$ 2

splits into two cubic irreducible representations, denoted $\Delta_X=\Delta_Z$ 3 and $\Delta_X=\Delta_Z$ 4. In an $\Delta_X=\Delta_Z$ 5-symmetric CFT, these two sectors are degenerate,

$\Delta_X=\Delta_Z$ 6

whereas in a genuinely cubic CFT they need not coincide. This makes the diagonal $\Delta_X=\Delta_Z$ 7 the natural geometric locus of symmetry enhancement in bootstrap plots (Kousvos et al., 7 Jul 2025).

The same decomposition logic underlies the importance of the cubic irrep $\Delta_X=\Delta_Z$ 8. In the cubic theory, the $\Delta_X=\Delta_Z$ 9-channel becomes the location where broken $O(3)$ 0 currents and quartic operators mix, and that mixing removes what would otherwise be a low-lying scalar primary. The cubic redundancy channel is therefore not a generic exclusion artifact; it is tied to a specific representation-theoretic mechanism in the symmetry breaking $O(3)$ 1 (Kousvos et al., 7 Jul 2025).

3. Redundant operators and the broken-current mechanism

In the paper’s formulation, a redundant operator is proportional to the equations of motion,

$O(3)$ 2

Operators of the form $O(3)$ 3, $O(3)$ 4, or $O(3)$ 5 have correlation functions that are pure contact terms and therefore do not appear as genuine primaries in the separated-point bootstrap spectrum (Kousvos et al., 7 Jul 2025).

The key cubic mechanism begins with the $O(3)$ 6 conserved currents

$O(3)$ 7

At the cubic fixed point these currents are no longer conserved, and their divergence is

$O(3)$ 8

In the same cubic $O(3)$ 9 representation there is also the quartic operator

$O(3)$ 0

The two relevant scalar structures in the $O(3)$ 1-channel,

$O(3)$ 2

mix so that one combination is redundant and the other is a descendant of the broken current. As a result, no genuine primary of $O(3)$ 3-type remains in the cubic $O(3)$ 4 spin-0 channel (Kousvos et al., 7 Jul 2025).

This mechanism is summarized by the mixing relation

$O(3)$ 5

with

$O(3)$ 6

In the cubic theory the leading physical $O(3)$ 7-scalar must therefore be of higher type, such as $O(3)$ 8, while in the $O(3)$ 9 theory there remains a low-lying $H$ 0-type primary in the corresponding sector (Kousvos et al., 7 Jul 2025).

Theory	Symmetry	Leading $H$ 1-scalar dimension
Cubic fixed point	$H$ 2	$H$ 3
$H$ 4 fixed point	$H$ 5	$H$ 6

These values, quoted from Padé-resummed $H$ 7-expansion estimates, are the numerical basis for imposing a bootstrap gap around $H$ 8 in the cubic theory while simultaneously excluding $H$ 9-symmetric solutions (Kousvos et al., 7 Jul 2025).

4. Bootstrap realization of the channel

In practical bootstrap computations, the central spectral assumption is a gap in the cubic $G\supset H$ 0-sector,

$G\supset H$ 1

The paper also studies mixed-correlator systems with further assumptions such as

$G\supset H$ 2

together with the Ward-identity condition

$G\supset H$ 3

and, in the full $G\supset H$ 4 system, a gap $G\supset H$ 5 above the stress tensor in the singlet spin-2 channel (Kousvos et al., 7 Jul 2025).

In the $G\supset H$ 6- $G\supset H$ 7 mixed system, scanning the $G\supset H$ 8 plane with the $G\supset H$ 9-gap produces a red strip of excluded points along $H$ 0. The paper reports this behavior at increasing derivative order $H$ 1, with the disallowed strip digging deeper into the diagonal as $H$ 2 increases. That diagonal is the $H$ 3-enhancement line, because $H$ 4 and $H$ 5 are the two cubic components of the same $H$ 6 tensor (Kousvos et al., 7 Jul 2025).

The full $H$ 7- $H$ 8- $H$ 9 system yields a more structured picture. With $G$ 0, the allowed region in $G$ 1 becomes a three-dimensional island whose $G$ 2 projection contains a cut along the diagonal. At $G$ 3, the $G$ 4 point

$G$ 5

falls inside the diagonal strip and is excluded, while the cubic estimate

$G$ 6

remains in the allowed region. When the $G$ 7-gap is removed, the diagonal cut disappears and the allowed region becomes symmetric about $G$ 8, with the $G$ 9 point again allowed (Kousvos et al., 7 Jul 2025).

5. Function in isolating the cubic fixed point

The cubic redundancy channel provides a nonperturbative criterion for distinguishing the cubic fixed point from the $B$ 00 model even though both satisfy the cubic crossing equations. The differentiation mechanism is representation-theoretic rather than purely numerical: the $B$ 01 theory requires a lighter scalar in the sector that descends to the cubic $B$ 02-channel, while the cubic theory does not, because the would-be low-lying operator is absorbed into a redundant-plus-descendant structure (Kousvos et al., 7 Jul 2025).

Operationally, the logic is direct. A cubic bootstrap with no redundancy-motivated gap permits both genuinely cubic data and $B$ 03-symmetric data. Imposing

$B$ 04

is justified by the cubic theory, whose leading physical $B$ 05-scalar is estimated near $B$ 06, but is incompatible with the $B$ 07 theory, whose corresponding scalar is estimated near $B$ 08. The disallowed strip created by that assumption is therefore the numerical imprint of the redundant-operator mechanism (Kousvos et al., 7 Jul 2025).

The paper places the cubic redundancy channel inside a broader program for isolating a genuinely cubic fixed point distinct from the $B$ 09 model. In addition to the $B$ 10-gap, it uses a separate redundancy argument based on broken stress tensors to exclude decoupled Ising theories via a gap in the cubic $B$ 11 spin-1 channel. Combined with gaps that rule out hidden sectors, these assumptions produce a finite region in $B$ 12 consistent with perturbative and Monte Carlo expectations for a fully coupled cubic CFT (Kousvos et al., 7 Jul 2025).

The cubic redundancy channel is a term from numerical conformal bootstrap, and its strength depends on the derivative order $B$ 13 and on which external operators are included. With only $B$ 14 and $B$ 15 external operators, the redundancy channel does not yet exclude the numerical $B$ 16 point at accessible $B$ 17; adding $B$ 18 improves the exclusion. The spectral gaps are chosen conservatively, guided by high-loop $B$ 19-expansion and Monte Carlo estimates, and the paper treats higher-precision isolation of the cubic island as future work (Kousvos et al., 7 Jul 2025).

The mechanism is also presented as general. Analogous redundancy-channel constructions are discussed for hypertetrahedral models, MN models, bifundamental models, and applications involving broken stress-energy tensors that distinguish fully coupled from decoupled theories. The underlying pattern is consistent across these cases: when symmetry breaks from $B$ 20 to $B$ 21, broken currents can mix with near-marginal operators to produce redundant combinations, and the resulting spectral gaps obstruct symmetry enhancement in bootstrap space (Kousvos et al., 7 Jul 2025).

The phrase should also be distinguished from unrelated uses of the words cubic, redundancy, and channel in other literatures. In CNN representation analysis, channel redundancy refers to overlap of neighborhood sets across channels in channel-wise NNK graphs, not to a bootstrap exclusion band (Bonet et al., 2021). In graph reliability, fixed redundancy and cubic graphs describe optimal network topologies with 3-regular cores, again in a different sense (Brand et al., 1 Mar 2026). In architectural compression for CNNs, the phrase is not used, even though the work studies redundancy along the channel dimension of $B$ 22 feature tensors (Qiu et al., 2020). In NR-U waveform design, cubic metric and frequency-domain repetition concern power-amplifier nonlinearity under repeated CAZAC signaling rather than operator redundancy in a CFT (Zhao et al., 2019). These contrasts matter because the cubic redundancy channel is specifically a bootstrap-geometric exclusion phenomenon produced by redundant operators in the cubic $B$ 23-sector (Kousvos et al., 7 Jul 2025).