Global Symmetry and Spinning Bootstrap Systems

Global symmetry and spinning external operators enlarge crossing from scalar equations to coupled tensor structures with representation projectors, parity sectors, conservation identities, and matrix positivity Poland, Rychkov, and Vichi 2019, §§III.C and VI.C. The main danger is structural: a numerically stable solver can faithfully solve an incomplete or sign-inconsistent system.

Required background. Mixed-Correlator Islands supply PSD OPE-vector geometry. Spinning operators and blocks supply tensor bases and seed normalization. Helpful background. Current and stress-tensor CFT data fix Ward-normalized conserved data.

Representation-resolved crossing

For external representation $r$, decompose

$$r\otimes r=\bigoplus_R R$$

with projectors normalized so their sum is the identity. Each $R$ has allowed spins and exchange parity determined by statistics. Mixed external representations require intertwiners and a fixed real structure. Projector normalization can be moved into OPE coefficients, but only through an explicit conversion.

Spinning four-point functions have a basis $\mathcal T_I$,

$$\langle\mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle =\sum_I\mathcal T_I\,\mathcal G_I(u,v).$$

Crossing acts by a matrix on $I$. Conservation imposes differential relations that reduce the independent components; impose them in a basis whose rank is checked at generic kinematics. Dimension-specific identities, parity-odd structures, and contact terms must not be removed using formulas from a different dimension.

Positivity after basis changes

If $B$ changes the three-point basis, an OPE vector transforms as $\boldsymbol\lambda'=B^{-1}\boldsymbol\lambda$ and a quadratic block matrix by congruence. PSD is preserved under a real nonsingular congruence, but numerical eigenvalues and conditioning can change dramatically. Store $B$, its precision, and the component order.

Conserved currents and stress tensors have Ward-fixed OPE coefficients only after generator, $C_J$, $C_T$, and tensor-structure normalizations are declared. Conservation alone does not imply positivity in a nonunitary theory, and parity-odd coefficients can enter blocks that are not separately nonnegative.

System checks

Structure	Exact check	Numerical check	Failure mode
Group projectors	idempotence, orthogonality, completeness	high-precision regeneration	missing representation sector
Tensor basis	generic-point rank and permutation closure	condition number across domain	spurious singular structure
Conservation	differential rank and Ward normalization	residual on direct blocks	over- or under-constrained system
Parity/statistics	exchange phases and orientation	sign-injected fixture	false PSD sector
Dimension identities	representation count at fixed $d$	compare symbolic and numeric ranks	imported generic-$d$ redundancy
PSD blocks	OPE-vector order and congruence	enclosed minimum eigenvalue	positive diagonals but negative matrix

Automated generation can reduce transcription errors only when these exact outputs are exposed and checked Go and Tachikawa 2019, §§2–4.

Numerical claim taxonomy

Claim class	Rigorous output	Interpretation	Assumptions	Convergence	Falsifier	Evidence requirement	Prohibited wording
Symmetry-resolved exclusion	represented spectra violate generated crossing and PSD conditions	a conditional sector-specific exclusion	complete projector and tensor system, correct signs	certificate and several block/functional settings	restored feasibility after a missing sector is added	serialized generator and verified certificate	absence of every CFT with that symmetry
Conserved-current bound	coefficient or gap is bounded in the fixed Ward basis	conditional restriction on conserved data	conservation rank and $C_J$ or $C_T$ convention	basis, cutoff, and precision refinements	bound changes under correct basis conversion	Ward-normalization record and convergence data	universal number without normalization
Spinning kink	stable feature in a declared tensor system	possible model feature	complete parity and dimension-specific basis	persists under block and functional refinements	feature vanishes after a structure is restored	dated curve data and independent model evidence	automatic model identity
Mixed spinning island	pointwise exclusions delimit a conditional component	possible candidate region	correlator closure and shared operators	topology and numerical refinements	missing component or assumption removal opens region	certificates and assumption-removal study	unique realized CFT
Candidate spectrum	no theorem beyond finite primal feasibility	approximate low data	extremality and degeneracy treatment	stable zeros, weights, and residuals	unstable zeros or negative OPE matrix	reconstruction record plus external checks	proof of existence

Failure tests

Projector test. Sum all representation projectors and compare with the identity before numerical evaluation.

Conservation test. Evaluate the divergence of each retained structure. A basis with the wrong rank must stop generation.

Congruence test. Change tensor basis and reproduce the same unscaled exclusion. If not, locate the normalization or conditioning error.

Proceed to Extremal Functionals, Navigators, and Spectrum Reconstruction only after the full system is stable.

References

• Go, Mocho, and Yuji Tachikawa. “autoboot: A Generator of Bootstrap Equations with Global Symmetry.” Journal of High Energy Physics 06 (2019): 084. DOI. Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF