Protected Data as Bootstrap Input

Protected supersymmetric information becomes conformal-bootstrap input only after it has been translated into ordinary CFT data: operator dimensions and representations, two- and three-point normalizations, OPE coefficients or linear combinations, and their uncertainties. A shortening label, index coefficient, or localization observable is not yet a crossing coefficient. The conversion must preserve its upstream convention, mixing and recombination status, and exactness claim. A concrete example of a protected subsector together with its nontrivial map back to higher-dimensional CFT data is Beem et al. 2015, §§2–4.

Required background. Mixed Correlators and Symmetry Sectors supplies matrix-valued crossing. Characters and Multiplet Counting supplies conformal-primary decomposition.

Which protected statements can enter crossing?

Useful inputs include:

• a scaling dimension fixed by a declared shortening condition;
• an exactly normalized stress-tensor or flavor-current coefficient;
• a protected OPE coefficient, or a known linear combination of coefficients;
• the absence or multiplicity of a protected multiplet after recombination is resolved;
• a cohomological or chiral-algebra correlator together with its map to a restricted higher-dimensional OPE;
• an integrated correlator from localization together with its kernel and contact subtraction.

The available short multiplets, their null states, and their recombination patterns depend on the spacetime dimension and superconformal algebra Córdova, Dumitrescu, and Intriligator 2019, abstract and §§2–4. Each statement has a different claim ceiling. Protection of $\Delta$ does not protect $\lambda$. An index can constrain a signed combination of short multiplets but usually cannot distinguish multiplets that share the same index class. A topological subsector determines only the OPE projection visible in its cohomology. In supersymmetric bootstrap systems, only the identified protected pieces are fixed; the remaining dimensions and OPE data stay as crossing variables Poland, Rychkov, and Vichi 2019, §8.

The supersymmetry algebra, shortening theorem, and protected calculation remain with the superconformal handoff, protection and mixing analysis, or another exact Volume 10 source. This page begins at their exported CFT datum.

What a usable protected input contains

A complete record specifies:

Field	Why crossing needs it
Theory and superconformal algebra	fixes dimension, R-symmetry, and allowed multiplets
Operator identity	names the superconformal primary, conformal component, charges, and representation
Protection statement	says exactly which dimension, multiplicity, or coefficient is fixed and why
Recombination status	prevents a short multiplet at threshold from being counted twice
Mixing basis	distinguishes a protected vector space from a preferred operator
Two-point convention	fixes what a quoted OPE coefficient means
Source version and content hash	freezes the upstream calculation used
Exact or numerical status	carries precision, uncertainty, and correlation information
Convention map and inverse	converts to this volume and verifies a round trip
Allowed conclusion	states which crossing equation or coefficient receives the datum

If any of these fields is missing, the input may still motivate an example, but it cannot support a substantive protected-sector bound or reconstruction.

Convention conversion

Suppose an upstream source normalizes operators by

$$\langle\mathcal O_i(x)\mathcal O_j(0)\rangle_{\rm src} =\frac{N_i\delta_{ij}}{|x|^{2\Delta_i}},$$

while crossing uses unit two-point functions. Define

$$\widehat{\mathcal O}_i=N_i^{-1/2}\mathcal O_i.$$

A three-point coefficient transforms as

$$\widehat\lambda_{ijk} =\frac{\lambda^{\rm src}_{ijk}}{\sqrt{N_iN_jN_k}}.$$

For a degenerate protected subspace, $N_i$ is replaced by a positive matrix $G_{ab}$. Whitening it requires a declared factorization $G=S^\dagger S$ and maps coefficient vectors by $S^{-1}$. A change of basis within the degenerate space changes individual components but not basis-invariant quadratic combinations.

A valid round trip applies the inverse transformation and recovers the source datum, including exact algebraic numbers or the full numerical covariance, within its stated tolerance. Agreement of a single decimal value is not enough.

Stop before crossing if the record is stale

The flow below contains an explicit stop: a missing version, normalization, mixing map, or inverse conversion prevents the protected datum from reaching Ward identities or superblocks.

Protected data enter ordinary conformal crossing only after source identity, exactness, normalization, mixing, recombination, and an invertible convention transformation have been verified. A failed or stale check stops the path; the diagram is schematic.

The same logic in structured form is:

Stage	Pass condition	Failure response
Source	canonical route, version, date, and hash exist	do not import
Multiplet	algebra, shortening, and recombination are identified	keep datum unassigned
Normalization	two-point and generator conventions are explicit	do not quote an OPE coefficient
Mixing	basis or invariant combination is supplied	retain only the unresolved subspace
Conversion	forward and inverse maps agree	reject transformed value
Crossing	protected contribution is isolated from unprotected unknowns	do not claim a complete solution

Physics output requires a frozen upstream record and its supporting evidence. A synthetic schema can test file parsing, but it must not be presented as protected-data physics.

Current evidence boundary

This interface was checked against the cited literature and repository inputs through 9 August 2026. No frozen Volume 10 protected-data record satisfying the full table above is materialized in this edition. Accordingly, this page gives conversion equations and rejection tests but asserts no dependent numerical superconformal bound. Once a record exists, its source version and evidence date—not the page date alone—control reuse.

Common pitfalls

Treating a protected dimension as a protected OPE coefficient. Shortening can fix one without fixing the other. Import only the quantity actually established upstream.

Whitening a mixed sector without retaining the matrix. The square root of a two-point matrix is part of the convention transform. Dropping it makes the conversion impossible to reverse.

Using an index coefficient as a multiplicity. Bosonic and fermionic short multiplets can cancel or recombine in an index. Inversion requires extra assumptions or fugacity information.

Exercises

An upstream operator has $\langle\mathcal O\mathcal O\rangle=N/|x|^{2\Delta}$ and $\langle\phi\phi\mathcal O\rangle=\lambda_{\rm src}$, with $\phi$ already unit normalized. Find the crossing coefficient.

Solution

The unit-normalized operator is $\widehat{\mathcal O}=\mathcal O/\sqrt N$, so $\widehat\lambda=\lambda_{\rm src}/\sqrt N$. For identical Hermitian $\phi$, its scalar crossing weight is $\widehat\lambda^2=\lambda_{\rm src}^2/N$.

References

• Beem, C., Lemos, M., Liendo, P., Peelaers, W., Rastelli, L., and van Rees, B. C. “Infinite Chiral Symmetry in Four Dimensions.” Communications in Mathematical Physics 336 (2015): 1359–1433. arXiv. DOI.
• Córdova, C., Dumitrescu, T. T., and Intriligator, K. “Multiplets of Superconformal Symmetry in Diverse Dimensions.” Journal of High Energy Physics 2019, 163 (2019). arXiv. DOI.
• Poland, D., Rychkov, S., and Vichi, A. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91, 015002 (2019), §8. arXiv. DOI.