Superconformal Bootstrap Interfaces

Superconformal bootstrap combines ordinary conformal crossing with extra representation theory and protected information. This chapter handles only the CFT-side interface: it accepts versioned supersymmetric data, solves the corresponding Ward reduction, assembles complete supermultiplet blocks, and states what localization or index outputs can constrain. Four-dimensional $\mathcal N=2,4$ chiral-primary correlators provide a classic example in which Ward identities reduce the four-point functions and organize their OPE into long and short supermultiplet contributions Dolan and Osborn 2002, abstract and §§4–6. Supersymmetry algebras, shortening proofs, localization calculations, indices, and duality claims remain in the supersymmetry volume. The representation-theoretic classification needed for this separation is given in Córdova, Dumitrescu, and Intriligator 2019, §§2–4.

Helpful background. Mixed Correlators and Symmetry Sectors supplies matrix crossing. Characters and Multiplet Counting supplies the decomposition into conformal primaries.

Enter this chapter

Need	Page	Result
Decide whether a protected datum is usable	Protected Data as Bootstrap Input	a normalization, mixing, recombination, and round-trip test
Reduce a BPS correlator	Superconformal Ward Identities	protected pieces plus independent dynamical functions
Build crossing vectors	Superconformal Blocks and Crossing	complete multiplet contributions and positivity sectors
Interpret exact observables	Localization and Index Data	integrated-correlator and graded-trace claim boundaries

The order is strict. A superblock cannot repair an ambiguous protected normalization, and a precise solver cannot convert an index coefficient into an OPE coefficient without an independently justified map.

The import test

Every protected input used for a substantive conclusion must supply:

1. a canonical Volume 10 source, version, evidence date, and content hash;
2. the spacetime dimension and superconformal algebra;
3. the external and exchanged multiplet labels, charges, and conformal components;
4. shortening and recombination status;
5. two- and three-point normalization, including mixing matrices;
6. exact or numerical status with uncertainty;
7. a forward conversion to this volume’s conformal conventions;
8. an inverse conversion that reproduces the source record.

Failure of any item stops the dependent claim. The canonical sources include the superconformal-algebra handoff, protected-data exports, sphere localization outputs, and index inversion limits.

From source datum to crossing

The interface has four transformations:

$$\text{protected source} \xrightarrow{\text{normalize and round-trip}} \text{ordinary CFT datum} \xrightarrow{\text{Ward reduction}} \text{independent functions} \xrightarrow{\text{superblocks}} \text{crossing system}.$$

At the final stage,

$$\vec V_{\rm protected} +\sum_{\mathcal M\in\mathrm{unknown}} \lambda_{i\mathcal M}\lambda_{j\mathcal M} \vec V_{\mathcal M}=0.$$

Only imported coefficients enter $\vec V_{\rm protected}$ as fixed numbers. Short multiplets whose coefficients are not known remain among the unknowns. For mixed correlators, the coefficient products form positive-semidefinite matrices only in the correctly normalized Hermitian basis.

Three distinctions to preserve

Protected representation versus protected coefficient

Shortening can fix $\Delta$ and charges while leaving an OPE coefficient dynamical. Protection must name the exact quantity.

Integrated observable versus local correlator

Localization derivatives contain integration kernels, curvature mixing, and contact terms. They constrain local data only after a demonstrated inversion or sum rule. Even when a supersymmetric sphere partition function computes a Kähler potential on a conformal manifold, the statement is dimension- and supersymmetry-specific and retains a Kähler-transformation ambiguity Gerchkovitz, Gomis, and Komargodski 2014, abstract.

Index class versus spectrum

An index is a graded trace. It is insensitive to recombining pairs and can combine several short multiplets into one coefficient. It never reveals a generic long spectrum.

Current interface state

The scientific interface and source literature were checked through 9 August 2026. This edition does not contain a frozen Volume 10 protected-data record satisfying the import test, so it makes no dependent numerical bound or model identification. Any future calculation must reject physics output until such a record and its supporting evidence are present.

This limitation is informative rather than cosmetic: it lets a reader practice rejecting incomplete inputs and prevents exact upstream methods from acquiring stronger downstream claims through an undocumented conversion.

Review the chapter

Normalize an imported coefficient

An upstream protected operator has two-point norm $N$ and a three-point coefficient $\lambda_{\rm src}$ with unit-normalized external operators. What enters crossing?

Solution

Rescale the protected operator by $N^{-1/2}$. The unit-normalized OPE coefficient is $\lambda=\lambda_{\rm src}/\sqrt N$, and an identical-scalar diagonal crossing equation receives $\lambda^2=\lambda_{\rm src}^2/N$.

Diagnose an index overclaim

An index coefficient equals five. May one insert five copies of a short superblock with fixed OPE coefficient?

Solution

No. The coefficient is a graded character combination and can include cancellations or several multiplet types. Even an unambiguous multiplicity would not fix their OPE coefficients. One needs a character/recombination inversion and independent local normalization data.

Check a superblock implementation

Name three independent tests.

Solution

Check the super-Casimir or Ward residual, the OPE expansion against the conformal-primary content, and the long-to-short recombination identity at the unitarity threshold. A fourth check is R-symmetry crossing under permutation generators.

Where to continue

• Continue to Large-N, Mellin, and Holographic CFT Interfaces for a separate assumption-to-inference boundary.
• Return to Numerical Bootstrap once a complete crossing system and certified inputs exist.
• Use Protected Correlators and Operator Algebras for the upstream protected construction.

References

• 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.
• Dolan, F. A., and Osborn, H. “Superconformal Symmetry, Correlation Functions and the Operator Product Expansion.” Nuclear Physics B 629 (2002): 3–73. arXiv. DOI.
• Gerchkovitz, E., Gomis, J., and Komargodski, Z. “Sphere Partition Functions and the Zamolodchikov Metric.” Journal of High Energy Physics 2014, 001 (2014). arXiv. DOI.