Superconformal Bootstrap Interfaces

Supersymmetry can supply exact information to a conformal-bootstrap problem, but each protected quantity must first be translated into the correlator being studied. This chapter explains that translation, the reduction by superconformal Ward identities, and the assembly of supermultiplet contributions. The underlying supersymmetry and localization constructions are developed in the supersymmetry volume.

Enter this chapter

Begin with Protected Data as Bootstrap Input to distinguish a known dimension or index from a normalized local OPE coefficient.

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

Topics

1. Protected Data as Bootstrap Input — Matches protected observables to the operator basis and normalization of crossing.
2. Superconformal Ward Identities — Reduces correlators using the chosen superconformal algebra and external multiplets.
3. Superconformal Blocks and Crossing — Combines ordinary conformal blocks into complete supermultiplet contributions.
4. Localization and Index Data in Bootstrap — Distinguishes integrated localization constraints from graded index information.

How the ideas fit together

A supermultiplet contains several conformal primaries whose contributions are linked by supersymmetry. Ward identities restrict their relative coefficients, so the block decomposition must include complete multiplets rather than independently varying each component. Four-dimensional chiral-primary examples make this reduction explicit. Dolan and Osborn 2002, §§ 4–6.

Shortening can protect an operator’s dimension without fixing every OPE coefficient involving it. Similarly, an index counts a signed combination of protected states; it does not generally resolve their individual couplings. Exact upstream information becomes a fixed crossing coefficient only after that particular conversion has been established.

Review the chapter

A protected exchanged operator has two-point norm $N>0$ and three-point coefficient $c$ with two unit-normalized identical Hermitian external operators. What coefficient enters its diagonal block contribution?

Solution

Rescaling the exchanged operator by $1/\sqrt N$ gives unit norm and three-point coefficient $c/\sqrt N$. In a real Hermitian convention its block weight is therefore $c^2/N$. The protection of a representation does not remove this normalization step.

Where to go next

Continue to Numerical Bootstrap once the crossing system and converted coefficients are specified. For their construction, use Protected Correlators and Operator Algebras.

References

• Dolan, F. A., and Osborn, H. “Superconformal Symmetry, Correlation Functions and the Operator Product Expansion.” Nuclear Physics B 629 (2002): 3–73. arXiv. DOI.