Localization and Index Data in Bootstrap

Localization and superconformal indices are exact tools for protected observables, but they produce different mathematical objects. A localized partition function can yield integrated correlators through source derivatives, including contact and mixing terms. An index is a graded trace over a supercharge cohomology and records signed protected combinations. Neither output is automatically a local OPE coefficient or a complete spectrum.

Required background. Protected Data as Bootstrap Input supplies the conversion test. Sphere Partition Functions and Universal CFT Data supplies source derivatives and counterterm qualifications.

Localization derivatives are integrated observables

Let $Z_{S^d}(m^a,\lambda^i)$ be a supersymmetric sphere partition function with mass and exactly marginal sources in a specified rigid supergravity background. Differentiation gives

$$\partial_a(-\log Z) =\int_{S^d}\sqrt g\, \langle\mathcal O_a\rangle,$$

and a second derivative gives an integrated connected two-point function plus local source contacts. Higher derivatives can encode integrated four-point functions or protected OPE combinations, but only after:

• identifying which flat-space operator sits in the background multiplet;
• undoing curvature-induced mixing with lower-dimensional operators;
• fixing finite supersymmetric counterterms;
• deconvolving the known integration kernel;
• matching the local two-point and OPE normalization.

For particular supersymmetric theories, sphere partition functions compute a Kähler potential on the conformal manifold up to Kähler transformations Gerchkovitz, Gomis, and Komargodski 2014. That result depends on the dimension and supersymmetry class; it is not a generic property of every localized sphere integral.

An index is a graded cohomological trace

Choose a supercharge $Q$ and commuting charges $\mathcal Q_A$. A superconformal index has the form

$$\mathcal I(\mathbf x) =\operatorname{Tr}_{\mathcal H(S^{d-1})} (-1)^F e^{-\beta\{Q,Q^\dagger\}} \prod_A x_A^{\mathcal Q_A}.$$

In the original four-dimensional $\mathcal N=1$ construction, radial quantization on $S^3\times\mathbb R$ counts chiral primaries and defines an index that also detects semishort multiplets Romelsberger 2006, abstract. States with $\{Q,Q^\dagger\}>0$ cancel in boson–fermion pairs, so the index is independent of $\beta$ under the usual discreteness and convergence assumptions. Its coefficients are signed character combinations of short representations. Recombining short multiplets can cancel from the index, and different collections of short multiplets can have the same index.

Consequently:

• a missing index term can sometimes prove that a protected cohomology class is absent once all possible cancellations are classified;
• a nonzero coefficient usually fixes only a linear combination of protected multiplicities;
• the index does not supply unprotected dimensions;
• the index contains no generic local OPE coefficients.

Refined fugacities and an explicit recombination basis can improve inversion, but every ambiguity must remain visible Kinney et al. 2007, §§2–3.

The protected and large-N evidence boundary

The shared comparison below records what may cross into an ordinary CFT claim. “Version required” means that no qualifying frozen source is present in this edition.

Imported datum	Canonical source and version	Conversion	Evidence carried	Allowed CFT conclusion	Optional later interpretation	Does not prove
Shortening-fixed dimension or multiplet	Versioned protected-data export; version required	algebra label, charges, two-point convention, recombination round trip	exact only if source says exact	fix the stated protected representation datum	none needed	its OPE coefficient or an unprotected gap
Protected correlator/cohomology class	Protected correlators and operator algebras; version required	restriction map and kernel to local CFT channel	exact or numerical as supplied	constrain the visible protected OPE projection	possible cross-dimensional algebraic description	the full higher-dimensional OPE
Sphere derivative	Sphere partition functions and matrix models; version required	background-source map, mixing subtraction, local normalization	integrated observable with contacts	impose the derived integrated sum rule or normalized coefficient	none needed	a pointwise correlator without deconvolution
Index coefficient	Index inversion and protected-spectrum limits; version required	character decomposition and recombination quotient	graded protected count	fix a proven multiplicity combination or absence	none needed	a raw degeneracy, OPE coefficient, or long spectrum
Large-N factorization	Large-N CFT data plus its source version	operator normalization and $1/N$ power counting	asymptotic CFT data	organize connected correlators and double-trace corrections	may motivate a bulk expansion after a separate dictionary	bulk locality or a Lagrangian
Large gap or Mellin criterion	Large-gap CFT tests with explicit gap, Regge, and contour data	Mellin/position-space normalization and limit order	conditional analytic or numerical evidence	state the corresponding CFT-side bound or scaling test	Volume 15 may assess a bulk interpretation	necessity as sufficiency, or an S-matrix

The last two rows are included because this table is shared with the following large-N interface chapter. They do not import a bulk conclusion into the present one.

A safe extraction workflow

1. Freeze the exact localized observable or index, including background, fugacities, regulator, and source version.
2. Identify the protected multiplets and possible recombination cancellations.
3. Derive the map to local conformal operators and their two-point convention.
4. Remove curvature mixing and state every contact counterterm.
5. Invert only the part of an integral transform or character sum that is mathematically determined.
6. Propagate exactness, numerical uncertainty, and unresolved degeneracy without improvement.
7. Insert the resulting datum into crossing and leave all other sectors unknown.

Current evidence boundary

Sources and interface claims were checked through 9 August 2026. The required frozen Volume 10 source record and supporting dossier are not materialized in this edition, so the table records admissible transformations rather than imported numerical values. No index-to-OPE or localization-to-local-correlator claim is made without that record.

Common pitfalls

Equating a partition-function derivative with a separated-point correlator. The derivative is integrated and contains contact terms. A kernel inversion and subtraction are additional steps.

Reading index coefficients as positive multiplicities. The trace is graded. Recombination and boson–fermion cancellation can erase or combine short multiplets.

Using exact localization to erase conversion uncertainty. The source observable may be exact while the map to a chosen local operator is obstructed by mixing or convention ambiguity.

Exercises

Why does $\mathcal I$ receive contributions only from $\{Q,Q^\dagger\}=0$ states?

Solution

States with positive $\{Q,Q^\dagger\}$ occur in $Q$-paired bosonic and fermionic states with equal commuting charges. Their contributions cancel because of $(-1)^F$. Unpaired cohomology classes with zero anticommutator remain.

References

• Gerchkovitz, E., Gomis, J., and Komargodski, Z. “Sphere Partition Functions and the Zamolodchikov Metric.” Journal of High Energy Physics 2014, 001 (2014). arXiv. DOI.
• Kinney, J., Maldacena, J. M., Minwalla, S., and Raju, S. “An Index for 4 Dimensional Super Conformal Theories.” Communications in Mathematical Physics 275 (2007): 209–254. arXiv. DOI.
• Romelsberger, C. “Counting Chiral Primaries in $\mathcal N=1$, $d=4$ Superconformal Field Theories.” Nuclear Physics B 747 (2006): 329–353. arXiv. DOI.