Protected N=1 SCFT Data and the Bootstrap Export

Supersymmetry can determine exact dimensions, anomaly coefficients, shortening types, and selected protected multiplicities. A bootstrap calculation can use those results only after their operator identities, normalizations, recombination status, mixing, provenance, and evidence ceiling have been frozen. An index coefficient is not automatically an operator degeneracy, and a duality-supported protected record is not a solution of crossing. The receiving problem is developed in Poland, Rychkov, and Vichi 2019, §8.

Required background. Accidental symmetries and a-maximization determines the exact R-current, while the superconformal-algebra handoff fixes representation labels. Helpful background. Conformal OPE data defines the consumer’s objects.

The N=1 export contract

A four-dimensional $\mathcal N=1$ SCFT is governed by $SU(2,2\lvert1)$. For a local superconformal primary, this page uses physical Lorentz spins in

$$(\Delta;j_{\mathrm{spin}},\bar j_{\mathrm{spin}};R;\mathcal R_F).$$

The machine record instead uses the integer Lorentz Dynkin labels of Córdova, Dumitrescu, and Intriligator,

$$j_{\mathrm{CDI}}=2j_{\mathrm{spin}}, \qquad \bar j_{\mathrm{CDI}}=2\bar j_{\mathrm{spin}},$$

and stores both the forward and inverse maps. A scalar chiral primary obeys

$$\bar Q_{\dot\alpha}\mathcal O=0, \qquad \bar j_{\mathrm{spin}}=0, \qquad \Delta=\frac32R.$$

Four-dimensional shortening and recombination data are tabulated in Córdova, Dumitrescu, and Intriligator 2019, §§2.2.1 and 4.5. Their general null-state construction is presented as a conjectural algorithm supported by extensive consistency checks, so an export should preserve that source status and name the particular rule it uses.

The exported operator is the gauge-invariant SCFT operator. A gauge-dependent ultraviolet field may appear only as provenance for its construction. If several operators have equal quantum numbers, the record supplies their two-point Gram matrix and a declared basis. When no preferred basis is known, it exports the protected subspace or a basis-invariant quadratic combination rather than inventing a unique operator.

The generic versioned-export contract defines the content-addressed envelope and cross-volume round trip. This page supplies its N=1 specialization: R-symmetry provenance, anomaly traces, N=1 multiplet labels, accidental-field separation, and the precise protected data derived from them.

Normalizations that survive a round trip

For traces over left-handed Weyl fermions, adopt

$$\begin{aligned} a&=\frac{3}{32}(3\operatorname{Tr}R^3-\operatorname{Tr}R),\\ c&=\frac{1}{32}(9\operatorname{Tr}R^3-5\operatorname{Tr}R). \end{aligned}$$

These nonperturbative relations between conformal central charges and R-current anomalies are derived in Anselmi, Freedman, Grisaru, and Johansen 1998, §§2–4.

Here $R$ is the fermion R-charge. If a chiral superfield has scalar charge $R(\Phi)$, its Weyl fermion has charge $R(\psi_\Phi)=R(\Phi)-1$.

After Euclidean continuation, define

$$I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2}$$

and use the current normalization

$$\langle J_\mu^a(x)J_\nu^b(0)\rangle =\frac{3\tau^{ab}}{4\pi^4} \frac{I_{\mu\nu}(x)}{|x|^6}.$$

For flavor generators satisfying

$$\operatorname{tr}_{\mathbf{fund}}T^aT^b =\frac12\delta^{ab},$$

supersymmetry gives

$$\tau^{ab}=-3\operatorname{Tr}(RT^aT^b).$$

These tensor and anomaly normalizations agree with Barnes et al. 2005, eqs. (1.1), (1.16), and (2.1). A symbol such as $C_J$ or $k_F$ is not interchangeable with $\tau$ until its tensor coefficient and generator normalization are supplied.

For the stress tensor, set

$$\begin{aligned} I_{\mu\nu,\rho\sigma}(x) &=\frac12\bigl(I_{\mu\rho}I_{\nu\sigma} +I_{\mu\sigma}I_{\nu\rho}\bigr) -\frac14\delta_{\mu\nu}\delta_{\rho\sigma},\\ \langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle &=C_T\frac{I_{\mu\nu,\rho\sigma}(x)}{|x|^8},\\ C_T&=\frac{40}{\pi^4}c. \end{aligned}$$

The export stores $c$, $C_T$, and both conversion directions. The tensor convention is the one used in Barnes et al. 2005, eq. (2.2).

A round-trippable operator entry

Each operator or multiplet entry states:

Field	Required content
Identity	Theory, point or stratum, operator name, aliases, and gauge-invariant definition
Representation	$\Delta$, CDI Lorentz labels, R-charge, faithful flavor representation, and exact shortening label
Counting	Representation dimension and copy multiplicity as separate fields; a bound or index-weighted coefficient is labeled as such
Recombination	Applicable threshold relation, partner multiplets, and quotient class
Normalization	Two-point Gram matrix, generator trace, tensor basis, and phase or sign convention
Mixing	Declared basis and covariance, or the unresolved protected subspace
OPE data	External normalization, tensor structure, coefficient or interval, and protection mechanism
Status	Algebraic, anomaly-derived, duality-dependent, index-inferred, perturbative, numerical, or unknown
Scope	Vacuum, chamber, global form, decoupled sectors, and parameter range
Provenance	Source version, exact locator, evidence cutoff, content hash, assumptions, and unresolved fields

This distinction matters immediately: one meson multiplet in a 36-dimensional flavor representation has copy multiplicity one, not multiplicity 36.

Worked SU(3) SQCD record

Consider the candidate interacting $SU(3)$ SQCD fixed point with $N_f=6$. For the chiral superfields and gauge-invariant generators,

$$R(Q)=R(\widetilde Q)=\frac12, \qquad R(M)=1, \qquad R(B)=R(\widetilde B)=\frac32,$$

where $M^i{}_{\tilde\jmath}=Q^i\widetilde Q_{\tilde\jmath}$ and $B\sim\varepsilon Q^3$. Therefore

$$\Delta(M)=\frac32, \qquad \Delta(B)=\Delta(\widetilde B)=\frac94.$$

The anomaly trace uses eight adjoint gauginos of charge one and 36 quark Weyl fermions of charge $-1/2$:

$$\begin{aligned} \operatorname{Tr}R &=8+36\left(-\frac12\right)=-10,\\ \operatorname{Tr}R^3 &=8+36\left(-\frac12\right)^3=\frac72. \end{aligned}$$

Substitution gives

$$a=\frac{123}{64}, \qquad c=\frac{163}{64}, \qquad C_T=\frac{815}{8\pi^4}.$$

For either $SU(6)$ flavor factor,

$$\operatorname{Tr}(RT^aT^b) =3\left(-\frac12\right)\frac12\delta^{ab} =-\frac34\delta^{ab},$$

so $\tau_{SU(6)_L}=\tau_{SU(6)_R}=9/4$ in the declared current convention.

At the level of flavor Lie algebras, the meson transforms as $(\mathbf6,\overline{\mathbf6})$. Its unit two-point convention is

$$\left\langle M^i{}_{\tilde\jmath}(x) (M^\dagger)_k{}^{\tilde\ell}(0)\right\rangle =\frac{\delta^i{}_k\delta^{\tilde\ell}{}_{\tilde\jmath}} {|x|^3}.$$

This fixes the meaning of later OPE coefficients but is a normalization choice, not a holomorphic prediction. The exact faithful flavor quotient is a separate mandatory theory-record field; the representation shorthand above does not determine it.

All values in this card are conditional on the existence of the proposed interacting SQCD fixed point and identification of its standard exact R-symmetry. Seiberg duality strongly supports that identification but is not a mathematical existence theorem.

Canonical record and consumer boundary

The bounded record is N=1 $SU(3)$, $N_f=6$ protected data v1.0.0, validated against QFT.org protected-SCFT export schema v1.0.0. Its publication tuple is

$$\begin{aligned} \bigl(&\texttt{schema\_version}=\texttt{1.0.0},\\ &\texttt{record\_version}=\texttt{1.0.0},\\ &\texttt{content\_hash}=\texttt{sha256:68a67e66c43e4d0a258f0d26b0b9c08c6164922be256a41112bb5761ee582910}\bigr). \end{aligned}$$

The third entry must begin with the literal prefix sha256: followed by exactly 64 lowercase hexadecimal characters; any other form is rejected. The digest is generator-computed from RFC 8785 canonical JSON rather than supplied by hand. The evidence cutoff is 24 August 2026, and the independently addressable convention subrecord has SHA-256 81124c40cb39131147e94235022cbcea42222795e2a4e4c4842936fc212bfd95.

The record exports anomaly traces, $a,c,C_T$, flavor-current coefficients, selected chiral-primary identities and dimensions, CDI label conventions, evidence status, free-factor status, and exact conversion maps. It deliberately exports no unit-normalized non-Ward OPE coefficient, unique index inversion, long-multiplet gap, crossing solution, or numerical bound. Volume 9 receives it through Protected Data as Bootstrap Input and Superconformal Blocks and Crossing.

Index and partition-function limits

The superconformal index is a graded trace over states annihilated by a chosen supercharge. Long multiplets cancel, and complete recombination packages represent zero. More precisely, the index factors through the quotient of the short-multiplet group by the subgroup generated by recombination relations. It therefore gives an equivalence class, not generally a unique nonnegative spectrum. The finite-order inverse problem and its assumptions are developed in Index Inversion and the Protected Spectrum, building on Romelsberger 2006, §§2–4 and Dolan and Osborn 2009, §§2 and 4–5.

An exported index entry must name the supercharge, fugacity map, exact infrared R-symmetry, descendant subtraction, expansion domain, and recombination quotient. It may be promoted to a multiplicity only when an inversion theorem or explicit supplementary assumptions remove the ambiguity.

A curved-space partition function is a background generating functional, not automatically a universal number. An export must freeze the full supersymmetric background, global sectors, contour or residue chamber, local counterterm and contact-term convention, and every divided-out free or topological factor. Only a convention-matched invariant combination, derivative, or integrated correlator may be passed onward. The classification is given in Partition Functions, Counterterms, and Universal Data, and the matching protocol in Defects, Instantons, and Duality Tests.

Protected products are not automatically physical OPE data

Ward identities can fix the coupling of a normalized scalar to current or stress-tensor multiplets in terms of its charges and $C_J$ or $C_T$. The exact tensor and block conventions must agree before that coefficient enters crossing.

A chiral-ring product has the form

$$\mathcal O_i\mathcal O_j =C_{ij}{}^k\mathcal O_k+\bar Q\text{-exact terms}.$$

Holomorphic rescaling changes $C_{ij}{}^k$. Thus a branch-qualified relation for a hypothetical generator, such as $X^2=0$, is meaningful without a Hermitian metric, whereas a numerical unit-normalized OPE coefficient generally is not. The SQCD meson should not be assigned the relation $M^2=0$; its actual chiral-ring relations depend on rank and branch.

Accidental fields and validation

If a chiral operator becomes free, export it as a separate multiplet with

$$\Delta=1, \qquad R=\frac23, \qquad (a,c)=\left(\frac1{48},\frac1{24}\right).$$

Subtract its contribution from the interacting anomalies and retain it in the total theory record. Combining total $a,c$ with an interacting-only spectrum violates Ward-identity consistency. The full iteration is summarized by the extremization-to-export flow.

Before handoff:

1. recompute every chiral dimension from $3R/2$;
2. recompute $a,c$ from stored fermion traces;
3. translate and invert the $C_T$ and flavor-current conventions;
4. verify CDI labels and every applicable recombination relation;
5. keep representation dimension distinct from copy multiplicity;
6. separate free, interacting, and topological factors;
7. propagate exact algebraic values or the full numerical covariance;
8. validate the schema and RFC 8785 content hash; and
9. assert no crossing feasibility, bound, or long gap that was not independently computed.

The mutable evidence and unresolved interfaces are tracked in the Supersymmetry and Duality Research map.

Common pitfalls

Using a scalar R-charge in a fermion anomaly trace. The anomaly formulas sum left-handed Weyl fermions, so a chiral multiplet contributes with $R(\psi)=R(\Phi)-1$.

Calling an index coefficient a degeneracy. Recombination and boson–fermion cancellation can make the inverse problem nonunique.

Mixing total and interacting central charges. Free accidental sectors must appear consistently in both the anomalies and operator list.

Exercises

Using the stored anomaly traces, reproduce $a$, $c$, and $C_T$. Then apply the inverse map $c=\pi^4C_T/40$. Which information is tested by this round trip, and which information is not?

Solution

The anomaly formulas give

$$a=\frac{3}{32}\left(3\frac72+10\right)=\frac{123}{64}, \qquad c=\frac1{32}\left(9\frac72+50\right)=\frac{163}{64}.$$

Therefore

$$C_T=\frac{40}{\pi^4}\frac{163}{64} =\frac{815}{8\pi^4}, \qquad \frac{\pi^4}{40}C_T=\frac{163}{64}.$$

The calculation checks the anomaly arithmetic and the declared normalization map. It does not test fixed-point existence, an OPE coefficient, index inversion, or crossing feasibility.

References

• Anselmi, Damiano, Daniel Z. Freedman, Marc T. Grisaru, and Andreas A. Johansen. “Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories.” Nuclear Physics B 526 (1998): 543–571. arXiv:hep-th/9708042.
• Barnes, Edwin, Elie Gorbatov, Ken Intriligator, Matt Sudano, and Jason Wright. “The Exact Superconformal R-Symmetry Minimizes $\tau_{RR}$.” Nuclear Physics B 730 (2005): 210–222. arXiv:hep-th/0507137.
• Córdova, Clay, Thomas T. Dumitrescu, and Kenneth Intriligator. “Multiplets of Superconformal Symmetry in Diverse Dimensions.” Journal of High Energy Physics 03 (2019): 163. arXiv:1612.00809.
• Dolan, F. A., and H. Osborn. “Applications of the Superconformal Index for Protected Operators and $q$-Hypergeometric Identities to $\mathcal N=1$ Dual Theories.” Nuclear Physics B 818 (2009): 137–178. arXiv:0801.4947.
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. arXiv:1805.04405.
• Romelsberger, Christian. “Counting Chiral Primaries in $\mathcal N=1$, $d=4$ Superconformal Field Theories.” Nuclear Physics B 747 (2006): 329–353. arXiv:hep-th/0510060.