Protected N=1 SCFT Data and the Bootstrap Export

Supersymmetry can determine exact dimensions, anomaly coefficients, shortening types, and selected protected multiplicities. These are valuable inputs to conformal bootstrap, but they must be exported with normalization, provenance, and status. An index coefficient is not automatically an operator degeneracy, and a duality-dependent protected datum is not a solution of crossing symmetry. The crossing, unitarity, and positivity problem to which these data are supplied is summarized in Poland and Simmons-Duffin 2016, pp. 535–539.

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 bootstrap consumer’s objects.

Export only intrinsic SCFT data

A four-dimensional $\mathcal N=1$ SCFT is governed by $SU(2,2|1)$. A local superconformal primary is labeled by

$$(\Delta;j,\bar j;R;\mathcal R_F),$$

where $(j,\bar j)$ are Lorentz spins and $\mathcal R_F$ is a representation of the faithful flavor group. A chiral primary obeys

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

The export should contain the gauge-invariant SCFT operator, not a gauge-dependent ultraviolet field unless the latter is explicitly identified only as a convenient constituent. Operator mixing must be resolved within each set of equal quantum numbers before assigning an OPE coefficient. Four-dimensional shortening conditions, recombination rules, and the complete multiplet tables are given in Córdova, Dumitrescu, and Intriligator 2019, §2.2 and §4.

Intrinsic protected data include:

• exact R- and flavor charges of identified short multiplets;
• protected dimensions fixed by shortening;
• $a$, $c$, and flavor-current anomaly coefficients;
• exact chiral-ring relations, with quantum and branch qualifications;
• protected index or cohomology data with its equivalence relation;
• symmetry, global form, and selection rules.

Couplings and duality frames are metadata about a presentation unless they label a point on a conformal manifold.

Normalization map for central data

Adopt

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

For flavor generators normalized by

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

one common current normalization is

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

The trace is over Weyl fermions and $R$ is their charge. The anomaly relations that determine supersymmetric central functions are derived in Anselmi, Freedman, Grisaru, and Johansen 1998, §§2–4. If a bootstrap code instead uses a two-point coefficient $C_J$ or level $k_F$, record the exact conversion used; these symbols differ by sphere-volume and generator conventions across the literature.

Likewise, if the stress-tensor two-point convention is

$$C_T=\frac{40}{\pi^4}c,$$

store both $c$ and $C_T$ together with this equation. Never export only a bare decimal called “central charge.” Round-tripping $C_T\to c\to C_T$ should reproduce the original exact value.

A round-trippable operator entry

Each operator or multiplet entry should state:

Field	Required content
Identity	Theory, point or stratum on the conformal manifold, operator name, and aliases
Representation	$\Delta$, $(j,\bar j)$, R-charge, faithful flavor representation, and shortening type
Normalization	Two-point function, generator trace, and phase/sign conventions
Multiplicity	Exact number, lower/upper bound, or index-weighted coefficient—never an unlabeled integer
OPE data	External-operator normalization, tensor structure, coefficient or interval, and whether protected
Status	Algebraic theorem, anomaly-derived, duality-dependent, index-inferred, perturbative, numerical, or unknown
Scope	Vacuum, chamber, global form, decoupled sectors, and parameter range
Source	Derivation or primary reference and date; unresolved mixing or assumptions

This schema allows a consumer to reject incompatible normalizations rather than silently combine them.

Worked SQCD export

Take the candidate $SU(3)$ SQCD fixed point with $N_f=6$. The protected data derived earlier are

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

and hence

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

The anomaly central charges are

$$a=\frac{123}{64}, \qquad c=\frac{163}{64}.$$

In the displayed $C_T$ convention,

$$C_T=\frac{815}{8\pi^4}.$$

An export should label these values conditional on the existence of the proposed interacting SQCD fixed point and its standard exact R-symmetry. Seiberg duality supplies a strong independent description but does not change that logical status into a mathematical existence theorem.

The meson transforms as $(\mathbf6,\overline{\mathbf6})$ under $SU(6)_L\times SU(6)_R$. Its two-point function can be normalized to

$$\langle M^i{}_j(x) M^{\dagger k}{}_{\ell}(0)\rangle =\frac{\delta^{ik}\delta_{j\ell}} {|x|^{3}},$$

which fixes the normalization of any exported OPE coefficients involving $M$. This normalization is a choice; the dimension and representation are invariant.

What an index can and cannot export

The superconformal index is a graded trace over states annihilated by a chosen supercharge. Long multiplets cancel, and distinct short multiplets can contribute the same monomial or recombine into a vanishing combination. Therefore an index coefficient generally determines

$$\text{an equivalence class of short-multiplet multiplicities},$$

not a unique list.

To export an actual multiplicity, one needs an inversion theorem, supplementary limits, or independent assumptions that remove recombination ambiguities. Otherwise label the datum “index-weighted coefficient” and include the fugacity convention and subtraction of descendants.

The same applies to chiral algebras or topological sectors: they determine the image under a protected functor. Operators in its kernel remain unconstrained.

Protected OPE information

Ward identities fix the coupling of a normalized scalar to conserved-current and stress-tensor multiplets in terms of its charges and $C_T$ or $C_J$. To use these coefficients, the bootstrap convention for conformal blocks and two-point functions must match the Ward-identity convention.

Chiral-ring products can identify which chiral primaries appear in a protected OPE,

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

but holomorphic rescaling of the operators changes $C_{ij}{}^k$. Physical OPE coefficients require unit-normalized two-point functions, which are generally Kähler data and may not be fixed by holomorphy.

Thus a ring relation such as $M^2=0$ is exportable without a metric, whereas a numerical unit-normalized coefficient usually is not.

Decoupled fields and accidental symmetries

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

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

Subtract its contribution from the interacting sector’s anomalies before exporting that sector. Record the accidental $U(1)$ and any mixed operator basis. Combining total $a,c$ with an interacting-only spectrum would violate Ward-identity consistency.

Validation before handoff

1. Recompute every chiral dimension from $3R/2$.
2. Recompute $a,c$ from the stored fermion anomaly traces.
3. Translate to the consumer’s $C_T,C_J$ conventions and round-trip.
4. Check unitarity and shortening labels.
5. Separate free, interacting, and topological factors.
6. Mark index coefficients that are not unique multiplicities.
7. Preserve exact rational or algebraic values and attach any uncertainty only to genuinely uncertain inputs.
8. Ensure the export makes no statement about crossing feasibility, numerical bounds, or long-multiplet gaps unless those were independently computed.

Common pitfalls

Exporting an R-charge as a dimension without the operator type. $\Delta=3R/2$ applies to chiral primaries, not arbitrary operators.

Calling an index coefficient a degeneracy. Recombination and cancellations can make the inverse problem nonunique.

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

Exercises

Using $c=163/64$ and $C_T=40c/\pi^4$, verify the value exported above. Then suppose a consumer uses $\widehat C_T=\pi^4C_T/40$; what should it receive?

Solution

$$C_T=\frac{40}{\pi^4}\frac{163}{64} =\frac{5}{8}\frac{163}{\pi^4} =\frac{815}{8\pi^4}.$$

The alternative convention gives

$$\widehat C_T=\frac{\pi^4}{40}C_T =\frac{163}{64}=c.$$

The consumer should receive the exact rational $163/64$ plus the explicit conversion, not a rounded decimal.

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.
• 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.
• Poland, David, and David Simmons-Duffin. “The Conformal Bootstrap.” Nature Physics 12 (2016): 535–539. arXiv:1602.07982.