Superconformal Algebras, Shortening Data, and the CFT Handoff

A superconformal algebra extends the conformal generators by Poincaré supercharges $Q$ and conformal supercharges $S$. Radial-quantization positivity turns their anticommutators into bounds on scaling dimensions; saturation creates null descendants, short multiplets, and recombination rules. The algebra exports these representation data to conformal field theory, but it does not determine which operators exist, their OPE coefficients, or a solution of crossing symmetry.

Required background. BPS bounds, shortening, and recombination supplies positive-matrix and null-quotient logic. Primaries, descendants, and conformal multiplets supplies radial quantization and conformal representation theory.

Helpful background. Superconformal Ward identities applies the exported algebra to correlators. Superconformal blocks and crossing develops block construction and crossing equations.

Superconformal structure

The bosonic conformal algebra in $d$ dimensions is $\mathfrak{so}(d,2)$ in Lorentzian signature. It contains translations $P_\mu$, special conformal transformations $K_\mu$, Lorentz generators $M_{\mu\nu}$, and dilatations $D$. A superconformal algebra adds

• $Q$, of scaling dimension $+\tfrac12$;
• $S$, of scaling dimension $-\tfrac12$; and
• an R-symmetry algebra that acts nontrivially on $Q$ and $S$.

Schematically,

$$\{Q,Q\}\sim P, \qquad \{S,S\}\sim K, \qquad \{Q,S\}\sim D+M+R.$$

In radial quantization, $P_\mu^\dagger=K_\mu$ and $Q^\dagger$ is an appropriate $S$. Consequently $\{Q,S\}$ is a positive norm matrix on descendants. A superconformal primary $|\mathcal O\rangle$ is annihilated by every $K_\mu$ and every $S$; acting with $Q$ and $P$ generates its descendants. This highest-weight structure, rather than the flat-space particle little group, is the representation space being classified.

The finite-dimensional superconformal algebras under the standard spinorial-odd-generator and compact-R assumptions were classified by Nahm Nahm 1978, pp. 149–166. Their existence as abstract real Lie superalgebras must not be confused with existence of an interacting unitary SCFT.

The d=3 through d=6 census

The table lists the algebras most directly used in unitary relativistic SCFT. Counts refer to Poincaré supercharges $Q$, not $Q$ plus $S$.

$d$	Superconformal algebra	Bosonic subalgebra	Real $Q$ count
3	$\mathfrak{osp}(\mathcal N\lvert4)$	$\mathfrak{so}(3,2)\oplus\mathfrak{so}(\mathcal N)$	$2\mathcal N$
4	$\mathfrak{su}(2,2\lvert\mathcal N)$; $\mathfrak{psu}(2,2\lvert4)$ for $\mathcal N=4$	$\mathfrak{so}(4,2)\oplus\mathfrak u(\mathcal N)$, with the central $\mathfrak u(1)$ removed in the projective $\mathcal N=4$ case	$4\mathcal N$
5	exceptional $\mathfrak f(4)$	$\mathfrak{so}(5,2)\oplus\mathfrak{su}(2)_R$	$8$
6	$\mathfrak{osp}(8^*\lvert2\mathcal N)$	$\mathfrak{so}(6,2)\oplus\mathfrak{usp}(2\mathcal N)$	$8\mathcal N$

Two-dimensional superconformal symmetry has left- and right-moving finite and infinite-dimensional families and is treated with the two-dimensional supersymmetric QFT chapter. Under the assumptions of the finite-dimensional classification there is no standard superconformal algebra for $d>6$. De Sitter, nonunitary, higher-spin, and noncompact-R real forms answer different questions and are not entries in this unitary SCFT table.

Even within the table, stress-tensor multiplet and interacting-theory requirements are stronger than algebra existence. For example, the modern multiplet analysis rules out interacting $d\geq4$ SCFTs with more than 16 Poincaré supercharges and identifies additional free-theory exceptions; see Córdova, Dumitrescu, and Intriligator 2019, §§ 5–6. This is a QFT restriction, not a change to Nahm’s algebra list.

A four-dimensional N=1 shortening calculation

The four-dimensional $\mathcal N=1$ algebra is $\mathfrak{su}(2,2\lvert1)$, with bosonic subalgebra $\mathfrak{so}(4,2)\oplus\mathfrak u(1)_R$. Label a superconformal primary by

$$[\Delta;(j,\bar j);r],$$

where $j,\bar j$ are $SU(2)\times SU(2)$ Lorentz spins and $r$ is normalized so that $R(Q_\alpha)=-1$ and $R(\bar Q_{\dot\alpha})=+1$.

For a scalar primary $|\mathcal O\rangle$ with $j=\bar j=0$, choose the $S$ normalization so that

$$\frac14\sum_{\dot\alpha=1}^2 \langle\mathcal O| \{\bar S^{\dot\alpha},\bar Q_{\dot\alpha}\} |\mathcal O\rangle =\left(\Delta-\frac32r\right) \langle\mathcal O|\mathcal O\rangle.$$

The left-hand side is a sum of descendant norms, hence the necessary level-one inequality

$$\Delta\geq\frac32r.$$

The conjugate $Q$ norm gives the companion inequality $\Delta\geq-3r/2$. Generic long multiplets obey additional, often stronger, conditions at higher descendant levels. If the scalar is chiral,

$$\bar Q_{\dot\alpha}|\mathcal O\rangle=0,$$

the norm vanishes and

$$\Delta=\frac32r.$$

An antichiral scalar instead obeys $Q_\alpha|\mathcal O\rangle=0$ and $\Delta=-\tfrac32r$. The coefficient $3/2$ depends on the declared $U(1)_R$ normalization; the invariant statement is that the appropriate $\{Q,S\}$ eigenvalue vanishes. General spinning primaries have additional $M_{\mu\nu}$ contributions and a richer set of $A$-, $B$-, and isolated shortening conditions. Minwalla derives representative unitarity bounds in Minwalla 1998, §§ 2–4; the comprehensive diverse-dimensional tables and null-state algorithm are in Córdova, Dumitrescu, and Intriligator 2019, §§ 2–4 and Appendices A–C.

This derivation shows exactly what is protected. The algebra fixes $\Delta$ in terms of the superconformal R-charge for an operator that remains in the same chiral short multiplet. It does not identify the correct infrared R-symmetry when accidental symmetries mix with it, nor prevent the operator from pairing and recombining when allowed.

Shortening and recombination data

As $\Delta$ approaches a unitarity threshold, a long multiplet develops null descendants. At the threshold its representation becomes reducible and decomposes into short multiplets. Conversely, compatible short multiplets can recombine into a long multiplet when $\Delta$ moves above the bound.

A usable recombination rule must specify

1. the spacetime dimension and real superconformal algebra;
2. the Lorentz and R-symmetry Dynkin labels;
3. the normalization of $D$ and every Abelian R-charge;
4. the long-multiplet inequality;
5. the exact saturation condition;
6. the level and quantum numbers of every primary null state;
7. the irreducible short multiplets appearing at threshold; and
8. the character convention used to verify equality.

A dimension equality alone is not enough. Two representations can have the same number of states but different null modules, spins, or R-charges. The constructive algorithm of Córdova, Dumitrescu, and Intriligator is explicitly described as conjectural in its most general form but is supported by extensive consistency checks; that status should accompany any imported table rather than being upgraded to a theorem by repetition.

The CFT export record

The following is the minimum algebraic payload a conformal calculation can safely import:

Field	Required content
Algebra	dimension, signature/radial real form, exact superalgebra, and R-symmetry normalization
Primary	$\Delta$, Lorentz labels, R and flavor labels, and conjugation
Shortening	annihilating $Q$ components, saturation equation, and null level
Recombination	all threshold summands with labels and character identity
Normalization	two-point function, generator normalization, and any central-charge convention
Protected datum	exactly which dimension, OPE coefficient, index contribution, or cohomology class is constrained
Failure boundary	accidental symmetry, multiplet pairing, contact terms, operator mixing, nonunitarity, or free-theory exception

For the scalar chiral example, the record says: four-dimensional $\mathcal N=1$; $\mathfrak{su}(2,2\lvert1)$; $j=\bar j=0$; $\bar Q_{\dot\alpha}\mathcal O=0$; $\Delta=3r/2$ in the declared $R(Q)=-1$ convention; and a conjugate antichiral multiplet with $r\to-r$. It does not provide the value of an OPE coefficient or prove that $\mathcal O$ occurs in a specific theory.

What conformal dynamics must still supply

The algebra fixes selection rules, descendant structure, null relations, and the tensor structures forced by symmetry. It does not fix

• the list and multiplicities of superconformal primaries;
• numerical scaling dimensions of long multiplets;
• OPE coefficients and central charges not protected by shortening;
• which algebraic representations coexist in a local, unitary SCFT;
• superconformal blocks in a chosen correlator normalization; or
• a crossing-symmetric, reflection-positive solution.

Superconformal Ward identities develops the correlator constraints after importing this page’s algebra and normalization. Superconformal blocks and crossing develops block decomposition, crossing equations, and bootstrap inference. The handoff is deliberately one-way: a numerical crossing solution does not redefine the superalgebra.

Common pitfalls

Equating an abstract algebra with an interacting SCFT. An allowed real Lie superalgebra is necessary representation data. Stress-tensor multiplets, locality, unitarity, and interacting existence impose additional restrictions.

Using a chiral dimension formula with the wrong R-charge. The relation $\Delta=3r/2$ uses the exact superconformal $U(1)_R$ and the stated normalization. A UV trial R-charge or an anomalous current cannot be substituted silently.

Calling shortening a complete bootstrap solution. Shortening organizes the exchanged representations and may fix protected pieces. Long dimensions and OPE coefficients remain dynamical data constrained, not determined, by crossing.

Check your understanding

Why does a scalar chiral primary obey $\Delta=3r/2$, and which hypothesis would fail if $r$ were only a trial ultraviolet charge?

Answer

Radial positivity makes the $\bar Q$-descendant norm proportional to $\Delta-3r/2$. Chirality sets that descendant to zero, so the eigenvalue saturates. The derivation uses the R-generator inside the infrared superconformal algebra; a trial UV current need not be that generator, especially when accidental symmetries mix with it.

References

• Clay Córdova, Thomas T. Dumitrescu, and Kenneth Intriligator, “Multiplets of Superconformal Symmetry in Diverse Dimensions,” Journal of High Energy Physics 2019 (2019), 163, DOI.
• Shiraz Minwalla, “Restrictions Imposed by Superconformal Invariance on Quantum Field Theories,” Advances in Theoretical and Mathematical Physics 2 (1998), 781–846, DOI.
• Werner Nahm, “Supersymmetries and Their Representations,” Nuclear Physics B 135 (1978), 149–166, DOI.