Superconformal Algebras, Shortening Data, and the CFT Handoff

A superconformal algebra combines conformal symmetry with Poincaré supercharges $Q$, conformal supercharges $S$, and R-symmetry. Radial-quantization positivity turns $\{Q,S\}$ into bounds on scaling dimensions; saturation produces null descendants, short multiplets, and exact recombination rules. These are indispensable inputs to a conformal-field-theory calculation, but they neither determine the operator spectrum and OPE coefficients nor solve 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.

Positive-energy superconformal representations

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

• $Q$, with scaling dimension $+\tfrac12$;
• $S$, with scaling dimension $-\tfrac12$; and
• a compact R-symmetry algebra acting nontrivially on the odd generators.

If $Q$ denotes the full real spinor, the schematic brackets are

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

The schematic first relation should not be misread in four-dimensional Weyl notation: there it is $\{Q_\alpha,\bar Q_{\dot\alpha}\}\sim P_{\alpha\dot\alpha}$, while same-chirality Poincaré brackets vanish in the superconformal algebra.

In radial quantization, $P_\mu^\dagger=K_\mu$ and each $Q^\dagger$ is the corresponding $S$. A superconformal primary $|\mathcal O\rangle$ is killed by all $K_\mu$ and $S$ generators; $Q$ and $P$ generate its descendants. Their Gram matrices are positive in a unitary theory. This positive-energy highest-weight problem replaces the massive or massless little-group problem used for particle multiplets.

Nahm classified the finite-dimensional superconformal algebras under standard assumptions on spinorial odd generators and compact R-symmetry Nahm 1978, pp. 149–166. An allowed abstract real Lie superalgebra is necessary representation data, not evidence that an interacting local SCFT with a stress tensor exists.

The d=3 through d=6 census

This table covers unitary relativistic superconformal symmetry for $d\geq3$. $N_Q$ counts real Poincaré supercharges only, not $Q$ plus $S$.

$d$	Real superconformal algebra	Even subalgebra	$N_Q$	QFT restriction
3	$\mathfrak{osp}(\mathcal N\lvert4;\mathbb R)$	$\mathfrak{so}(3,2)\oplus\mathfrak{so}(\mathcal N)_R$	$2\mathcal N$	$\mathcal N\geq9$ SCFTs are free; genuine $\mathcal N=7$ enhances to $\mathcal N=8$
4	$\mathfrak{su}(2,2\lvert\mathcal N)$ for $\mathcal N\neq4$; $\mathfrak{psu}(2,2\lvert4)$ for $\mathcal N=4$	$\mathfrak{so}(4,2)\oplus\mathfrak{su}(\mathcal N)_R\oplus\mathfrak u(1)_R$, with no $\mathfrak u(1)$ in the projective case	$4\mathcal N$	no SCFT for $\mathcal N\geq5$; the central extension $\mathfrak{su}(2,2\lvert4)$ has no stress-tensor multiplet
5	the superconformal real form of $\mathfrak f(4)$	$\mathfrak{so}(5,2)\oplus\mathfrak{su}(2)_R$	8	this is the unique possibility; a 16-supercharge 5d Poincaré theory is not itself superconformal
6	$\mathfrak{osp}(8^*\lvert2\mathcal N)$	$\mathfrak{so}(6,2)\oplus\mathfrak{usp}(2\mathcal N)_R$	$8\mathcal N$	only chiral $(\mathcal N,0)$ supersymmetry occurs; no SCFT exists for $\mathcal N\geq3$

Some classification tables denote the six-dimensional family by $\mathfrak{osp}(6,2\lvert\mathcal N)$ while taking its R-symmetry to be $\mathfrak{sp}(2\mathcal N)$; this is the same family with a different naming convention. The algebra census and real forms are summarized by Córdova, Dumitrescu, and Intriligator 2019, § 1.2, pp. 6–8. Their stress-tensor-multiplet analysis gives the $d=4$, $d=6$, central-extension, and free-only restrictions Córdova, Dumitrescu, and Intriligator 2019, § 5.1.4, pp. 98–103, including the separate $\mathcal N=7\to8$ enhancement result Córdova, Dumitrescu, and Intriligator 2019, § 5.4.7, pp. 116–117.

Two-dimensional superconformal symmetry has independent left- and right-moving finite and infinite-dimensional families; the necessary Virasoro background begins in two-dimensional CFT. Under the finite-dimensional assumptions above, there is no standard superconformal algebra for $d>6$. Nonunitary, de Sitter, noncompact-R, and higher-spin extensions answer different classification questions.

Four-dimensional N=1 labels and the first norm

The four-dimensional $\mathcal N=1$ algebra is $\mathfrak{su}(2,2\lvert1)$. We use the Córdova–Dumitrescu–Intriligator notation

$$[j;\bar j]_\Delta^{(r)},$$

where $j,\bar j\in\mathbb Z_{\geq0}$ are Lorentz Dynkin labels, so the physical Lorentz spins are $(j/2,\bar j/2)$. The R-charge convention is

$$Q_\alpha:[1;0]_{1/2}^{(-1)}, \qquad \bar Q_{\dot\alpha}:[0;1]_{1/2}^{(+1)}.$$

For a scalar primary $|\mathcal O\rangle$ with labels $[0;0]_\Delta^{(r)}$, 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 side is a sum of descendant norms, hence

$$\Delta\geq\frac32r.$$

The conjugate $Q$ norm gives $\Delta\geq-3r/2$. If $\bar Q_{\dot\alpha}\mathcal O=0$, the scalar is chiral and saturates $\Delta=3r/2$; an antichiral scalar obeys $Q_\alpha\mathcal O=0$ and $\Delta=-3r/2$.

These are necessary first-level conditions, not the complete scalar classification. Higher descendant norms separate isolated short branches from the long continuum and create forbidden gaps. Minwalla develops the positive-energy method in Minwalla 1998, §§ 2–4, pp. 788–824; the exact labels below follow Córdova, Dumitrescu, and Intriligator 2019, § 2.2.1, pp. 29–32.

Complete scalar branches and forbidden gaps

In the stated convention, every unitary scalar superconformal primary belongs to one of the following branches:

$$\begin{array}{rll} L\bar L[0;0]_\Delta^{(r)} &:& \Delta>2+\tfrac32\lvert r\rvert,\\ A_2\bar L[0;0]_{2-\frac32r}^{(r)} &:& r<0,\\ L\bar A_2[0;0]_{2+\frac32r}^{(r)} &:& r>0,\\ A_2\bar A_2[0;0]_{2}^{(0)} &:& \text{conserved-current multiplet},\\ B_1\bar L[0;0]_{-\frac32r}^{(r)} &:& r<-\tfrac23,\\ L\bar B_1[0;0]_{\frac32r}^{(r)} &:& r>\tfrac23,\\ B_1\bar A_2[0;0]_{1}^{(-2/3)},\quad A_2\bar B_1[0;0]_{1}^{(+2/3)} &:& \text{free antichiral/chiral endpoints},\\ B_1\bar B_1[0;0]_{0}^{(0)} &:& \mathbf1. \end{array}$$

For example, at fixed $r>0$ a scalar may lie on the chiral line $\Delta=3r/2$ when $r\geq2/3$, or on and above the semishort/long threshold $\Delta=2+3r/2$. No unitary scalar multiplet occupies the open interval between those branches. The endpoint $r=2/3$, $\Delta=1$ is a free chiral multiplet, not a generic interacting chiral. The identity is the isolated $r=\Delta=0$ case.

This branch structure also sharpens the word protected. Generic $L\bar B_1$ scalar chirals with $2/3<r<2$ are absolutely protected because no long-multiplet recombination rule can supply them with compatible partners; the $r=2/3$ free $A_2\bar B_1$ endpoint is also absolutely protected. At $r=2$, the chiral can appear in the neutral long threshold together with a current multiplet and its antichiral conjugate; for $r>2$, a charged long parent can also supply it. Chirality fixes a dimension while the multiplet remains short, but it does not by itself guarantee absolute protection.

Explicit scalar recombination

At the neutral scalar threshold, the normalized long family decomposes as

$$\begin{aligned} L\bar L[0;0]_{\Delta\to2^+}^{(0)} \longrightarrow{}& A_2\bar A_2[0;0]_{2}^{(0)}\\ &\oplus B_1\bar L[0;0]_{3}^{(-2)} \oplus L\bar B_1[0;0]_{3}^{(+2)}. \end{aligned}$$

There are three summands, not a naive four: the simultaneous left and right null structures meet in the conserved-current multiplet. In any one fixed complete fugacity convention, the rule implies

$$\begin{aligned} \lim_{\Delta\downarrow2}\chi_{L\bar L[0;0]_\Delta^{(0)}} ={}&\chi_{A_2\bar A_2[0;0]_2^{(0)}} +\chi_{B_1\bar L[0;0]_3^{(-2)}}\\ &+\chi_{L\bar B_1[0;0]_3^{(+2)}}. \end{aligned}$$

The charged scalar rules are

$$\begin{aligned} r>0:\quad L\bar L[0;0]_{\Delta\to(2+\frac32r)^+}^{(r)} &\longrightarrow L\bar A_2[0;0]_{2+\frac32r}^{(r)} \oplus L\bar B_1[0;0]_{3+\frac32r}^{(r+2)},\\ r<0:\quad L\bar L[0;0]_{\Delta\to(2-\frac32r)^+}^{(r)} &\longrightarrow A_2\bar L[0;0]_{2-\frac32r}^{(r)} \oplus B_1\bar L[0;0]_{3-\frac32r}^{(r-2)}. \end{aligned}$$

Every label is part of the result: dimension, Lorentz Dynkin labels, R-charge, null side, and threshold direction. The shared shortening map visualizes the positive-norm logic, while the comparison table prevents this radial-quantization threshold from being confused with a massless little-group rank change or a particle BPS wall.

For multiplets beyond this scalar example, an imported rule must state the spacetime dimension and real superalgebra, all Lorentz and R-symmetry Dynkin labels, the long inequality, exact saturation, levels and labels of primary nulls, every threshold summand, and the character convention. A state-count equality is insufficient. When a source marks a general null-state construction as conjectural or consistency-tested, preserve that status rather than silently upgrading it.

Data needed for a CFT calculation

An algebraic result becomes usable in a block or crossing calculation only after its conventions, normalization, and domain of validity have been stated. For the scalar chiral example, specify:

Datum	What must be fixed
Theory and algebra	4d unitary $\mathcal N=1$ SCFT; $\mathfrak{su}(2,2\lvert1)$; radial adjoint; $R(Q)=-1$
Operator labels	basis label, flavor representation, conjugate, and either $L\bar B_1[0;0]_{3r/2}^{(r)}$ for $r>2/3$ or the free $A_2\bar B_1[0;0]_1^{(2/3)}$ endpoint
Shortening and protection	$\bar Q_{\dot\alpha}\mathcal O=0$, null level $1/2$, $\Delta=3r/2$, and whether the charge lies in an absolutely protected range
Recombination	every applicable threshold partner with full labels; when a chiral child has $r_{\mathcal O}=2$, its possible parent is the neutral $r_{\mathrm{parent}}=0$ long multiplet in the three-summand rule
Degenerate sectors and mixing	the operator basis in every degenerate quantum-number sector and its transformation under basis changes
Normalization conversion	two-point matrix, generator normalization, explicit rescaling to the block convention, inverse rescaling, and a round-trip check
Source and scope	the precise cited classification; whether the statement is exact, numerical, conjectural, or consistency-tested; and every free-theory or accidental-symmetry exception
Permitted conclusion	the precise selection rule or protected datum that a block or crossing calculation may use—and what remains dynamical

For a diagonal source basis, suppose

$$\langle O_i(x)O_j^\dagger(0)\rangle_{\mathrm{src}} =\frac{N_i\delta_{ij}}{\lvert x\rvert^{2\Delta_i}}, \qquad N_i>0.$$

If $\lambda^{\mathrm{src}}_{ijk}$ denotes the corresponding three-point coefficient, the forward normalization map is

$$\widehat O_i=N_i^{-1/2}O_i, \qquad \widehat\lambda_{ijk} =\frac{\lambda^{\mathrm{src}}_{ijk}}{\sqrt{N_iN_jN_k}}.$$

The inverse must be recorded and tested as well:

$$O_i=\sqrt{N_i}\,\widehat O_i, \qquad \lambda^{\mathrm{src}}_{ijk} =\sqrt{N_iN_jN_k}\,\widehat\lambda_{ijk}.$$

Applying the forward and inverse maps must recover the source data, including any mixing matrix in a degenerate sector. The protected-data bootstrap input develops the downstream use. The logical dependence is directional—crossing data do not redefine the superalgebra—but convention validation is bidirectional: every imported normalization map needs an inverse and a successful round trip.

What conformal dynamics must still supply

The algebra fixes representation labels, shortening equations, descendant relations, and selection rules; it constrains correlator tensor structures. It does not fix

• which superconformal primaries occur or with what multiplicities;
• scaling dimensions of long multiplets;
• unprotected OPE coefficients and central charges;
• 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 correlator constraints after importing defined labels and normalization. Superconformal blocks and crossing develops block decomposition and crossing equations; neither page should have to reconstruct missing shortening or convention data.

Common pitfalls

Confusing Dynkin labels with physical spins. In the classification notation, $[j;\bar j]$ uses integers. The associated physical Lorentz spins are $(j/2,\bar j/2)$.

Treating the first descendant norm as the full unitarity region. The inequality $\Delta\geq3r/2$ detects the chiral zero mode, but higher-level positivity creates the long threshold and the forbidden gap.

Equating chiral with absolutely protected. A chiral dimension is fixed while the multiplet remains chiral. Absolute protection additionally requires the absence of every compatible recombination channel.

Equating an algebra with an interacting SCFT. A real Lie superalgebra can exist even when no unitary stress-tensor multiplet, or only a free realization, is possible.

Exporting a one-way convention conversion. Without an inverse and round-trip test, a normalization mismatch can masquerade as an OPE or block discrepancy.

Exercises

1. Locate the forbidden gap. For a scalar with $r=1$, list the chiral point, the semishort threshold, and the long region. What changes at $r=2/3$?

Solution

At $r=1$, the chiral multiplet lies at $\Delta=3/2$, the $L\bar A_2$ threshold is $\Delta=2+3/2=7/2$, and long multiplets have $\Delta>7/2$. No unitary scalar multiplet exists for $3/2<\Delta<7/2$. At $r=2/3$, the chiral dimension is $\Delta=1$ and the multiplet acquires the additional $A_2$ shortening characteristic of a free chiral field.

2. Check the neutral recombination labels. Verify the dimensions and R-charges of all three summands in the $r=0$, $\Delta\to2^+$ rule, and identify the chiral pair.

Solution

The threshold multiplet $A_2\bar A_2[0;0]_2^{(0)}$ retains the parent’s scalar labels and contains the conserved current. The two other primaries occur one unit of dimension higher, at $\Delta=3$, with charges $-2$ and $+2$. They obey $\Delta=-3r/2$ and $\Delta=3r/2$, respectively, so they are an antichiral $B_1\bar L$ multiplet and its chiral $L\bar B_1$ conjugate.

3. Test a normalization round trip. Starting from positive $N_i$ and $\lambda^{\mathrm{src}}_{ijk}$, apply the forward map to hatted variables and then the inverse. Why would omitting the inverse be dangerous in a bootstrap import?

Solution

Substitution gives $\sqrt{N_i}(N_i^{-1/2}O_i)=O_i$ and multiplies the hatted coefficient by exactly $\sqrt{N_iN_jN_k}$, recovering $\lambda^{\mathrm{src}}_{ijk}$. Without the inverse, one cannot distinguish a convention mismatch from a physical difference in OPE data, especially after basis mixing in a degenerate sector.

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, arXiv.
• Shiraz Minwalla, “Restrictions Imposed by Superconformal Invariance on Quantum Field Theories,” Advances in Theoretical and Mathematical Physics 2 (1998), 781–846, DOI, arXiv.
• Werner Nahm, “Supersymmetries and Their Representations,” Nuclear Physics B 135 (1978), 149–166, DOI.