Global Symmetry and Spinning Bootstrap Systems

A symmetry-resolved spinning bootstrap problem is not obtained by attaching group labels to a scalar crossing equation. The group channels, three- and four-point tensor structures, permutation phases, conservation kernels, Ward-normalized exchanges, and positive OPE matrices must be assembled as one system. The central result is a matrix crossing equation whose positivity follows from reflection positivity in a declared Hermitian basis. This page builds that equation and works through its group, conservation, and Ward-identity data for four conserved currents in a three-dimensional CFT.

We work with separated-point Euclidean correlators in a unitary CFT at fixed dimension $d>2$. External operators are Hermitian or paired with their Hermitian conjugates, their two-point metric is positive, and every change of tensor basis is recorded. These hypotheses are what make positive-semidefinite (PSD) OPE matrices available; conservation by itself does not Poland, Rychkov, and Vichi 2019, § III.E, pp. 10–13. The spinning-block, global-symmetry, conserved-current, and crossing ingredients are reviewed in Poland, Rychkov, and Vichi 2019, §§ III.F.7–III.I, pp. 19–24.

Required background. Mixed-Correlator Islands supply PSD OPE-vector geometry. Spinning operators and blocks supply tensor bases, seed normalization, and the block-generation pipeline. Helpful background. Current and stress-tensor CFT data fix the two-point and Ward conventions used below.

Method cutoff: 2026-09-01. Generator capabilities and numerical implementations evolve. The equations and stop rules below are method-level statements; no current numerical bound for a named CFT is quoted.

Symmetry channels include multiplicity data

For external representations $r_i$ and $r_j$ of a compact global-symmetry group, the general decomposition is

$$r_i\otimes r_j \simeq \bigoplus_R\left(V_R\otimes M^R_{ij}\right), \qquad N^R_{ij}=\dim M^R_{ij}.$$

$V_R$ carries the irrep $R$; the multiplicity space $M^R_{ij}$ counts independent copies of that irrep in the tensor product. When $N^R_{ij}>1$, an irrep label is not enough: crossing acts on intertwiner indices in $M^R_{ij}$ as well. The matrix unit $P_{R,mn}$ maps copy $n$ of $R$ to copy $m$ and obeys

$$P_{R,mn}P_{S,pq} =\delta_{RS}\delta_{np}P_{R,mq}.$$

Its diagonal units resolve the identity,

$$\sum_R\sum_{m=1}^{N^R_{ij}}P_{R,mm} =\mathbf1_{r_i\otimes r_j}.$$

For complex representations, the relevant OPEs can involve $r\otimes\bar r$ as well as $r\otimes r$; a mixed system uses $r_i\otimes r_j$ channel by channel. Writing only $r\otimes r=\bigoplus_R R$ silently assumes identical external representations and usually a multiplicity-free, self-conjugate situation.

A non-Abelian current example

Let $J_\mu^a$ be a Hermitian conserved current for $SU(2)$ in $d=3$. The adjoint is the real three-dimensional representation, so

$$\mathbf3\otimes\mathbf3 =\mathbf1_s\oplus\mathbf3_a\oplus\mathbf5_s.$$

With $\delta^{ab}$ as the adjoint metric, a convenient multiplicity-free projector basis is

$$\begin{aligned} (P_{\mathbf1})^{ab}{}_{cd} &=\frac13\delta^{ab}\delta_{cd},\\ (P_{\mathbf3})^{ab}{}_{cd} &=\frac12\left(\delta^a{}_c\delta^b{}_d-\delta^a{}_d\delta^b{}_c\right),\\ (P_{\mathbf5})^{ab}{}_{cd} &=\frac12\left(\delta^a{}_c\delta^b{}_d+\delta^a{}_d\delta^b{}_c\right) -\frac13\delta^{ab}\delta_{cd}. \end{aligned}$$

Direct contraction gives $P_RP_S=\delta_{RS}P_R$, $\sum_RP_R=\mathbf1_{\mathbf3\otimes\mathbf3}$, and traces $1$, $3$, and $5$. These are exact fixtures for a projector generator.

Crossing already mixes these channels. Order them as $(\mathbf1,\mathbf5,\mathbf3)$ and define the $14\to32$ pair exchange by

$$(P_R)^{ad}{}_{cb} =\sum_S M_{RS}(P_S)^{ab}{}_{cd}.$$

Direct substitution gives

$$M= \begin{pmatrix} \frac13&\frac13&-\frac13\\ \frac53&\frac16&\frac56\\ -1&\frac12&\frac12 \end{pmatrix}, \qquad M^2=\mathbf1.$$

The nonsymmetric appearance comes from using idempotent projectors with different traces; dimension-normalized channel tensors give an equivalent orthogonal convention. Derive this $SU(2)$ matrix directly. Generic-$SU(N)$ formulas contain channels that collapse when $N=2$ and need not have a nonsingular term-by-term specialization.

For identical scalars in the same adjoint representation, the symmetric $\mathbf1$ and $\mathbf5$ channels contain even spin while the antisymmetric $\mathbf3$ channel contains odd spin. That familiar rule is only a warm-up. For currents, exchanging the first two operators also acts on the spacetime tensor-structure index. The allowed sector is determined by the combined group, tensor, spin, and Bose-statistics eigenvalue, not by the group projector alone.

Four adjoint currents probe the Lie algebra and adjoint OPE sector. By themselves they cannot distinguish the global forms $SU(2)$ and $SO(3)$ or reveal center-charged doublet operators; that requires correlators of center-sensitive operators.

Tensor structures turn OPE coefficients into matrices

For the multiplicity-free $SU(2)$ example, remove a declared scalar prefactor $\mathcal K(x_i)$ and write the current correlator in the $12\to34$ channel as

$$\langle J_\mu^aJ_\nu^bJ_\rho^cJ_\sigma^d\rangle =\mathcal K(x_i) \sum_R(P_R)^{ab}{}_{cd} \sum_I\mathcal T^I_{\mu\nu\rho\sigma}(x_i)\, \mathcal G_I^R(u,v).$$

$I$ labels four-point tensor structures. An exchanged primary $\mathcal O$ can couple to $JJ$ through several three-point structures $t_{JJ\mathcal O}^{(p)}$. Its contribution therefore has the quadratic form

$$\mathcal G_I^R(u,v) \supset \sum_{p,q} \lambda_{\mathcal O,p}^{*}\, G_{\mathcal O,I}^{R;pq}(u,v)\, \lambda_{\mathcal O,q}.$$

Here the first block index is the row index and the second is the column index, so the expression is $\boldsymbol\lambda^\dagger G\boldsymbol\lambda$. In a multiplicity-bearing problem, keep group-copy indices $m,n$ separate from three-point-structure indices $p,q$; the block is then a matrix on their tensor product rather than an object with an ambiguous composite label.

Crossing has two coupled pieces. Group recoupling rewrites, for example, $(P_R)^{ab}{}_{cd}$ in the $14\to32$ projector basis. A kinematic permutation rewrites $\mathcal T_I(1,2,3,4)$ in the permuted tensor basis. Their product, together with the scalar prefactor ratio and statistics sign, produces the matrix-valued crossing vector $\vec V_{\mathcal O}^R$. Permutation matrices must satisfy the relevant symmetric-group relations, not merely reproduce one chosen exchange.

This is why the automated global-symmetry equation generation of Go and Tachikawa 2019, §§ 2–4 is valuable but not a blanket spinning solution: the published autoboot construction treats arbitrary numbers of scalar external operators. Spinning systems additionally need independently verified tensor structures, conservation maps, and spinning blocks.

Conservation is a rank reduction

For fixed $R$, exchanged quantum numbers $(\Delta,\ell)$, and a raw $JJ\mathcal O$ basis, separated-point conservation of the two external currents gives a linear system. In a parity-preserving three-dimensional CFT it is useful to display both spacetime parity $\wp=\pm$ and the group-exchange eigenvalue $\sigma_R$:

$$D^{R,\wp,\sigma_R}_{\Delta,\ell}\, \boldsymbol\lambda=0, \qquad \sigma_{\mathbf1}=\sigma_{\mathbf5}=+1, \quad \sigma_{\mathbf3}=-1.$$

Choose a full-column-rank matrix $Q^{R,\wp,\sigma_R}_{\Delta,\ell}$ whose columns span the kernel. Suppressing these labels when no confusion can arise, define physical coordinates by

$$\boldsymbol\lambda=Q^R\widehat{\boldsymbol\lambda}, \qquad \widehat{\vec V}^{\,R} =Q^{R\dagger}\vec V^{\,R}Q^R.$$

The dagger matters in a complex basis; it becomes a transpose in a real basis. Conservation has now reduced the OPE-coordinate space before positivity is imposed.

For different left and right OPE pairs, the reduction is $Q_{12}^{\dagger}\vec VQ_{34}$; a Hermitian PSD interpretation is available only after the correlator ordering pairs conjugate coefficient spaces.

The rank of $D^R$ can jump at shortening values of $\Delta$, at special spin, or in special integer dimensions. Reusing a generic matrix $Q^R$ across such a jump deletes or invents structures. Recompute the kernel in each exceptional sector and compare symbolic rank with a generic-kinematics numerical rank. The construction of spinning structures and their conservation equations is developed in Costa et al. 2011, §§ 4.2–5, pp. 15–31.

The antisymmetric, parity-even, spin-one $JJ\mathcal O$ sector supplies an exact rank-jump fixture. In the four-structure basis of He and collaborators, a generic nonconserved spin-one operator has

$$\boldsymbol\lambda_{\mathrm{generic}} =a \begin{pmatrix} \Delta+3\\1\\1\\\Delta-1 \end{pmatrix}.$$

When the exchanged operator is itself a conserved current, $(\Delta,\ell)=(2,1)$ and an additional null state lowers the rank:

$$\boldsymbol\lambda_J =a \begin{pmatrix}1\\0\\0\\0\end{pmatrix} +b \begin{pmatrix}0\\1\\1\\1\end{pmatrix}.$$

The first kernel imposes conservation only at the two external current insertions. The second also incorporates shortening of the exchanged current. Poles in a rational kernel chart are another reason to recompute the polynomial nullspace rather than substitute blindly He, Rong, Su, and Vichi 2024, appendix B.1, tables 3–6.

Separated-point conservation is homogeneous. Ward identities also contain contact terms and fix only particular combinations after the generator normalization, $C_J$, $C_T$, and tensor basis are declared. For $SU(2)$, take Hermitian fundamental generators $T^a=\sigma^a/2$, so

$$\operatorname{tr}(T^aT^b)=\frac12\delta^{ab}, \qquad [T^a,T^b]=i\epsilon^{abc}T^c, \qquad f^{abc}=\epsilon^{abc}.$$

In the current and embedding-tensor convention of He et al. 2024, § II.A, Eqs. (6)–(7),

$$\langle J_\mu^a(x)J_\nu^b(0)\rangle =C_J\frac{\delta^{ab}I_{\mu\nu}(x)}{x^4},$$

the same parity-even current basis gives

$$\boldsymbol\lambda_J = \begin{pmatrix} c-5t\\-t\\-t\\-t \end{pmatrix}, \qquad c=\frac{3C_J}{4\pi}, \quad t=\lambda_{JJJ}.$$

Thus the current exchange is not wholly Ward-fixed: one three-current parameter remains. The $JJT$ vector likewise retains a convention-dependent stress-tensor shape parameter. At each chosen parameter point, form the complete outer product of the affine Ward vector. Splitting its “fixed” and “free” coordinates into unrelated crossing terms would discard their linear cross terms and enlarge the feasible set.

The vector above is a three-point-function coefficient in the stated current convention. If the conformal blocks assume a unit-normalized exchanged primary, convert it to $\boldsymbol\lambda_J/\sqrt{C_J}$ before forming the outer product; apply the analogous $C_T$ conversion to stress-tensor exchange.

We retain the physical external currents rather than replacing them by $j=J/\sqrt{C_J}$. Thus, if $\vec V_{\mathrm{id}}^{(0)}$ denotes the identity crossing vector with unit external two-point metric, the identity term below is $\vec V_{\mathbf1}=C_J^2\vec V_{\mathrm{id}}^{(0)}$. The division by $\sqrt{C_J}$ in the previous paragraph normalizes the exchanged current only. One may instead divide the complete crossing equation by $C_J^2$, but then every distinguished term and every spectral density must be rescaled together.

Reflection positivity produces Hermitian OPE matrices

In a parity-preserving three-dimensional CFT, let several primaries $\mathcal O_\alpha$ have the same $R$, spacetime parity $\wp$, $\Delta$, and $\ell$. Define their reduced OPE density matrix in the conserved three-point basis by

$$\rho_{pq}^{R,\wp,\Delta,\ell} =\sum_\alpha \widehat\lambda_{\alpha,p} \widehat\lambda_{\alpha,q}^{*}.$$

For every vector $z$, $z^\dagger\rho z=\sum_\alpha\lvert\widehat{\boldsymbol\lambda}_\alpha^\dagger z\rvert^2\ge0$, so $\rho$ is Hermitian PSD. A unique exchanged operator gives a rank-one matrix. Off-diagonal entries can be negative or complex; positivity constrains the complete quadratic form, not each component.

The full crossing system can now be written compactly as

$$\vec V_{\mathrm{fixed}} +\sum_{R,\wp}\sum_{(\Delta,\ell)\in\mathcal S_{R,\wp}} \operatorname{Tr}\!\left( \rho^{R,\wp,\Delta,\ell}\, \widehat{\vec V}^{\,R,\wp}_{\Delta,\ell} \right)=0, \qquad \rho^{R,\wp,\Delta,\ell}\succeq0.$$

The trace is taken component by component in the crossing vector, with $\operatorname{Tr}(\rho V)=\sum_{p,q}\rho_{pq}V_{qp}=\boldsymbol\lambda^\dagger V\boldsymbol\lambda$ for a rank-one density. The matrices $\widehat{\vec V}$ are not themselves required to be PSD. A separating functional $\alpha$ certifies exclusion when, in one consistent sign convention,

$$\alpha[\vec V_{\mathrm{fixed}}]>0, \qquad \alpha[\widehat{\vec V}^{\,R,\wp}_{\Delta,\ell}]\succeq0 \quad\text{for every allowed sector and every allowed }\Delta.$$

Applying $\alpha$ to crossing would then make a zero equal to a strictly positive number. Positivity at a few sampled dimensions is not enough: the block approximation or an interval/rational enclosure must control the continuous $\Delta$ domain used by the claim.

What a basis change must preserve

Define a new coefficient basis by $\widehat{\boldsymbol\lambda}'=S\widehat{\boldsymbol\lambda}$ with $S$ invertible. The same quadratic crossing contribution is represented by

$$\widehat{\vec V}' =S^{-\dagger}\widehat{\vec V}S^{-1}, \qquad \rho'=S\rho S^\dagger, \qquad \operatorname{Tr}(\rho'\widehat{\vec V}') =\operatorname{Tr}(\rho\widehat{\vec V}).$$

This transformation is a congruence, not a similarity transformation. It preserves Hermitian PSD because $z^\dagger\rho'z=(S^\dagger z)^\dagger\rho(S^\dagger z)\ge0$. A singular $S$ is not a basis change: it loses directions and can change the feasible problem. Record $S$, component order, phases, precision, and condition number with the generated blocks.

Assemble the three-dimensional SU(2) current system

For the worked example, take identical Hermitian currents with $\Delta_J=2$, positive $C_J$, and a parity-preserving spectrum. The group channels are $R=\mathbf1,\mathbf3,\mathbf5$. After the tensor and conservation reductions, separate the identity, the distinguished current, and the stress tensor from the remaining spectral sum:

$$\vec V_{\mathbf1} +\vec V_J(C_J,t) +\vec V_T(C_J,C_T,\gamma) +\sum_{R,\wp}'\sum_{(\Delta,\ell)\in\mathcal S_{R,\wp}} \operatorname{Tr}\!\left( \rho_{\Delta,\ell}^{R,\wp} \widehat{\vec V}_{\Delta,\ell}^{\,R,\wp} \right)=0.$$

The prime omits the explicitly displayed operators. Here $t=\lambda_{JJJ}$ and $\gamma$ denotes the remaining $JJT$ shape parameter in the chosen basis; no numerical range for either is assumed. At a fixed $(C_J,t,C_T,\gamma)$ point, $\vec V_J$ and $\vec V_T$ use the complete Ward-constrained OPE vectors, including every cross term. All other allowed structures remain visible in the PSD sum.

Four-point conservation makes the reduction concrete. Per group channel, four unconstrained vector insertions begin with $41$ conformal-frame helicity structures. Pairwise permutations organize them into $17$ independent functions. External-current conservation gives $14$ first-order equations, which may be represented by $14\times17$ operators acting on the function column. The time-derivative matrix has rank $12$: five bulk-function vectors must be known throughout the cross-ratio plane. Conservation also requires two point-function vectors initialized at $z=\bar z=\tfrac12$, but crossing relates their values there through $M$, leaving one independent point-function vector. The remaining functions are reconstructed by conservation. Since global symmetry commutes with the spacetime equations, the same reduction applies in each $R$ and the recoupling matrix $M$ mixes the resulting crossing data He et al. 2024, appendix B.2.d, Eqs. (B45)–(B46), and appendix C.

A complete non-Abelian current implementation must perform the following steps.

1. Freeze the external contract. Record $d=3$, current normalization, $SU(2)$ generator convention, Hermitian conjugation, parity content, external ordering, OPE prefactor, and block normalization.
2. Generate group recoupling exactly. Verify the three projectors above, derive every channel-crossing matrix, and test the permutation-group relations over exact rationals before numerical conversion.
3. Enumerate tensor structures in the physical dimension. Form the $JJ\mathcal O$ and $JJJJ$ bases, quotient three-dimensional identities, and compute exchange matrices. Do not import a generic-$d$ count.
4. Impose conservation sector by sector. Construct $D^R(\Delta,\ell)$ and $Q^R(\Delta,\ell)$, with separate branches wherever the rank changes. Check divergences directly on generated blocks away from coincident points.
5. Separate distinguished and generic data. Insert the identity and the complete affine Ward vectors for $J$ and $T$ in the chosen $C_J$ and $C_T$ convention. Retain every unfixed current, stress-tensor, and parity-odd parameter explicitly.
6. Build the Hermitian crossing matrices. Apply group and kinematic crossing, then verify that the reduced matrices have the declared adjoint relation in the reflection-positive ordering.
7. Define each spectral sector. State its irrep, exchange eigenvalue, spin, spacetime parity, unitarity bound, shortening exceptions, gap assumptions, and whether degeneracy is represented by a full PSD matrix.
8. Enclose the numerical problem. Check block recursions, spin tails, continuous-$\Delta$ positivity, precision, and the independent certificate verifier.

The complete non-Abelian current construction, including conservation, permutation reduction, and linearly independent crossing equations, is worked out in He, Rong, Su, and Vichi 2024, §§ II–IV and appendices B–D. The stress-tensor analogue shows why nonredundant tensor equations must be derived before the semidefinite program is formed Dymarsky et al. 2018, §§ 2–4.

Keep exchange, spatial parity, and reality separate

Three notions that are often called “parity” answer different questions.

• Exchange symmetry is the action of permuting identical operators. It combines Bose or Fermi statistics with a matrix acting on group and tensor structures.
• Spatial parity is an orientation-reversing spacetime transformation. In three dimensions, epsilon-tensor structures can be parity odd even when their exchange behavior is even.
• Reality is fixed by Hermitian conjugation and the Euclidean tensor basis. A parity-odd structure may carry a conventional factor of $i$; absorb that phase into the basis if a real SDP basis is desired, then transform blocks and OPE coefficients together.

For identical bosonic currents, the kinematic $JJ\mathcal O$ exchange eigenvalue must equal the group eigenvalue $\sigma_R$. Spatial parity is a separate label: parity-odd structures can occur with more than one spin parity and exchange sector. In a parity-violating CFT, $\wp$ is not a superselection label; combine even and odd structures in one OPE vector and retain their cross-Gram entries.

Never impose elementwise positivity on a parity-even or parity-odd block component. Reflection positivity applies to the Hermitian OPE Gram matrix after all fixed phases have been propagated. Spinning fermion systems provide a useful independent sign fixture because the permutation matrix and the Grassmann sign must both be present Iliesiu et al. 2018, § 3.

Acceptance tests before a solver run

Every generated system should stop on the first failed required acceptance test.

• Group completeness: matrix-unit multiplication, completeness, irrep dimensions, and every recoupling identity are exact.
• Correlator closure: all OPE pairings required by crossing are present, including conjugate channels and shared operators in mixed systems.
• Tensor completeness: generic-kinematics rank, physical-dimension identities, and permutation-group relations agree in two independent representations.
• Conservation: symbolic kernel dimension, direct divergence residual, Ward normalization, and every special-$\Delta$ rank branch agree.
• Reality and positivity: the two-point metric is positive, crossing matrices have the declared adjoint relation, and basis round trips preserve the unscaled equation.
• Block accuracy: Casimir or recursion residuals, leading OPE tensors, shortening limits, radial order, and spin-tail controls pass in each sector.
• Continuous positivity: the minimum eigenvalue is enclosed on the full represented $\Delta$ interval; a sampled grid is only a diagnostic.
• Certificate independence: a separate reader reconstructs the unscaled crossing problem and verifies residuals and PSD margins without reusing solver state.

Rychkov and Su 2024, §§ 2.1–2.2, pp. 2–3 and § 3.2.3, pp. 8–10 review current spinning and global-symmetry software and explain why conserved-current bootstraps require specialized equation generation rather than a scalar-only front end.

Numerical claim taxonomy

This is the canonical structured taxonomy shared by the numerical-bootstrap pages. Choose the row matching the object actually verified, then apply the symmetry-and-spin assembly checks above as a precondition. If a projector, tensor, conservation, reality, or positivity test fails, the output does not qualify for any row.

On wide layouts, focus the table region and use the horizontal scrollbar or keyboard arrow keys. On narrow screens and in print, each row reflows into a labeled card.

Numerical-bootstrap outputs, supported interpretations, and stop rules

Claim class	Verified output	Supported interpretation	Required assumptions	Convergence study	Failure or downgrade trigger	Evidence and freshness record	Unsupported inference
Certified exclusion	An independently verified infeasibility certificate for the declared finite represented problem.	The tested spectral hypothesis fails in that represented problem.	External data, sectors, gaps, block and tail representation, positivity basis, and sign convention.	Certificate residuals and positivity margins across block, functional, spin, and precision settings.	An independently verified feasible witness, failed sign check, or failed positivity enclosure.	Serialized input, certificate, unscaled residuals, independent-verifier result, release date, and software environment.	“No exact CFT” without complete approximation control.
Conditional bound	One-sided certified exclusions establish a bound in the represented problem; a quoted boundary location also requires a declared unresolved or feasible-side bracket.	A finite conditional upper or lower limit under the stated assumptions.	Normalization, spectral assumptions, monotonicity of the tested hypothesis, search protocol, and represented parameter domain.	For a one-sided exclusion bound, verify certificate margins across functional, block, spin-tail, and precision refinements; for a quoted boundary location, also require bracket and interpolation stability.	A feasible witness in the excluded region or motion outside the reported refinement envelope.	Retain the trial record, exclusion certificates, and full refinement table; when quoting a boundary location, also retain its feasible-side or unresolved bracket history and cutoff date.	“The boundary point exists” or an assumption-free universal number.
Stable feature or kink	A dated, algorithmically defined feature persists within its reported envelope under the declared numerical and representation refinements.	A reproducible geometric feature and possible change of the active spectrum.	Curve definition, interpolation, parameterization, and complete represented system.	Location and shape remain within the stated envelope as the mesh, functional, blocks, spin tail, and precision are refined.	The feature smooths, drifts beyond its envelope, or disappears after a missing sector is restored.	Dated curve data, underlying certificates, feature definition, and current independent physics checks with an evidence cutoff.	Automatic identification with a named model.
Scan-delimited component	A certified connected barrier or excluded-cell cover encloses a declared not-excluded component, with every unresolved boundary cell recorded.	A conditional component within the scanned domain and resolution.	Correlator closure, shared-operator assumptions, search domain, and topology protocol.	Adaptive boundary refinement, topology checks, and assumption-removal scans.	A missed passage or component, failed enclosing certificate, or an opened region after relaxing an assumption.	Boundary certificates, dated scan map, unresolved cells, topology diagnostics, and search environment.	“Complete island,” realization of every interior point, or uniqueness without separate proof.
Navigator evaluation	The value or sign of a declared finite navigator objective.	Position relative to that deformed finite feasible set.	Objective, deformation, normalization, finite representation, and sign convention.	Precision and cutoff studies plus direct feasibility checks near sign changes.	A direct certificate contradicts the sign or a convention round trip changes it.	Objective definition, evaluation point, dated solver record, environment, and feasibility cross-check.	A physical distance to exact theory space.
Navigator search or optimum	A declared search returned a termination point satisfying its recorded termination tolerances; stationarity or local optimality only if independently checked.	A candidate region for more direct feasibility and reconstruction tests.	Search domain, starting points, optimizer, stopping rule, and navigator definition.	Multiple starts, boundary traces, objective variations, and direct certificate checks.	A lower point, missed component, unstable active set, or failed direct feasibility check.	Dated search trajectories, starts, gradients, termination data, environment, and nearby certificates.	A unique theory or globally closest exact CFT without proof.
Extremal-functional support	Zeros and support conditions of a finite extremal functional.	Candidate exchanged dimensions and active constraints at a represented boundary.	Extremality, functional basis, sector convention, and endpoint or tangency rule.	Zero positions, multiplicities, derivative conditions, and positivity persist across cutoffs.	Zero drift, lost tangency, failed positivity, or incompatible primal data.	Saved functional, dated zero-finding record, derivative tests, cutoff sequence, and software environment.	An exact operator spectrum.
Primal spectrum or OPE reconstruction	A scalar-nonnegative or matrix-PSD finite truncated solution with a verified residual in the original unscaled equations.	Approximate low-lying dimensions and invariant OPE data.	Selected support, degeneracy treatment, basis convention, and tail model.	Unscaled residuals and low invariant data remain stable across cutoffs and support changes.	A negative OPE-matrix eigenvalue, ill-conditioned instability, or nondecreasing controlled residual.	Dated primal-dual record, residual in the original basis, conditioning data, environment, and external checks.	A full exact crossing solution or proof of model identity.
Model compatibility	Named-model predictions and the underlying verified exclusions, bounds, or reconstructed observables agree within declared tolerances after applying an explicit convention map.	A named model is compatible with the tested data and conventions.	An explicit convention dictionary and a declared identification hypothesis.	Several observables, correlator systems, and independent methods remain compatible.	Conflicting symmetry, spectrum, OPE data, anomaly, or independently measured observable.	Current analytic, lattice, experimental, or independent bootstrap evidence with an explicit cutoff and convention map.	An existence or uniqueness theorem.

Download the structured taxonomy (JSON).

Global stop rule. Every conclusion stops at the declared finite represented problem. Failure to exclude, primal feasibility, a scan-delimited component, a navigator minimum, stable extremal data, or agreement with a candidate model does not establish realization, existence, uniqueness, completeness, or model identity. Any stronger conclusion requires separate evidence whose hypotheses and conventions are stated alongside it.

For a symmetry-resolved spinning result, the additional precondition is absolute: projector completeness, tensor-basis rank, conservation and Ward data, parity and statistics signs, Hermitian structure, block control, and PSD congruence must all pass before any taxonomy row applies.

Common pitfalls

A symmetric group channel does not automatically mean even spin. That shortcut applies to identical scalars in a one-dimensional tensor-structure space. For spinning operators, diagonalize the complete exchange action.

Conservation does not fix every conserved three-point coefficient. It gives a homogeneous kernel at separated points; Ward identities fix selected combinations after normalization and contact-term conventions are supplied.

Positive diagonals do not make a matrix PSD. Check the full Hermitian spectrum or an exact factorization. In a continuous $\Delta$ sector, enclose the smallest eigenvalue rather than sampling it.

A well-conditioned basis is not a new theory. An invertible congruence can improve numerical conditioning, but the unscaled crossing equation and exclusion must survive the round trip.

Exercises

Verify the $SU(2)$ projectors above, derive the displayed recoupling matrix $M$, and check $M^2=\mathbf1$. Then determine the even- or odd-spin selection rule for four identical adjoint scalars.

Solution

For example, contraction of the antisymmetric projector gives

$$(P_{\mathbf3})^{ab}{}_{ef} (P_{\mathbf3})^{ef}{}_{cd} =(P_{\mathbf3})^{ab}{}_{cd};$$

contracting it with either symmetric projector gives zero. The same calculation for all three channels yields $P_RP_S=\delta_{RS}P_R$ and

$$P_{\mathbf1}+P_{\mathbf3}+P_{\mathbf5} =\delta^a{}_c\delta^b{}_d.$$

Exchanging two identical scalars contributes $(-1)^\ell$ from the exchanged spin. Bose symmetry therefore keeps even $\ell$ in $\mathbf1$ and $\mathbf5$, and odd $\ell$ in $\mathbf3$. For currents, this conclusion cannot be reused until the tensor-structure exchange eigenvalue is included.

To obtain $M$, replace $(a,b;c,d)$ by $(a,d;c,b)$ in each projector and expand the result in the ordered basis $(P_{\mathbf1},P_{\mathbf5},P_{\mathbf3})$. The three coefficient rows are

$$\left(\frac13,\frac13,-\frac13\right), \quad \left(\frac53,\frac16,\frac56\right), \quad \left(-1,\frac12,\frac12\right).$$

Their matrix product gives $M^2=\mathbf1$, as required because applying the pair exchange twice restores the original ordering.

Let $\rho=\operatorname{diag}(1,4)$ and

$$S=\begin{pmatrix}1&-1\\0&1\end{pmatrix}.$$

Compute $\rho'=S\rho S^\dagger$. Why does its negative off-diagonal entry not violate positivity, and why would a singular $S$ be unacceptable as a basis change?

Solution

The transformed matrix is

$$\rho'=\begin{pmatrix}5&-4\\-4&4\end{pmatrix}.$$

Its trace is $9$ and determinant is $4$, so both eigenvalues are positive. More generally, $z^\dagger S\rho S^\dagger z=(S^\dagger z)^\dagger\rho(S^\dagger z)\ge0$. PSD is a quadratic-form statement, not elementwise positivity. If $S$ were singular, $S^{-1}$ and therefore the compensating block transformation would not exist; coefficient directions would be lost and the represented feasibility problem could change.

In the parity-even antisymmetric spin-one sector, external-current conservation gives

$$\boldsymbol\lambda_{\mathrm{generic}} =a(\Delta+3,1,1,\Delta-1)^T.$$

At the exchanged-current point $(\Delta,\ell)=(2,1)$, the kernel is instead spanned by $e_1=(1,0,0,0)^T$ and $v=(0,1,1,1)^T$. Explain the rank jump. Then set $c=3C_J/(4\pi)$, $t=\lambda_{JJJ}$ and expand the Ward vector $\boldsymbol\lambda_J=ce_1+t(-5,-1,-1,-1)^T$ into its full outer product.

Solution

For a generic nonconserved spin-one operator the conserved space is one-dimensional. At $\Delta=2$, conservation of the exchanged operator introduces an additional null relation among descendants, lowers the constraint rank, and makes the kernel two-dimensional. Substituting $\Delta=2$ into the generic vector recovers only $(5,1,1,1)^T$, one direction in the larger kernel; it cannot discover the new direction.

Let $w=(-5,-1,-1,-1)^T$. In a real tensor basis,

$$\boldsymbol\lambda_J\boldsymbol\lambda_J^T =c^2e_1e_1^T +ct\left(e_1w^T+we_1^T\right) +t^2ww^T.$$

The middle term is the interference between the Ward-fixed and unfixed pieces. Treating $ce_1$ as a fixed exchange while placing $tw$ in an unrelated PSD variable would omit this term and solve a different, artificially enlarged problem.

Continue the calculation

Use Extremal Functionals, Navigators, and Spectrum Reconstruction only after the full symmetry-and-spin system passes the exact checks above. Before treating any result as reproducible evidence, finish with Benchmark Reproduction and Data Provenance.

References

• Costa, Miguel S., João Penedones, David Poland, and Slava Rychkov. “Spinning Conformal Correlators.” Journal of High Energy Physics 2011, no. 11 (2011): 071. DOI. Open PDF
• Dymarsky, Anatoly, Filip Kos, Petr Kravchuk, David Poland, and David Simmons-Duffin. “The 3d Stress-Tensor Bootstrap.” Journal of High Energy Physics 2018, no. 2 (2018): 164. DOI. Open PDF
• Go, Mocho, and Yuji Tachikawa. “autoboot: A Generator of Bootstrap Equations with Global Symmetry.” Journal of High Energy Physics 2019, no. 6 (2019): 084. DOI. Open PDF
• He, Yin-Chen, Junchen Rong, Ning Su, and Alessandro Vichi. “Non-Abelian Currents Bootstrap.” Journal of High Energy Physics 2024, no. 3 (2024): 175. DOI. Open PDF
• Iliesiu, Luca, Filip Kos, David Poland, Silviu S. Pufu, David Simmons-Duffin, and Ran Yacoby. “Bootstrapping 3D Fermions with Global Symmetries.” Journal of High Energy Physics 2018, no. 1 (2018): 036. DOI. Open PDF
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI. Open PDF
• Rychkov, Slava, and Ning Su. “New Developments in the Numerical Conformal Bootstrap.” Reviews of Modern Physics 96 (2024): 045004. DOI. Open PDF