Protection, Q-Cohomology, Recombination, and Operator Mixing

Protection is a statement about a particular equivalence class of operators or observables, not a permanent label attached to a bare symbol. A short operator can mix with others carrying the same quantum numbers, become null, or join a long multiplet under recombination. A reliable construction first forms Q-cohomology, then quotients exact relations and null states, and finally resolves the two-point-function mixing matrix.

Required background. Use BPS bounds and recombination and the renormalized definition of local composite operators.

Helpful background. Index inversion shows why an alternating protected trace cannot reconstruct every short multiplet.

Q-cohomology and the selected sector

Choose an odd symmetry $Q$ satisfying $Q^2=0$ on the gauge-invariant sector of interest. The local cohomology is

$$H_Q=\frac{\ker Q}{\operatorname{im}Q}.$$

A representative $\mathcal O$ is closed if $[Q,\mathcal O\}=0$. Replacing it by

$$\mathcal O\longmapsto\mathcal O+[Q,\Lambda\}$$

does not change its class. In a correlator of separated Q-closed insertions, a Q-exact change vanishes if the vacuum and measure are Q-invariant and no boundary term or contact singularity is crossed Witten 1988, §§2–3.

If a translation generator along a coordinate $x$ is Q-exact,

$$\widehat P_x=\{Q,G_x\},$$

then translated representatives satisfy

$$\frac{d}{dx}[\mathcal O(x)]_Q=0.$$

If only the antiholomorphic translation is Q-exact, correlators are meromorphic rather than position independent. The protection applies only along these twisted directions.

Shortening and recombination

In a unitary superconformal theory, positivity of $\{Q,Q^\dagger\}$ gives shortening bounds. Saturation removes descendants and fixes the scaling dimension in terms of spins and R-charges. This protects the dimension while the shortening condition holds.

At a threshold, however, a long multiplet decomposes:

$$L\big|_{\Delta_*}=A_1\oplus A_2\oplus\cdots.$$

Moving away from the threshold reverses the process. The individual short multiplets disappear from protected cohomology or cancel in an index, although the long representation remains. Thus existence of one short representative is stable only when charges, selection rules, or an anomaly forbid its recombination partners Córdova, Dumitrescu, and Intriligator 2019, §§2–4.

Resolving operator mixing

Let $\{\mathcal O_i\}$ be renormalized Q-closed operators with identical conserved quantum numbers. Their separated two-point functions define a Hermitian form

$$G_{i\bar j}(x) =|x|^{2\Delta} \left\langle\mathcal O_i(x)\overline{\mathcal O}_j(0)\right\rangle.$$

Exact relations and Q-exact descendants span a subspace $N$ generated by vectors $R_A{}^i\mathcal O_i$. When the two-point function is genuinely cohomological, the Ward identity at separated points implies

$$R_A{}^iG_{i\bar j}=0$$

for every $A$ and $j$. Thus $N$ lies in the radical of $G$, and $G$ descends to a well-defined form on the quotient. If this condition fails, a contact term, boundary contribution, or non-Q-invariant prescription remains; one must repair that observable before treating its matrix as a metric on cohomology. To obtain a physical protected basis:

1. compute the relation subspace $N$ and verify that it lies in the radical of the separated two-point form;
2. quotient the Q-closed operator space by $N$;
3. use the induced form on the quotient and remove any remaining zero-norm classes;
4. diagonalize or orthonormalize the positive matrix;
5. express OPE coefficients and duality maps in that same basis.

If $G$ depends on exactly marginal couplings, parallel transport around the conformal manifold can rotate this basis by a Berry connection. Fixed dimensions do not imply a globally constant choice of operators.

A simple two-operator example makes the point. If an exact relation is

$$\mathcal O_1-\lambda\mathcal O_2=\{Q,\Lambda\},$$

then cohomology contains one class, not two. Provided $(1,-\lambda)$ lies in the radical of $G$, choose any vector independent of it, pass to its coset, and normalize that coset with the induced one-dimensional form. Adding a multiple of the exact vector changes the representative but neither the class nor its norm. An auxiliary positive metric may select a convenient complement, but it is not the physical cohomological two-point form.

Scope of the protected claim

Protected statement	What may still vary
Scaling dimension fixed by shortening	Basis, normalization, and allowed OPE coefficients
Nonzero Q-cohomology class	Representative and Q-exact contact terms
Chiral-ring product	Kähler metric and nonchiral correlators
Topological correlator	Embedding into the parent spacetime and unprotected deformations
Holomorphic coupling dependence	Kähler and nonholomorphic terms
Index contribution	Individual multiplet multiplicities within a recombination class
Anomaly coefficient	Nonanomalous contact terms and generic dynamics

The protected label should name one row, not imply all rows simultaneously.

Accidental symmetries and null states

An accidental infrared current can change the exact R-symmetry and hence the shortening condition. Operators that appeared protected under ultraviolet charges may be regraded or pair differently. The cohomology and mixing analysis must therefore use the infrared superconformal algebra.

Null states require equal care. A state can vanish because of an equation of motion, a gauge identity, a quantum ring relation, or zero norm at a special parameter value. These mechanisms have different behavior under deformations. Record the explicit relation rather than deleting the state silently.

Failure modes

Bare-operator protection. Renormalization mixes every operator with all operators of equal quantum numbers, including descendants and multi-traces. Protection belongs to the resolved class.

Ignoring contact terms. Q-exact insertions decouple at separated points; collisions can generate contact contributions that modify integrated observables.

Using an index as a multiplicity. An index is signed and invariant under recombination. A positive coefficient need not count one irreducible short multiplet.

Assuming global triviality. A protected vector bundle over parameter space can have nontrivial connection and monodromy even when its rank is constant.

Exercises

Suppose $\mathcal O_1-\lambda\mathcal O_2$ is Q-exact and the separated two-point form has precisely the corresponding one-dimensional radical. How many protected classes remain?

Solution

The relation spans one null direction in the two-dimensional space, so the quotient has dimension one. A normalized class is obtained by choosing any vector independent of $(1,-\lambda)$ and normalizing its coset with the induced two-point form.

References

• Córdova, C., T. T. Dumitrescu, and K. Intriligator. “Multiplets of Superconformal Symmetry in Diverse Dimensions.” Journal of High Energy Physics 2019, no. 3 (2019): 163. DOI; Open PDF.
• Witten, E. “Topological Quantum Field Theory.” Communications in Mathematical Physics 117 (1988): 353–386. DOI.