Protected Correlators and Operator Algebras

Protected operator algebras arise when translations in some directions are Q-exact. Correlators of Q-cohomology classes then become position independent along a line or meromorphic on a plane, and the parent OPE descends to an associative lower-dimensional product. The reduced algebra is exact but generally forgets tensor structures, long multiplets, and the unique lift of a class back to the parent QFT.

Required background. Use the mixing-resolved Q-cohomology construction and the product structure of chiral rings.

Helpful background. Conformal OPE data clarify which parent-theory structures are removed by the reduction.

Twisted translations

Let $Q$ be nilpotent and let $\mathcal O(0)$ represent a Q-cohomology class. Suppose a combination of an ordinary translation and an internal-symmetry generator is exact:

$$\widehat P=\{Q,G\}.$$

Define the twisted translate

$$\widehat{\mathcal O}(x) =e^{x\widehat P}\mathcal O(0)e^{-x\widehat P}.$$

For separated Q-closed insertions,

$$\frac{\partial}{\partial x_i} \left\langle \widehat{\mathcal O}_1(x_1)\cdots \widehat{\mathcal O}_n(x_n) \right\rangle =0,$$

because differentiating inserts a Q-exact operator. If only $\widehat P_{\bar z}$ is Q-exact, then

$$\partial_{\bar z_i}\langle\widehat{\mathcal O}_1\cdots\widehat{\mathcal O}_n\rangle=0$$

away from collisions, producing a meromorphic algebra. The topological version is the local-operator counterpart of the Q-exact stress-tensor construction in cohomological field theory Witten 1988, §§2–3.

The internal twist is essential. Translating the bare operator without rotating its R-symmetry polarization need not remain in the same cohomology class.

The descended OPE

At separated points the parent OPE can be projected to cohomology:

$$[\widehat{\mathcal O}_i(x)]_Q [\widehat{\mathcal O}_j(0)]_Q =\sum_k C_{ij}{}^k(x) [\widehat{\mathcal O}_k(0)]_Q.$$

In a one-dimensional topological sector, $C_{ij}{}^k$ is constant within each ordering chamber. Crossing $x=0$ can change the ordering, so the product may be noncommutative. On a holomorphic plane, the coefficients are Laurent series in $z$ and define a vertex algebra.

Associativity follows from the parent OPE and Q-cohomology, provided collision contact terms are included consistently:

$$(\mathcal O_i\star\mathcal O_j)\star\mathcal O_k =\mathcal O_i\star(\mathcal O_j\star\mathcal O_k).$$

This is the protected remnant of crossing symmetry. It is a powerful constraint, but it involves only operators and tensor structures surviving the quotient.

One-dimensional deformation quantization

Three-dimensional $N=4$ theories contain a Higgs-branch topological sector on a line. Twisted Higgs-branch operators form an ordered product whose semiclassical expansion has the form

$$f\star g =fg+\frac{\hbar}{2}\{f,g\}_{\mathrm{PB}} +O(\hbar^2).$$

The commutator therefore recovers the holomorphic symplectic Poisson bracket:

$$[f,g]_\star =\hbar\{f,g\}_{\mathrm{PB}}+O(\hbar^2).$$

Here $\hbar$ is set by the sphere radius and normalization of operators in the localized construction. The exact star product is a quantization of the Higgs-branch coordinate ring, while the full three-dimensional theory contains many operators outside it Dedushenko, Pufu, and Yacoby 2018, §§3–5.

Normalization and mixing

Choose a basis using protected two-point functions before quoting structure constants. If

$$\langle\mathcal O_i(x)\mathcal O_j(0)\rangle=G_{ij},$$

then a basis change $\mathcal O_i\mapsto M_i{}^j\mathcal O_j$ changes both $G$ and $C_{ij}{}^k$. Basis-independent statements include associativity, representation content, central terms after current normalization, and isomorphism class of the algebra.

Contact terms can shift products of integrated operators or the coincident prescription. A regulator preserving Q and the relevant twisted translations is part of the algebra definition. Null classes must be quotiented; otherwise the two-point form is degenerate and OPE coefficients are not uniquely defined.

Lost information and lift ambiguity

The reduction can discard:

• every Q-exact operator and all long multiplets without representatives;
• transverse position dependence and parent tensor structures;
• OPE coefficients that multiply classes vanishing in cohomology;
• positivity, because the reduced conjugation need not be the parent Hermitian conjugation;
• a unique parent operator, since several operators can define one class;
• global data not acting on the protected sector.

Consequently, isomorphic protected algebras do not by themselves imply isomorphic parent QFTs. A lift requires additional spectrum, correlation, anomaly, and global information.

Exercises

Show that a Q-exact translation makes a separated correlator position independent.

Solution

Differentiation brings down $[\widehat P,\widehat{\mathcal O}_i]=[\{Q,G\},\widehat{\mathcal O}_i]$. Using Q-closure and the graded Jacobi identity, this is Q-exact. Its expectation value with other separated Q-closed insertions vanishes when the vacuum and measure are Q-invariant and there is no boundary or contact contribution.

References

• Dedushenko, M., S. S. Pufu, and R. Yacoby. “A One-Dimensional Theory for Higgs Branch Operators.” Journal of High Energy Physics 2018, no. 3 (2018): 138. DOI; Open PDF.
• Witten, E. “Topological Quantum Field Theory.” Communications in Mathematical Physics 117 (1988): 353–386. DOI.