Four-Dimensional Chiral-Algebra Sectors and Lost Information

Every unitary four-dimensional $N=2$ SCFT contains a meromorphic two-dimensional operator algebra obtained from a nilpotent supercharge on a plane. Its operators are Schur classes, its vacuum character is the Schur index, and its central terms are fixed by four-dimensional anomalies. The construction is many-to-one: it discards long multiplets, identifies cohomologous operators, and does not reconstruct the full four-dimensional OPE.

Required background. Use the general protected operator-algebra construction and the superconformal shortening conventions.

Helpful background. The superconformal-index page fixes trace and Casimir-factor conventions.

Schur operators on a plane

Choose a plane with complex coordinates $(z,\bar z)$ and the nilpotent combination

$$\mathbb Q=Q^1_-+\widetilde S^{2\dot -}$$

in a standard four-dimensional $N=2$ superconformal convention. The relevant local operators obey the Schur shortening conditions

$$E-(j_1+j_2)-2R=0, \qquad r+j_1-j_2=0,$$

where $R$ is the $SU(2)_R$ weight and $r$ the $U(1)_r$ charge. Their twisted translates use an $SU(2)_R$ polarization that depends on $\bar z$. The antiholomorphic translation is $\mathbb Q$-exact, so cohomology classes depend meromorphically on $z$ Beem et al. 2015, §§2–3.

The two-dimensional holomorphic weight is

$$h=E-R=\frac{E+j_1+j_2}{2},$$

where the equality uses the Schur condition. Four-dimensional operators with different data can map to the same $h$ and flavor representation, so mixing must be resolved before reduction.

Meromorphic OPE and vertex algebra

For cohomology classes $\chi[\mathcal O_i](z)$, the OPE is

$$\chi[\mathcal O_i](z) \chi[\mathcal O_j](0) \sim\sum_{k,n} \frac{C_{ij}^{k,n}\, \partial^n\chi[\mathcal O_k](0)} {z^{h_i+h_j-h_k-n}}.$$

Associativity follows from the parent four-dimensional OPE restricted to the plane. The resulting structure is a vertex operator algebra after quotienting null classes and fixing the vacuum and stress tensor.

The $SU(2)_R$ current in the four-dimensional stress-tensor multiplet maps to the two-dimensional stress tensor. In standard anomaly conventions,

$$c_{2d}=-12c_{4d}.$$

A four-dimensional flavor-current multiplet maps to an affine current with

$$k_{2d}=-\frac12k_{4d}.$$

The negative signs explain why the chiral algebra of a unitary interacting four-dimensional theory is generally nonunitary as a two-dimensional theory. This is not a contradiction: two-dimensional Hermitian conjugation is not inherited as an ordinary positive radial conjugation.

Free hypermultiplet benchmark

A free four-dimensional hypermultiplet has

$$c_{4d}=\frac1{12},$$

so its chiral algebra has $c_{2d}=-1$. It is generated by symplectic bosons $X$ and $Y$ of weight $1/2$ with OPE

$$X(z)Y(0)\sim\frac1z, \qquad Y(z)X(0)\sim-\frac1z.$$

The corresponding stress tensor has central charge $-1$, confirming $c_{2d}=-12c_{4d}$. This elementary example also shows explicitly how a unitary parent produces a nonunitary meromorphic algebra.

The Schur index as a vacuum character

With matched fugacity and normalization conventions,

$$\mathcal I_{\mathrm{Schur}}(q,\mathbf x) =\operatorname{Tr}_{\mathrm{VOA}} q^{L_0-c_{2d}/24}\mathbf x^{\mathbf F}$$

up to the conventional vacuum-energy prefactor. The equality identifies the protected state-counting function with the vacuum character. It does not say that the complete four-dimensional Hilbert space is the VOA module.

Other defects or modules can produce nonvacuum characters. Their relation to four-dimensional line or surface defects requires a separate construction; character matching alone does not identify the defect.

Information lost in reduction

The chiral algebra retains exact meromorphic OPE coefficients for Schur classes, flavor levels, $c_{4d}$ through $c_{2d}$, null relations, and the Schur index. It loses or compresses:

• all long multiplets and short multiplets without Schur operators;
• the four-dimensional $a$ anomaly unless supplied independently;
• transverse spacetime dependence and most tensor structures;
• OPE coefficients involving discarded operators;
• distinctions between operators that become cohomologous;
• the global line spectrum and many discrete choices;
• a unique four-dimensional completion of the same VOA.

Even $c_{4d}$ is recovered only after assuming the standard stress-tensor map and normalization. A VOA presented without its four-dimensional embedding does not intrinsically label a unique parent SCFT.

Consistency checks

1. Verify the Schur shortening conditions and exact infrared R-charges.
2. Quotient Q-exact and zero-norm operators before computing the OPE.
3. Check $c_{2d}=-12c_{4d}$ and every affine level.
4. Match the vacuum character to the Schur index in the same fugacity convention.
5. Test null relations against four-dimensional multiplet recombination.
6. State which four-dimensional data were supplied externally rather than reconstructed.

Exercises

Compute the two-dimensional central charge of a free hypermultiplet.

Solution

The four-dimensional anomaly is $c_{4d}=1/12$. Applying $c_{2d}=-12c_{4d}$ gives $c_{2d}=-1$, the central charge of one symplectic-boson pair.

References

• Beem, C., M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees. “Infinite Chiral Symmetry in Four Dimensions.” Communications in Mathematical Physics 336 (2015): 1359–1433. DOI; Open PDF.