Superspace invariants and 3-point correlators in 3d $\mathcal{N}=3,4$ SCFTs

Abstract: We use the auxiliary polarization spinor formalism together with superspace techniques to construct a complete and minimal list of 3-point invariant structures in three-dimensional superconformal field theories (SCFTs) with $\mathcal{N}=3$ and $\mathcal{N}=4$ superconformal symmetry. The existence of non-abelian $R$-symmetry for $\mathcal{N}=3,4$ gives rise to novel invariant structures built from the antisymmetric invariant tensor. These invariants are used to enumerate the structural form of spinning 3-point correlators of general as well as conserved spinning superfield operators in 3d SCFTs. For conserved operators, we find that the $\mathcal{N}=3$ correlators are fixed upto one parity-even and one parity-odd structure, while $\mathcal{N}=4$ conserved correlators admit two parity-even structures and one parity-odd structure, with the second parity-even structure associated with mirror symmetry breaking.

• The paper develops a systematic framework to enumerate three-point superconformal invariants in 3d N=3,4 SCFTs using auxiliary spinor formalism and superspace techniques.
• It classifies bosonic, fermionic, and epsilon-type invariants with explicit consideration of parity and permutation symmetries, imposing conservation constraints on spinning operator correlators.
• The results offer crucial insights for superconformal bootstrap applications and mirror symmetry analysis, paving the way for higher-dimensional extensions.

Superspace Invariants and Three-Point Correlators in 3d $\mathcal{N}=3,4$ SCFTs

Overview

The paper develops a systematic framework for enumerating and constructing all three-point superconformal invariants in three-dimensional $\mathcal{N}=3$ and $\mathcal{N}=4$ superconformal field theories (SCFTs), focusing on their implications for spinning operator correlators. Utilizing the auxiliary polarization spinor formalism in conjunction with the superspace approach, the authors deliver an explicit and minimal set of building blocks for three-point structures, including both bosonic and fermionic (Grassmann) invariants. An important contribution is the exhaustive classification of new invariant structures unique to the non-abelian $R$-symmetry present for $N=3,4$, especially those arising from the antisymmetric $SO(N)$ epsilon tensor. The consequences for the structure of three-point functions of both unconserved and conserved operators are analyzed thoroughly, with explicit parameter count and permutation symmetry constraints. The parity structure and the correlation with mirror symmetry in the $N=4$ case are established.

Superspace Formalism and Invariant Construction

The superspace formalism for $N$-extended SCFTs, following the Park-Osborn procedure, is applied and extended. The basic superspace $\{ x^\mu, \theta^{a\alpha} \}$ incorporates $N$ copies of Majorana supercharges (with $\mathcal{N}=3$0), transforming under the internal $\mathcal{N}=3$1 $\mathcal{N}=3$2-symmetry. Superinversion transformations and the $\mathcal{N}=3$3-matrix (i.e., $\mathcal{N}=3$4-symmetry dressing) are central, as is the polarization spinor $\mathcal{N}=3$5 technique to index-free encoding of symmetric traceless Lorentz representations.

All $\mathcal{N}=3$6 three-point functions are expressible as polynomials in a finite set of basic invariants:

• Bosonic invariants are generalizations of known conformal structures: $\mathcal{N}=3$7, $\mathcal{N}=3$8 (parity-even), and $\mathcal{N}=3$9 (parity-odd).
• Fermionic invariants originate from contractions of covariant Grassmannian building blocks (arising from the $\mathcal{N}=4$0 structures) with invariant $\mathcal{N}=4$1 tensors: for $\mathcal{N}=4$2, only contractions with Kronecker delta are possible; for $\mathcal{N}=4$3, the full antisymmetry allows new epsilon-type structures.
• $\mathcal{N}=4$4-invariants arise by saturating free $\mathcal{N}=4$5-symmetry indices with the $\mathcal{N}=4$6 tensor. Their explicit enumeration is performed at each order in auxiliary spinor degree and parity, and all nontrivial linear relations are worked out.

Nilpotency and permutation symmetry induce non-linear relations among the invariants, which are mathematically catalogued. Point-permutation (e.g., $\mathcal{N}=4$7) symmetries or antisymmetries are tracked explicitly at the invariant level, crucial for allowed three-point structure selection.

Invariants in $\mathcal{N}=4$8 and $\mathcal{N}=4$9 SCFTs

For $R$0, the $R$1 $R$2-symmetry leads to invariants built from the rank-3 $R$3. At each auxiliary spinor degree, parity-even and parity-odd invariants are classified and their linear dependencies addressed. The total count is 13 parity-even and 13 parity-odd $R$4-invariants.

For $R$5, $R$6 decomposes as $R$7, and the rank-4 $R$8 expands the space of invariants. The count increases substantially: 22 parity-even and 21 parity-odd structures, with several nontrivial relations among them due to overcounting from nilpotency and symmetry.

Importantly, $R$9-invariants exhibit precise transformation properties under point permutations, sometimes symmetric, sometimes antisymmetric, and sometimes invariant. Many vanish by antisymmetry or due to explicit contraction patterns.

Three-Point Correlators: Structure and Conservation Constraints

The three-point function $N=3,4$0 is an explicit sum of monomials in the enumerated invariants, with polynomial degree set by the operator spins. Parity symmetry dictates that each parity-odd invariant enters at most linearly. For fully unconstrained operators, all invariant monomials consistent with spin and permutation symmetry can occur.

Operator conservation—i.e., supersymmetric shortening—enforces linear relations among OPE coefficients by annihilating the three-point structure with supercovariant derivatives. For $N=3,4$1 and $N=3,4$2, the outcome is:

• For $N=3,4$3: The three-point function of conserved higher-spin currents is completely characterized by one parity-even and one parity-odd structure. The parity-odd structures arise from the same formalism as in non-supersymmetric Chern-Simons-matter theories, but nilpotency, supersymmetry, and triangle inequalities restrict their appearance.
• For $N=3,4$4: Conservation allows two independent parity-even structures and one parity-odd structure—the second parity-even structure is entirely due to $N=3,4$5-invariants and signals possible $N=3,4$6 mirror symmetry breaking.

Non-conserved operators lead to more general structures, but constrained by permutation symmetry and fermionic nilpotency. For instance, all half-integer spin correlators vanish for $N=3,4$7 due to the even fermion number required.

Parity, Mirror Symmetry, and Physical Implications

A direct consequence revealed is the connection between nontrivial parity-even $N=3,4$8-invariants and mirror symmetry breaking in $N=3,4$9 SCFTs. While one parity-even structure is universal, the other occurs only in mirror-nonsymmetric (i.e., $SO(N)$0) theories. This agrees with previous superspace and component computations for supercurrent three-point functions.

Strong statements are made for the enumeration of all possible three-point structures for arbitrary (including non-conserved) superfields, going beyond earlier works which were restricted to supercurrents and flavor current multiplets.

Numerical Results and Explicit Solutions

For each operator configuration (specified by spins and operator statistics), the allowed superconformal invariant structures are listed. Conservation conditions are solved explicitly, and all undetermined OPE coefficients are parameterized. Exceptional cases where all correlators vanish due to nilpotency or symmetry are categorized.

For example, in $SO(N)$1 the supercurrent three-point function is always fixed to two structures, one parity-even and one parity-odd, whereas in $SO(N)$2 the existence of two parity-even structures (but only one in mirror-symmetric theories) is made fully explicit at the level of invariant monomial counting.

Future Directions

Several directions are suggested:

• Construction of generating functions for all spinning three-point functions, analogous to the non-supersymmetric higher-spin/free theory case, which is obstructed for large $SO(N)$3 due to the rich $SO(N)$4-invariant structure.
• Application to superconformal bootstrap, since three-point structures feed directly into the construction of crossing-symmetric superconformal blocks.
• Potential for supersymmetrization of the Maldacena-Zhiboedov theorem relating higher-spin current existence to free field theory in three dimensions, now that all superspace structures are explicitly available.
• Generalization to four-dimensional extended SCFTs, especially $SO(N)$5 super Yang-Mills, which should possess analogous $SO(N)$6-type structures in their superconformal correlators.

Conclusion

The paper completes the explicit enumeration of three-point superconformal invariants in 3d $SO(N)$7 SCFTs, showing the central role of $SO(N)$8-symmetry and $SO(N)$9-tensors in the structure of allowed correlators. For all spins and both conserved and unconserved operators, the necessary invariants and their permutation properties are specified, and conservation constraints are solved completely. The strong connection to mirror symmetry and the parity structure of spinor correlators provides both theoretical insight and practical input for the analytic and numerical superconformal bootstrap, and the framework is poised to facilitate further study in higher dimension and higher supersymmetry.

Reference: "Superspace invariants and 3-point correlators in 3d $N=4$0 SCFTs" (2607.10133)