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

Technical Summary: Superspace Invariants and 3-point Correlators in 3d N = 3, 4 SCFTs

Problem Statement
The paper addresses the systematic enumeration of 3-point correlation functions for spinning superfield operators in three-dimensional superconformal field theories (SCFTs) with $N=3$ and $N=4$ supersymmetry. While the constraints of conformal symmetry and supersymmetry largely fix 2- and 3-point functions up to scaling dimensions and Operator Product Expansion (OPE) coefficients, the kinematic problem for spinning operators remains non-trivial. Specifically, one must construct a complete and minimal basis of superconformally invariant structures to describe general 3-point correlators. Previous works had established this framework for $N=1$ and $N=2$ theories, but the $N=3$ and $N=4$ cases present unique challenges due to their non-abelian R-symmetry groups ($SO(3)$ and $SO(4)$, respectively). The authors aim to extend the existing program to these higher-supersymmetry cases, including the analysis of conserved currents (shortening conditions) and the identification of novel invariant structures arising from the non-abelian R-symmetry.

Methodology
The authors employ a combination of superspace techniques and the auxiliary polarization spinor formalism.

1. Superspace Formalism: Utilizing the Park-Osborn formalism for 3d $N$-extended SCFTs, the authors work in superspace coordinates $z^A = \{x^\mu, \theta^a_\alpha\}$, where $\theta^a_\alpha$ are Majorana spinors transforming under the R-symmetry group $SO(N)$. They utilize superinversion transformations to construct covariant building blocks, including the supersymmetric interval $\tilde{x}_{ij}$ and the $V$-matrix, which ensures R-symmetry covariance.
2. Polarization Spinors: To handle operators with spin $s$, the authors introduce auxiliary commuting spinors $\lambda_\alpha$ to contract with the symmetric, traceless spinor indices of the superfield operators, rendering them index-free ($O_s(x, \theta, \lambda)$).
3. Invariant Construction: The core of the methodology involves constructing superconformal invariants by contracting the polarization spinors $\lambda$ and the Grassmannian structures $\Theta^a_i$ with the covariant building blocks ($X_{ij\pm}$, $\bar{x}_{ij}$).
   • Bosonic and Fermionic Invariants: The paper utilizes known parity-even and parity-odd bosonic invariants ($P_i, Q_i, S_i$) and fermionic invariants ($\bar{R}_i, R', T'$) established in previous $N=1, 2$ studies.
   • Novel $\epsilon$-invariants: A significant methodological step is the construction of new invariants by contracting the free R-symmetry indices of the fermionic structures using the totally antisymmetric invariant tensors $\epsilon_{a_1 \dots a_N}$ of $SO(N)$. This is only possible for $N \ge 3$. The authors systematically enumerate these $\epsilon$-invariants for $N=3$ (using $\epsilon_{abc}$) and $N=4$ (using $\epsilon_{abcd}$), classifying them by their parity (even/odd) and homogeneity in the polarization spinors $\lambda$.
4. Constraints: The authors impose permutation symmetry (exchange of operators) and conservation constraints (shortening conditions $D_{\alpha_1} J^{\alpha_1 \dots \alpha_{2s}} = 0$) to reduce the general list of invariants to the specific structures allowed for conserved currents.

Key Contributions and Results

1. Complete Enumeration of Invariants:

   • The paper provides a complete and minimal list of superconformal invariants for $N=3$ and $N=4$ 3d SCFTs.
   • $N=3$: The authors identify 13 parity-even and 13 parity-odd $\epsilon$-invariants, categorized by their $\lambda$-homogeneity (e.g., $U_{100}, \tilde{U}_{300}$).
   • $N=4$: The authors identify 22 parity-even and 21 parity-odd $\epsilon$-invariants. Notably, this includes a unique $\lambda$-free parity-even $\epsilon$-invariant, $U' \equiv \epsilon \Omega_1 \Omega_1$, which has no analogue in lower $N$.
   • The authors derive non-linear relations among these invariants (e.g., products of two parity-odd invariants reduce to parity-even ones, and squares of certain $\epsilon$-invariants vanish or reduce to products of $N=2$ invariants), ensuring the basis is minimal.

2. Structure of 3-Point Correlators:

   • The authors apply these invariants to enumerate the tensor structures of 3-point correlators for various combinations of spinning operators ($O_s O_{s'} O_{s''}$).
   • General (Non-Conserved) Operators: The paper lists the allowed structures for general spinning operators, distinguishing between parity-even and parity-odd contributions.
   • Conserved Operators: By imposing conservation constraints, the authors determine the number of independent OPE coefficients for conserved currents.
     • $N=3$ Conserved Correlators: The results show that conserved 3-point functions are fixed by one parity-even and one parity-odd structure. This aligns with previous findings for the supercurrent in $N=3$ theories.
     • $N=4$ Conserved Correlators: A novel result is found for $N=4$: conserved correlators admit two parity-even structures and one parity-odd structure.
       • The first parity-even structure is present in all $N=4$ SCFTs.
       • The second parity-even structure arises exclusively from the $\epsilon$-invariants. The authors link this structure to the breaking of mirror symmetry. In $N=4$ theories, the R-symmetry $SO(4) \cong SU(2)_L \times SU(2)_R$ admits a $\mathbb{Z}_2$ mirror map exchanging $SU(2)_L$ and $SU(2)_R$. The $\epsilon$-invariant structure is the superspace lift of this asymmetry; it is non-zero only in theories that are not invariant under the mirror map. This explains why free hypermultiplets (which are mirror-symmetric) only saturate the first structure.

3. Specific Correlator Constraints:

   • The paper details specific constraints for various spin combinations (e.g., $\langle J_1 J_1 O_0 \rangle$, $\langle J_2 J_2 O_0 \rangle$).
   • For $N=3$, conserved correlators involving half-integer spins often vanish or are fixed to a single structure.
   • For $N=4$, the parity-odd sector often vanishes upon imposing conservation for certain spin combinations (consistent with spin triangle inequalities), while the parity-even sector retains the two independent structures mentioned above.

Significance and Claims
The paper claims to complete the systematic enumeration of spinning 3-point correlators in 3d SCFTs for $N=1$ through $N=4$. The primary significance lies in:

• Handling Non-Abelian R-Symmetry: Successfully incorporating the non-abelian R-symmetry of $N=3$ and $N=4$ theories, which introduces the novel $\epsilon$-invariants absent in $N \le 2$.
• Mirror Symmetry Breaking: Providing a precise kinematic characterization of the second parity-even structure in $N=4$ conserved correlators, identifying it as the signature of mirror symmetry breaking. This offers a structural explanation for the difference between mirror-symmetric and non-mirror-symmetric $N=4$ theories.
• Foundation for Bootstrap: The enumerated structures serve as necessary ingredients for constructing superconformal blocks, which are essential for the superconformal bootstrap program in 3d $N=3$ and $N=4$ theories.
• Extension of GPY Formalism: The work extends the Giombi-Prakash-Yin (GPY) program of enumerating invariants to higher supersymmetry, confirming that while the $N=1, 2$ results are robust, the $N \ge 3$ landscape is qualitatively different due to the antisymmetric R-symmetry tensors.

The authors note that superspace methods are not expected to be as useful for $N > 4$ in 3d, and they suggest future directions including the construction of generating functions for these correlators, the application of these results to the superconformal bootstrap, and potential extensions to 4d $N \ge 2$ SCFTs.