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Novel R-symmetry invariants fix the form of spinning three-point functions in 3d N=3 and N=4 superconformal theories, with conservation leaving two parity-even structures only for N=4.

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2026-07-14 14:02 UTC pith:UJFRB23K

load-bearing objection Solid completion of the 3d SCFT spinning 3-pt program for N=3,4; the new epsilon-invariants are real and the conserved counts match known supercurrent results.

arxiv 2607.10133 v1 pith:UJFRB23K submitted 2026-07-11 hep-th

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

Aditya Jain , Amin A. Nizami This is my paper
classification hep-th
keywords 3d SCFTssuperspace invariantsspinning correlatorsR-symmetryepsilon-invariantsconservation constraintsmirror symmetrypolarization spinors
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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Three-dimensional superconformal field theories with three or four supersymmetries possess non-abelian R-symmetry, which admits an antisymmetric tensor that can be used to build new three-point invariants. This paper systematically constructs a complete, minimal basis of all such superconformal invariants (bosonic, fermionic, and the new epsilon-type structures) by combining superspace methods with auxiliary polarization spinors that encode operator spin. The resulting invariants are then used to write the most general spinning three-point correlators of both ordinary and conserved superfield operators. After the shortening conditions of conservation are imposed, N=3 correlators are fixed by a single parity-even and a single parity-odd structure, while N=4 correlators admit two independent parity-even structures and one parity-odd structure; the extra even structure is generated by the epsilon-invariants and is present only when mirror symmetry is broken. The enumeration therefore supplies the full kinematic scaffolding needed for higher-spin correlators, bootstrap blocks, and mirror-symmetry diagnostics in these theories.

Core claim

The complete and minimal set of three-point superconformal invariants for 3d N=3 and N=4 SCFTs includes novel epsilon-invariants built from the antisymmetric R-symmetry tensor. When these invariants are used to construct correlators of conserved spinning superfields, conservation fixes N=3 correlators to one parity-even plus one parity-odd structure, while N=4 correlators retain two parity-even structures (the second arising solely from the epsilon-invariants and signalling mirror-symmetry breaking) plus one parity-odd structure.

What carries the argument

Epsilon-invariants: superconformal scalars obtained by contracting three (N=3) or four (N=4) copies of the same Grassmannian covariant Theta_i with the totally antisymmetric SO(N) tensor, then saturating the free spinor indices with auxiliary polarization spinors lambda. These structures, together with the ordinary bosonic and fermionic invariants, form the complete basis that enumerates every allowed three-point tensor structure.

Load-bearing premise

The claim rests on the assertion that the exhaustive listing of candidate epsilon-structures at each spin order, reduced by the listed linear relations among Grassmannian building blocks, yields a complete and independent basis; any overlooked algebraic identity would change the final count of free coefficients.

What would settle it

An explicit free-field or interacting computation of a conserved three-point function (for example the N=4 supercurrent correlator) that produces a different number of independent tensor structures, or that shows a linear dependence among the claimed independent epsilon-invariants, would falsify the enumeration.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper constructs a complete and minimal basis of superconformal 3-point invariants for 3d N=3 and N=4 SCFTs by combining the Park–Osborn superspace formalism with auxiliary polarization spinors. Non-abelian R-symmetry produces novel ε-invariants (built from the SO(N) antisymmetric tensor) that are absent for N≤2; these are enumerated at each λ-order, reduced by linear relations (Appendix B), and subjected to product and permutation identities. The resulting invariants are used to write the most general spinning 3-point correlators of both non-conserved and conserved superfield operators. After imposing conservation, N=3 correlators are fixed by one parity-even and one parity-odd structure, while N=4 correlators admit two parity-even structures and one parity-odd structure, the second even structure arising from the ε-invariants and associated with mirror-symmetry breaking. The lowest-spin results reproduce the known supercurrent correlators of Buchbinder–Kuzenko–Samsonov.

Significance. The work completes the systematic enumeration of spinning 3-point structures for all 3d N-extended SCFTs with N≤4, extending the authors’ earlier N=1,2 results and the non-supersymmetric polarization-spinor analysis of Giombi–Prakash–Yin. The novel ε-invariants supply a concrete superspace realization of the second parity-even structure previously identified for N=4 supercurrents and link it to L–R asymmetry. The tabulated free-coefficient counts after conservation, together with the explicit product reductions that enforce Grassmann nilpotency, provide ready-to-use kinematic input for superconformal bootstrap and for possible supersymmetric extensions of Maldacena–Zhiboedov theorems. The construction is fully algebraic and reproduces independent literature results for the lowest spins, giving high confidence that the higher-spin counts are reliable.

minor comments (4)
  1. In §3.2–3.3 the ε-invariants are written with R-symmetry indices suppressed; a single explicit example with free indices restored (e.g., for U100) would make the contraction with εabc/εabcd completely unambiguous for readers reconstructing the basis.
  2. Equation (3.16) and the subsequent N=3,4 replacements (3.17)–(3.18) are important for the product algebra; a short remark that the same pattern holds for the other bilinear covariants (π,σ,ω) would avoid the need to consult the earlier N=2 paper.
  3. In several places in §4 the kinematic prefactor of (4.1) is written with mixed |x| and |¯x| notation (e.g., after (4.7)); consistent use of ¯x throughout would eliminate a minor source of confusion.
  4. The discussion of future directions (§5) mentions generating functions and superconformal blocks; a one-sentence pointer to the precise spin ranges already covered in §4 would help readers who wish to extract blocks immediately.

Circularity Check

0 steps flagged

No significant circularity: invariants and free-coefficient counts are derived from superinversion covariance and linear algebra on Grassmann building blocks, not forced by definition or self-citation.

full rationale

The paper constructs 3-point superconformal invariants from first principles: superspace coordinates, superinversion transformations of the 2- and 3-point blocks Xij± and Θa_i (Sec. 2), auxiliary polarization spinors λ, and contraction of free R-symmetry indices with δab and the SO(N) antisymmetric tensors ε (Sec. 3). The novel ε-invariants for N=3,4 are enumerated order-by-order in λ-homogeneity; Appendix B lists all candidate structures and the linear relations that reduce them to one independent parity-even and one parity-odd structure per order. Product relations (3.19)–(3.24) follow from Grassmann nilpotency and the decomposition of two ε-tensors into δ’s. Correlator structures (Sec. 4) are then monomials in this basis, truncated by permutation symmetry and by the conservation condition Dα J=0 (which produces linear relations among OPE coefficients). Self-citations to the authors’ N=1,2 papers supply only the bosonic/fermionic building blocks already derived there by the same method; they are not load-bearing for the new ε-sector or for the final free-coefficient counts (one even+one odd for conserved N=3; two even+one odd for N=4). Consistency with the independent supercurrent results of Buchbinder–Kuzenko–Samsonov is a cross-check, not an input. No parameter is fitted, no uniqueness theorem is imported from the authors’ prior work to forbid alternatives, and no ansatz is smuggled in. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

Purely kinematic superconformal construction; no free parameters are fitted. All background is standard superspace geometry or previously published N=1,2 results. The only new objects are the epsilon-invariants themselves, which are derived rather than postulated.

axioms (4)
  • domain assumption 3d N-extended superconformal algebra and the Park–Osborn superspace transformation rules under superinversion (including the V-matrix).
    Taken as given from Park (2000) and Osborn; used throughout §2 to define covariant building blocks.
  • domain assumption Auxiliary commuting polarization spinors λ encode traceless symmetric spin-s operators and transform as Lorentz scalars of weight −s.
    Standard Giombi–Prakash–Yin formalism, already used for N=1,2; invoked from §3 onward.
  • standard math Products of two parity-odd invariants reduce to sums of parity-even invariants; epsilon-invariants appear at most linearly.
    Follows from nilpotency of Grassmann variables and the decomposition of ε×ε into δ’s; stated in §3.4 and used to truncate monomials.
  • domain assumption Conservation of a superfield current implies the differential constraint ∂/∂λ^α D_α J_s = 0 and fixes au = 1.
    Standard shortening condition; applied in §4 to obtain linear relations among coefficients.
invented entities (1)
  • epsilon-invariants U_ijk and Ũ_ijk (and U' for N=4) independent evidence
    purpose: Provide the additional superconformal invariants that exist only when the R-symmetry admits a totally antisymmetric tensor; they generate the second parity-even structure in conserved N=4 correlators.
    Constructed explicitly by contracting three (N=3) or four (N=4) copies of the same Θ_i with ε and closing spinor indices with λ’s; independent evidence is internal consistency with known supercurrent results and the mirror-symmetry interpretation.

pith-pipeline@v1.1.0-grok45 · 31173 in / 2767 out tokens · 24905 ms · 2026-07-14T14:02:06.765214+00:00 · methodology

0 comments
read the original 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.

discussion (0)

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Reference graph

Works this paper leans on

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