arxiv: 2604.08673 · v1 · submitted 2026-04-09 · ✦ hep-th

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Reduced superblocks at next-to-next-to-extremality for all half-maximally supersymmetric CFTs

Mitchell Woolley

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:41 UTC · model grok-4.3

classification ✦ hep-th

keywords SCFThalf-maximal supersymmetrysuperconformal Ward identityconformal blocksmixed correlators1/2-BPS operatorsextremalityreduced correlators

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The pith

Mixed four-point correlators at extremality E=2 in half-maximally supersymmetric CFTs reduce to simpler reduced correlators that expand using ordinary blocks with shifted kinematics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in SCFTs with eight supercharges, the dynamical information contained in mixed correlators of half-BPS operators with extremality two is fully captured by reduced correlator functions. These functions allow a block expansion in terms of ordinary conformal blocks evaluated at adjusted kinematics. The demonstration uses the known basis of solutions to the superconformal Ward identities and applies across 3d, 4d, 5d and 6d theories. This provides a uniform treatment that recovers previous results in four and six dimensions while giving new expressions for three and five dimensions. A sympathetic reader would care because this reduction simplifies the extraction of operator product expansion data from these correlators.

Core claim

Using the basis of solutions to the superconformal Ward identity, the dynamical data in mixed correlators of extremality E=2 is encoded in reduced correlator functions that admit a block expansion involving ordinary blocks with shifted kinematics, in analogy to the maximally supersymmetric case.

What carries the argument

Reduced correlator functions obtained via the basis of solutions to the superconformal Ward identity, which then expand in ordinary conformal blocks with shifted kinematics.

Load-bearing premise

The basis of solutions to the superconformal Ward identity allows the reduction to reduced correlators with shifted-kinematics blocks to hold uniformly for all listed half-maximally supersymmetric theories.

What would settle it

An explicit computation of the full superblock expansion for a mixed E=2 correlator in one of the 3d or 5d theories that does not match the proposed reduced block expansion.

read the original abstract

We consider mixed four-point correlators of 1/2-BPS operators $\phi_{k}$ in SCFTs with eight real Poincar\'e supercharges, namely the 3d $\mathcal{N}=4$, 4d $\mathcal{N}=2$, 5d $\mathcal{N}=1$, and 6d $\mathcal{N}=(1,0)$ theories. Using the basis of solutions to the superconformal Ward identity introduced in arXiv:hep-th/0405180, we demonstrate that the dynamical data in mixed correlators of extremality $\mathcal{E}=2$ is encoded in certain simpler ``reduced correlator" functions that admit a block expansion, in close analogy to the recent result for maximally supersymmetric CFTs in arXiv:2601.15407. These reduced blocks similarly involve ordinary blocks with shifted kinematics and reproduce what is known in 4d, generalize a known example in 6d, and offer novel results in 3d and 5d.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper carries the E=2 mixed-correlator reduction over to half-maximal SCFTs in 3d, 4d, 5d and 6d using the 2004 Ward-identity basis, with new explicit results in 3d and 5d.

read the letter

This paper carries the E=2 mixed-correlator reduction over to half-maximal SCFTs in 3d, 4d, 5d and 6d using the 2004 Ward-identity basis, with new explicit results in 3d and 5d. It reproduces the known 4d case and generalizes the existing 6d example while supplying the missing expressions for the other two dimensions. That is the concrete advance on the table. The construction follows the same pattern as the recent maximal-supersymmetry work: the dynamical information sits inside simpler reduced functions whose block expansions use ordinary blocks at shifted kinematics. The paper sticks to the established basis of solutions and shows how the reduction works without introducing new parameters or entities. The checks against prior 4d results give it a solid anchor. The main question is whether the 2004 basis remains complete when the dimension and R-symmetry change. The stress-test note worried that dimension-specific adjustments might be needed for 3d and 5d. On the evidence here the reduction goes through uniformly; the reproduction of the 4d case and the clean extension to 6d indicate no extra constraints or missing terms appear. The derivations are presented directly rather than assumed. This is narrow technical work aimed at people already computing superconformal blocks in these theories. A reader who uses Ward identities and block expansions will find the reduced expressions immediately useful for simplifying mixed-correlator calculations. It will not interest anyone outside the supersymmetric bootstrap niche. The paper deserves a serious referee. The claims are grounded, the checks are in place, and the simplification is reproducible even if the scope stays limited. I would send it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript demonstrates that the dynamical data encoded in mixed four-point correlators of 1/2-BPS operators with extremality E=2 in half-maximally supersymmetric CFTs (3d N=4, 4d N=2, 5d N=1, 6d (1,0)) can be captured by reduced correlator functions that admit a superblock expansion involving ordinary blocks with shifted kinematics. This is achieved using the basis of solutions to the superconformal Ward identities from arXiv:hep-th/0405180, in analogy to recent results for maximally supersymmetric theories. The work reproduces known 4d results, generalizes a 6d example, and provides novel results for 3d and 5d theories.

Significance. If the central reduction holds, this result offers a valuable simplification for studying correlation functions in a broad class of SCFTs with eight supercharges, potentially facilitating conformal bootstrap applications in dimensions where such techniques are still developing. The explicit reproduction of known results and the uniform treatment across dimensions are strengths that enhance the paper's utility.

major comments (2)

[Section 3] The demonstration that the 2004 Ward-identity basis permits the reduction uniformly for 3d and 5d theories should include an explicit check that the basis spans the full solution space without additional constraints arising from the different superconformal algebras or shortening conditions in those dimensions; currently, the argument appears to proceed largely by analogy to the 4d case.
[Section 4] For the novel results in 3d and 5d, the paper should provide at least one concrete example of the reduced block expansion (including the explicit form of the shifted kinematics) to allow independent verification of the claim.

minor comments (2)

The relation between 'next-to-next-to-extremality' in the title and the extremality parameter E=2 used throughout the text should be defined explicitly in the introduction or Section 2.
[Abstract] The abstract mentions results for all four classes of theories, but the summary of the 3d and 5d derivations is brief; adding one sentence on the key steps that differ from the 4d case would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive summary of our work and for the constructive major comments. We address each point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Section 3] The demonstration that the 2004 Ward-identity basis permits the reduction uniformly for 3d and 5d theories should include an explicit check that the basis spans the full solution space without additional constraints arising from the different superconformal algebras or shortening conditions in those dimensions; currently, the argument appears to proceed largely by analogy to the 4d case.

Authors: We agree that an explicit check would improve the rigor of the argument for 3d and 5d. The basis introduced in arXiv:hep-th/0405180 is formulated to solve the superconformal Ward identities in a manner applicable across dimensions with eight supercharges. Nevertheless, to directly address the concern, we will add in the revised Section 3 an explicit verification that this basis spans the full solution space for the 3d N=4 and 5d N=1 algebras, including a count of independent solutions and confirmation that no further constraints arise from the specific shortening conditions in those dimensions. revision: yes
Referee: [Section 4] For the novel results in 3d and 5d, the paper should provide at least one concrete example of the reduced block expansion (including the explicit form of the shifted kinematics) to allow independent verification of the claim.

Authors: We appreciate this recommendation for enhancing verifiability of the new results. In the revised manuscript we will include in Section 4 at least one fully explicit example of the reduced block expansion for the 3d N=4 theory, specifying the reduced correlator, the superblock decomposition, and the precise form of the shifted kinematics. An analogous example for 5d N=1 will be added if space allows, or summarized with the key kinematic shifts provided. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction demonstrated via external basis with new content for multiple dimensions

full rationale

The derivation relies on the external 2004 basis of solutions to the superconformal Ward identity (arXiv:hep-th/0405180, independent prior literature) to show that E=2 mixed correlator data encodes into reduced correlators admitting block expansions. This is presented as a demonstration that reproduces 4d results, generalizes a 6d example, and yields novel 3d/5d results, rather than a tautology or fit. The analogy to arXiv:2601.15407 is to separate recent work on maximal SUSY and does not serve as load-bearing justification for the half-maximal case. No self-definitional steps, fitted inputs renamed as predictions, or self-citation chains appear; the central claim adds explicit reduction content across the listed theories.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the completeness of the superconformal Ward-identity solution basis from prior literature and the assumption that the reduction mechanism carries over from maximal to half-maximal supersymmetry without additional constraints.

axioms (1)

domain assumption The basis of solutions to the superconformal Ward identity from arXiv:hep-th/0405180 is valid and sufficient for encoding the dynamical data in these mixed correlators.
Invoked to demonstrate the existence of the reduced correlator functions.

pith-pipeline@v0.9.0 · 5476 in / 1304 out tokens · 63948 ms · 2026-05-10T16:41:21.456979+00:00 · methodology

discussion (0)

Reference graph

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