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arxiv: 2604.19885 · v2 · submitted 2026-04-21 · ✦ hep-th · math-ph· math.MP

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On non-relativistic integrable models and 4d SCFTs

Rotem Ben Zeev , Anirudh Deb , Hee-Cheol Kim , Shlomo S. Razamat

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Pith reviewed 2026-05-10 01:30 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords non-relativistic limitsintegrable modelsSchur index4d SCFTsclass S theorieselliptic Jack functionsDeligne-Cvitanovic seriesE-string theory
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The pith

Non-relativistic limits of integrable models correspond to generalized Schur indices of four-dimensional superconformal field theories.

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

The paper establishes that the generalized Schur index of N=2 SCFTs arises from the non-relativistic limit of the elliptic Ruijsenaars-Schneider model. It expresses these indices for class S theories using elliptic Jack functions, which serve as eigenfunctions in the integrable systems. This approach verifies mappings between theories in the Deligne-Cvitanović series by checking identities among eigenfunctions of associated Calogero-Moser models. A reader would care because it bridges two fields, allowing computation of SCFT indices through known solutions of integrable models and suggesting similar constructions for N=1 theories. The authors argue that for N=1 SCFTs the Schur index relates to the free fermionic limit of non-relativistic models, as illustrated with the Inozemtsev model for E-string compactifications.

Core claim

The generalized Schur index of N=2 SCFTs in four dimensions can be obtained as the non-relativistic limit of the elliptic Ruijsenaars-Schneider model. In particular, indices of class S theories are expressed in terms of elliptic Jack functions, reducing in the A1 case to eigenfunctions of the Lamé equation. This permits checking that generalized Schur indices of different Deligne-Cvitanović theories map onto each other, leading to identities on sums of eigenfunctions for different root systems. The non-relativistic limits of various integrable models, including the Inozemtsev model as the limit of the van Diejen model, produce generalized Schur-like limits for classes of N=1 SCFTs such asコンパ

What carries the argument

The non-relativistic limit of the elliptic Ruijsenaars-Schneider model, which produces the generalized Schur index expressed via elliptic Jack functions as eigenfunctions.

Load-bearing premise

The non-relativistic limit of the elliptic Ruijsenaars-Schneider model and its relatives exactly reproduces the generalized Schur index without additional corrections or assumptions.

What would settle it

A mismatch between the explicit eigenfunction expression for the generalized Schur index of a specific class S theory and its independently computed value would falsify the correspondence.

read the original abstract

We elaborate on the relation between the generalized Schur index of $N=2$ SCFTs in four dimensions and the non-relativistic limit of the elliptic Ruijsenaars-Schneider model. In particular we discuss explicitly how to express generalized Schur indices of theories of class $S$ in terms of elliptic Jack functions. For example, in the $A_1$ case the indices are given naturally in terms of eigenfunctions of the Lam\'{e} equation. We use the expression in terms of eigenfunctions to further check the recent observation that the generalized Schur indices of different theories in the Deligne-Cvitanovi\'{c} series can be mapped onto each other. This mapping implies non trivial identities on unrefined sums of eigenfunctions of non-relativistic elliptic Calogero-Moser models associated to different root systems. We claim then that the non-relativistic limits of various integrable models give rise naturally to generalized Schur-like limits of classes of $N=1$ SCFTs. As an example we discuss the relation of the Inozemtsev model, the non relativistic limit of the van Diejen model, and compactifications of the rank $Q$ E-string theory. We argue that in general the ``Schur index'' of $N=1$ $4d$ SCFTs can be understood as being related to the free fermionic limit of a non-relativistic integrable model.

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.

Referee Report

2 major / 3 minor

Summary. The paper elaborates the connection between generalized Schur indices of 4d N=2 SCFTs (especially class S theories) and the non-relativistic limit of the elliptic Ruijsenaars-Schneider model, expressing the indices via elliptic Jack functions (with the A1 case using Lamé eigenfunctions). It uses these expressions to verify mappings between Deligne-Cvitanović series theories, which imply identities among eigenfunction sums for different root systems. The paper then claims that non-relativistic limits of various integrable models naturally yield generalized Schur-like limits for classes of N=1 SCFTs, with the Schur index of general N=1 4d SCFTs understood as the free-fermionic limit of a non-relativistic integrable model, illustrated via the Inozemtsev model for rank-Q E-string compactifications.

Significance. If the central correspondences hold, the work supplies an integrable-systems viewpoint on SCFT indices that yields explicit expressions for class S theories and non-trivial identities among eigenfunctions of elliptic Calogero-Moser models associated to different root systems. The explicit A1 treatment and Deligne-Cvitanović checks constitute a concrete contribution that could enable new computations and cross-field insights. The proposed N=1 extension is suggestive but rests on analogy and a single example rather than a general derivation.

major comments (2)
  1. The section on N=1 SCFTs and the Inozemtsev model: the general assertion that the Schur index of arbitrary N=1 4d SCFTs is the free-fermionic limit of a non-relativistic integrable model is advanced without a general construction or derivation that applies beyond the rank-Q E-string example. The step from the N=2 class S results to this broad N=1 claim therefore remains an analogy rather than a demonstrated equivalence, which is load-bearing for the paper's title and final claim.
  2. The discussion of the A1 case and Deligne-Cvitanović mappings: while the abstract states that explicit checks are performed using the eigenfunction expressions, the manuscript does not display the full derivations, intermediate steps, or verification that the non-relativistic limit of the elliptic RS model precisely reproduces the generalized Schur index or that the observed mappings follow directly from the eigenfunctions without extra assumptions. This affects assessment of the claimed identities on unrefined sums of eigenfunctions for different root systems.
minor comments (3)
  1. The distinction between the ordinary Schur index and the generalized Schur index should be defined explicitly in the introduction, including the precise fugacity assignments and limits involved.
  2. Additional references to prior literature on Schur indices of class S theories and on the elliptic Ruijsenaars-Schneider and Inozemtsev models would help situate the new results.
  3. Equation labels and figure captions could be expanded to make the precise mapping between the integrable-model eigenfunctions and the index expressions clearer to readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript arXiv:2604.19885. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: The section on N=1 SCFTs and the Inozemtsev model: the general assertion that the Schur index of arbitrary N=1 4d SCFTs is the free-fermionic limit of a non-relativistic integrable model is advanced without a general construction or derivation that applies beyond the rank-Q E-string example. The step from the N=2 class S results to this broad N=1 claim therefore remains an analogy rather than a demonstrated equivalence, which is load-bearing for the paper's title and final claim.

    Authors: We agree that the N=1 discussion relies on the specific example of the Inozemtsev model for rank-Q E-string compactifications and presents the general statement as a natural extension rather than a fully derived equivalence. The manuscript argues this based on the pattern observed from the N=2 results and the integrable model limits. To address this, we will revise the relevant section, abstract, and conclusion to emphasize that this is a proposed generalization supported by the example, and we will add further discussion on the analogy to make the scope clearer. This will involve partial revisions to tone down the generality of the claim while preserving the suggestive nature of the connection. revision: partial

  2. Referee: The discussion of the A1 case and Deligne-Cvitanović mappings: while the abstract states that explicit checks are performed using the eigenfunction expressions, the manuscript does not display the full derivations, intermediate steps, or verification that the non-relativistic limit of the elliptic RS model precisely reproduces the generalized Schur index or that the observed mappings follow directly from the eigenfunctions without extra assumptions. This affects assessment of the claimed identities on unrefined sums of eigenfunctions for different root systems.

    Authors: For the A1 case, the manuscript expresses the indices using Lamé eigenfunctions and states that these are used to check the Deligne-Cvitanović mappings, leading to identities among eigenfunction sums. We acknowledge that the full step-by-step derivations and verifications are not explicitly shown in the main text. In the revised version, we will add detailed calculations, perhaps in an appendix, demonstrating how the non-relativistic limit of the elliptic RS model yields the generalized Schur index for A1, and explicitly verify the mappings for the Deligne-Cvitanović series using the eigenfunctions, ensuring no extra assumptions are needed. This will strengthen the presentation of these checks. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit identities and natural extensions between independent objects

full rationale

The paper derives explicit expressions for N=2 class S generalized Schur indices as elliptic Jack/Lamé eigenfunctions of the non-relativistic elliptic Ruijsenaars-Schneider model, then uses those expressions to verify Deligne-Cvitanović mappings via implied eigenfunction sum identities. These steps equate independently defined quantities (SCFT indices computed from superconformal algebra vs. integrable model eigenfunctions) without defining one in terms of the other. The N=1 extension is presented as a claim and argument by analogy to the N=2 case and specific examples (e.g., Inozemtsev model for E-string), without any fitted parameters, self-definitional loops, or load-bearing self-citations that reduce the central claim to its inputs. No step matches the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central correspondence between Schur indices and non-relativistic integrable limits is taken as a domain assumption rather than derived from first principles within the paper.

axioms (1)
  • domain assumption Generalized Schur index of N=2 SCFTs equals non-relativistic limit of elliptic Ruijsenaars-Schneider model
    This identification is the starting point elaborated throughout the abstract.

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discussion (0)

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Reference graph

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