Elliptic Generalization of Cherednik-Macdonald-Mehta identities

Shamil Shakirov

arxiv: 2605.25908 · v1 · pith:AJ6EGZ77new · submitted 2026-05-25 · 🧮 math.QA · hep-th· math-ph· math.MP· math.RT

Elliptic Generalization of Cherednik-Macdonald-Mehta identities

Shamil Shakirov This is my paper

Pith reviewed 2026-06-29 19:18 UTC · model grok-4.3

classification 🧮 math.QA hep-thmath-phmath.MPmath.RT

keywords Cherednik-Macdonald-Mehta identitieselliptic generalizationShiraishi functionstheta functionsMacdonald polynomialselliptic matrix modelssuperintegrabilityintegrable systems

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The pith

Cherednik-Macdonald-Mehta identities extend to an elliptic setting by replacing Vandermonde products with theta functions and Macdonald polynomials with Shiraishi functions, at least to first order.

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

The paper proposes an elliptic generalization of the Cherednik-Macdonald-Mehta integral identities, in which trigonometric Vandermonde products are replaced by products of theta functions. Macdonald polynomials are lifted to Shiraishi functions, which arise as solutions to the non-stationary Ruijsenaars problem, as characters in equivariant K-theory of affine Laumon spaces, and as partition functions for surface defects in five-dimensional super Yang-Mills theory. The proposal is realized as an elliptic matrix model that possesses a superintegrability property. The resulting identities are proven to hold when the elliptic parameter is expanded to linear order.

Core claim

The central claim is that the Cherednik-Macdonald-Mehta identities admit an elliptic lift obtained by substituting theta-function products for the trigonometric Vandermonde factors and replacing Macdonald polynomials by Shiraishi functions; the lifted identities continue to hold at least through the first order in the elliptic deformation parameter.

What carries the argument

Shiraishi functions, defined as the distinguished elliptic lift of Macdonald polynomials that preserve the structure of the integral identities under the theta-function replacement.

If this is right

The elliptic matrix model inherits a superintegrability property from the original CMM identities.
Refined Chern-Simons theory and the associated refined knot invariants acquire an elliptic deformation controlled by the theta-function replacement.
The first-order proof supplies a perturbative bridge between the trigonometric DAHA setting and elliptic integrable systems.
Shiraishi functions furnish explicit integral representations that match known avatars in algebraic geometry and gauge theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the identities hold to all orders, they would supply a non-perturbative definition of elliptic refined knot invariants.
The linear-order verification suggests that a full proof might proceed by showing that both sides satisfy the same elliptic difference equations.
The construction may extend to higher-rank or multivariable cases by iterating the rank-one verification.

Load-bearing premise

Shiraishi functions are the correct elliptic analogues of Macdonald polynomials such that the theta-function substitution yields identities that remain valid beyond the first order in the elliptic parameter.

What would settle it

Direct expansion of both sides of a low-rank identity to second order in the elliptic parameter and comparison of the quadratic coefficients.

read the original abstract

Integral identities for Macdonald polynomials play an important role in modern mathematics and mathematical physics. Especially interesting are the Cherednik-Macdonald-Mehta (CMM) identities, with profound connections to Double Affine Hecke Algebras (DAHA) and representation theory of quantum groups. These identities are central in refined Chern-Simons theory, where they lead to refined S and T matrices and ultimately to refined knot invariants. We suggest an elliptic generalization of CMM identities, where trigonometric Vandermonde products are replaced by theta functions. At the same time Macdonald polynomials are promoted to Shiraishi functions -- distinguished elliptic functions with several interesting avatars, from the non-stationary Ruijsenaars problem in integrable systems, to equivariant K-theory characters of the affine Laumon space in algebraic geometry, to surface defect partition functions in 5d super Yang-Mills theory. From the perspective of matrix models, we present an elliptic matrix model with a superintegrability property. We prove the suggested identities to the first order in the elliptic parameter.

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

The paper suggests an elliptic CMM identity via theta functions and Shiraishi functions but only verifies it to first order in the elliptic parameter.

read the letter

The central point is that this paper replaces the trigonometric Vandermonde with a product of theta functions and Macdonald polynomials with Shiraishi functions, then checks that the resulting integral identity holds through linear order in the elliptic nome q. That first-order expansion is the only verification supplied.

The construction itself is new. The elliptic matrix model with superintegrability, the explicit link to Shiraishi functions across integrable systems, K-theory, and 5d SYM, and the direct perturbative check do not appear in the earlier CMM literature. The calculation at O(q) is concrete and can be inspected term by term.

The limitation is straightforward: nothing is shown for O(q^2) or higher. Shiraishi functions arise from the non-stationary Ruijsenaars operator, which does not automatically inherit the orthogonality or shift properties that make the original CMM identities work under DAHA action. A first-order cancellation therefore gives no guarantee that the identity survives exactly. The full non-perturbative statement remains a suggestion rather than an established result.

The paper is aimed at people already working on elliptic lifts of DAHA structures, refined Chern-Simons, or 5d gauge theory. A reader who wants a complete proof will be disappointed; someone looking for a concrete new ansatz to test or extend may find it useful.

It is worth sending to peer review. The idea connects active areas and the partial evidence is explicit enough that referees can assess whether the conjecture is plausible and worth pursuing.

Referee Report

2 major / 2 minor

Summary. The paper proposes an elliptic generalization of the Cherednik-Macdonald-Mehta (CMM) identities, replacing trigonometric Vandermonde products by theta functions and Macdonald polynomials by Shiraishi functions. It introduces an elliptic matrix model with a superintegrability property and proves the suggested identities to first order in the elliptic parameter q via direct expansion of both sides.

Significance. If the full non-perturbative identities hold, the work would furnish a natural elliptic extension of identities central to DAHA representation theory, refined Chern-Simons theory, and equivariant K-theory of affine Laumon spaces, with potential implications for 5d super Yang-Mills surface defects and knot invariants. The first-order verification supplies concrete supporting evidence and the matrix-model perspective is a useful organizing device.

major comments (2)

[Abstract] Abstract and the section presenting the proof: the central claim is the elliptic generalization of the CMM identities, yet only a perturbative verification to O(q) is supplied by direct expansion. No recursion, residue computation, or independent argument is given showing that the O(q^2) and higher coefficients continue to match, which is load-bearing for the suggested non-perturbative identities.
[Introduction / definition of Shiraishi functions] The paragraph introducing Shiraishi functions as the elliptic lift: the first-order cancellation does not automatically extend because Shiraishi functions are eigenfunctions of the non-stationary Ruijsenaars operator and are not known to obey the same orthogonality or shift relations that Macdonald polynomials satisfy under the DAHA action; an explicit check or structural argument is required.

minor comments (2)

The definition of the elliptic matrix model measure and the precise statement of its superintegrability property would benefit from an explicit formula or reference to the relevant equation.
A short table or explicit expansion of the first few coefficients on both sides of the identity would make the O(q) verification easier to inspect.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below, clarifying the scope of our results as a proposal supported by first-order verification.

read point-by-point responses

Referee: [Abstract] Abstract and the section presenting the proof: the central claim is the elliptic generalization of the CMM identities, yet only a perturbative verification to O(q) is supplied by direct expansion. No recursion, residue computation, or independent argument is given showing that the O(q^2) and higher coefficients continue to match, which is load-bearing for the suggested non-perturbative identities.

Authors: We agree with the observation. The manuscript proposes the elliptic generalization of the CMM identities as a conjecture and supplies an explicit verification to first order in q by direct expansion of both sides. No non-perturbative proof is claimed or provided. We will revise the abstract, introduction, and proof section to state explicitly that the identities are conjectural, with the O(q) check serving as supporting evidence rather than a full demonstration. revision: yes
Referee: [Introduction / definition of Shiraishi functions] The paragraph introducing Shiraishi functions as the elliptic lift: the first-order cancellation does not automatically extend because Shiraishi functions are eigenfunctions of the non-stationary Ruijsenaars operator and are not known to obey the same orthogonality or shift relations that Macdonald polynomials satisfy under the DAHA action; an explicit check or structural argument is required.

Authors: The referee correctly identifies that Shiraishi functions are eigenfunctions of the non-stationary Ruijsenaars operator and do not possess the same established DAHA orthogonality and shift relations as Macdonald polynomials. Our verification proceeds via direct term-by-term expansion to O(q), where the matching occurs due to the explicit series forms. We will add a remark in the introduction clarifying this distinction in properties and noting that the result is a computational first-order check. No additional structural argument is available. revision: partial

Circularity Check

0 steps flagged

No circularity; first-order verification is independent expansion

full rationale

The paper proposes replacing Vandermonde products by theta functions and Macdonald polynomials by Shiraishi functions, then states that the resulting identities are proved to first order in the elliptic parameter by direct expansion of both sides. No quoted step equates a claimed prediction to a fitted input, renames a known result, or reduces the central claim to a self-citation chain whose content is unverified. The first-order match is presented as an explicit perturbative check rather than a tautology or self-definition, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or newly invented entities; the construction rests on the prior existence of Shiraishi functions and the perturbative expansion in the elliptic parameter.

pith-pipeline@v0.9.1-grok · 5713 in / 1125 out tokens · 35178 ms · 2026-06-29T19:18:10.715862+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages · 4 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv
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work page arXiv 2020
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work page internal anchor Pith review Pith/arXiv arXiv 2015
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work page arXiv 2024
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work page arXiv 2019
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work page arXiv 2020
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S. Arthamonov and Sh. Shakirov,An Elliptic Generalization ofA 1 Spherical DAHA atK= 2, IMRN 19 (2024) 13046–13084, arxiv.org/abs/2306.00215

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A.Mironov,Seiberg-Witten theory and duality in integrable systems, arxiv.org/abs/hep-th/0011093

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

P.Koroteev and Sh.Shakirov,The quantum DELL system, Lett. Math. Phys. 110 (2020) 969-999, arxiv.org/abs/1906.10354

work page arXiv 2020

[3] [3]

I.Cherednik,Difference Macdonald-Mehta Conjecture, IMRN 10 (1997) 449–467, arxiv.org/abs/q- alg/9702022

work page arXiv 1997

[4] [4]

On Cherednik-Macdonald-Mehta identities

P. Etingof and A. Kirillov-Jr.,On Cherednik-Macdonald-Mehta identities, ERA of the AMS, 4 (1998) 43–47, arxiv.org/abs/q-alg/9712051

work page internal anchor Pith review Pith/arXiv arXiv 1998

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Knot Homology from Refined Chern-Simons Theory

M. Aganagic and Sh. Shakirov,Knot Homology from Refined Chern-Simons Theory, Com- mun.Math.Phys. 333 (2015) 1, 187-228, arxiv.org/abs/1105.5117

work page internal anchor Pith review Pith/arXiv arXiv 2015

[6] [6]

Mironov, A

A. Mironov, A. Morozov and A. Popolitov,CMM formula as superintegrability property of unitary model, Phys. Rev. D 110 (2024) 126026, arxiv.org/abs/2410.03175

work page arXiv 2024

[7] [7]

Shiraishi,Affine screening operators, affine Laumon spaces and conjectures concerning non- stationary Ruijsenaars functions, J.Integrable.Syst

J. Shiraishi,Affine screening operators, affine Laumon spaces and conjectures concerning non- stationary Ruijsenaars functions, J.Integrable.Syst. 4 (2019) xyz010, arXiv:1903.07495

work page arXiv 2019

[8] [8]

E.Langmann, M.Noumi and J.Shiraishi,Basic Properties of Non-Stationary Ruijsenaars Functions, SIGMA 16 (2020) 105, arxiv.org/abs/2006.07171 13

work page arXiv 2020

[9] [9]

Arthamonov and Sh

S. Arthamonov and Sh. Shakirov,An Elliptic Generalization ofA 1 Spherical DAHA atK= 2, IMRN 19 (2024) 13046–13084, arxiv.org/abs/2306.00215

work page arXiv 2024

[10] [10]

Mironov, A

A. Mironov, A. Morozov and Sh. Shakirov,5D AGT conjecture for circular quivers, Phys.Lett. B875 (2026) 140308, arxiv.org/abs/2512.23878

work page arXiv 2026

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Defects and Quantum Seiberg-Witten Geometry

M. Bullimore, H.-C. Kim and P. Koroteev,Defects and Quantum Seiberg-Witten Geometry, JHEP 05 (2015) 095, arxiv.org/abs/1412.6081

work page internal anchor Pith review Pith/arXiv arXiv 2015

[12] [12]

I. G. Macdonald,Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathe- matics (Vol. 157), Cambridge University Press, 2003 14

2003