Peter Koroteev · Mathematics

The Diamond

Integrable systems, opers, and their algebras. Select a vertex to explore the results and the papers behind them.

Help complete the Diamond. Experts are invited to contribute references, constructions, and corrections, especially for entries marked “?” in the picture. The adjacent descriptions give known results, candidate constructions, and questions for contributors. Send a contribution with the relevant vertex or arrow and a paper link or explanation.

November 2024 edition2024 handout (PDF)Companion reviewReference notes

Magnetic-frame correspondencesArrows = degeneration limits
The Diamond: nine integrable systems November 2024 Diamond, including DDCA at vertex 9. Select a numbered vertex for its opers, Baxter relation, spin system, algebras, and paper links. The last line of each oval names its large-rank algebra. Arrows lead to the cited degeneration table. DELL → dual eRS DELL → eRS dual eRS → dual eCM dual eRS → tRS eRS → tRS eRS → eCM dual eCM → rRS tRS → rRS tRS → tCM eCM → tCM rRS → rCM tCM → rCM 1 DELL Opers: ? Spin: ? ? 2 dual eRS e(G, q)-opers XYZ · Q𝔔 eDIM! 3 eRS Opers: ? Spin: ? eDIM 4 dual eCM eG-opers eGaudin ? 5 tRS (G, q)-opers XXZ · QQ DIM 6 eCM Opers: ? Spin: ? ? 7 rRS tG-opers tGaudin · qqᵗ ? 8 tCM (G, ε)-opers XXX · Qq Affine Yangian 9 rCM rG-opers rGaudin · qq DDCA
Ordinary Laumon diagonalAffine Laumon diagonal? = a gap to fill
Numbering and colour groups follow the November 2024 handout. Each oval lists the many-body system, opers, magnetic-frame spin/Baxter data, and its large-rank algebra on the last line. Select a vertex for complete entries, eigenfunctions, related connections, and sources. ℰ, ℂ×, and ℂ denote elliptic, multiplicative, and additive coordinate types; the phase-space pair is shorthand. On a small screen, scroll the diagram horizontally or use the vertex selector below.
Vertex 1 · DELL

Double elliptic system

ℰp × ℰx

Elliptic dependence on both momentum and position. This is the top vertex of the hierarchy.

Integrable System

Original construction. Braden–Marshakov–Mironov–Morozov (1999/2000), §5, construct the classical two-particle double-elliptic Hamiltonian. Koroteev–Shakirov (2019/2020) propose the quantum many-body operators. [102] [31]

Integrability. After removing the centre of mass, the original classical system has one degree of freedom. For the many-body theta-function approach, Aminov–Braden–Mironov–Morozov–Zotov prove the identities underlying three-particle Poisson commutativity (§4). [102] [104]

Quantum status. Commutativity of the general Koroteev–Shakirov family is Conjecture 1.1 in its construction paper, with perturbative checks. A proof of that full statement is unknown in the sources collected here; contributions are welcome. [31]

Opers
unknown
Baxter relation
unknown
Spin system
unknown
Finite-rank algebra
unknown
Large-rank algebra
unknown

Quantum Hamiltonians and the spectral problem

Conjecture in the review

Koroteev–Shakirov introduce quantum DELL Hamiltonians and an elliptic-genus construction of candidate eigenfunctions. The review keeps the general spectral statement as Conjecture 3.8 and reports perturbative evidence for commutativity.

[31]

Lax and large-N constructions

Construction

Grekov–Zotov develop a characteristic determinant, a Manakov triple, and a large-N operator construction. The double-elliptic DAHA entry is unknown.

[20] [21]

Classical origins and limits

Background

The classical constructions arise from generalized Mukai–Sklyanin algebras and compactified six-dimensional gauge theories. Double Inozemtsev limits and the status review give further context.

[1] [2] [27] [19] [39]

DELL spectral surface and categorical algebras

Candidate

Braden–Hollowood’s four-torus is an abelian surface containing the spectral curve. Rains’s elliptic DAHA sheaf algebras and noncommutative surface constructions provide candidate frameworks. Identifying their parameters, surface geometry, and commuting operators with DELL is an open problem; the rationally ruled geometry of the 2019 construction requires an identification or extension.

[2] [31] [78] [79]

Quantization of symplectic surfaces

Geometric framework

Etingof–Oblomkov construct formal symmetric-power deformations under a vanishing hypothesis. Dolgushev–Etingof formulate the broader unobstructedness statement. Kontsevich distinguishes formal and semi-formal quantization, including the obstruction for projective abelian varieties with nonzero Poisson structure. An affine, sheaf, or categorical realization must be specified.

[91] [92] [93]
Question for contributors: which surface and algebra or category realize the DELL operators and their known limits? The target is a distinguished commuting Hamiltonian family; at a classical specialization the full center can describe the entire phase space. The oper, Baxter, and spin entries remain unknown.
2024 handoutReview §3.2Permalink to vertex 1
Vertex 2 · dual eRS

Dual elliptic Ruijsenaars–Schneider

ℰp × ℂx×

The elliptic q-oper vertex, with XYZ in the magnetic frame.

Integrable System

Original construction. Braden–Marshakov–Mironov–Morozov (1999/2000), §4, give the classical two-particle Hamiltonian dual to elliptic Ruijsenaars–Schneider: elliptic in momentum and trigonometric in position. [102]

Integrability. The explicit canonical duality establishes the reduced one-degree-of-freedom case. Marshakov, §3.3, describes the extension of the theta-function commutativity argument to the elliptic-RS spectral curve; the detailed derivation in §3.2 treats elliptic CM. [102] [103]

Quantum construction. Mironov–Morozov give explicit dual elliptic-RS Hamiltonians and explain their relation to the Koroteev–Shakirov degeneration. A general quantum integrability proof for the full dual family is unknown in the sources collected here. [39]

Opers
e(G, q)-opers [22]
Baxter relation
Q𝔔-system [22]
Spin system
XYZ [22]
Finite-rank algebra
el.trig.DAHAℏ,c,p
Large-rank algebra
eDIM!ℏ,q,p

Elliptic q-opers, XYZ, and dual eRS

Announced in the review

The review announces a nondegenerate SL(n) XYZ/elliptic q-oper correspondence (Theorem 2.12), and a dual-eRS description for a special class of opers (Proposition 2.13). Both are attributed to “Elliptic opers,” listed as forthcoming.

[22]

Elliptic algebra background

Related construction

Saito constructs elliptic Ding–Iohara operators and commuting elliptic Macdonald families. These are relevant sources for eDIM, but do not establish the particular mirror-marked eDIM! or every identification at this vertex.

[57] [58]

Elliptic DAHA sheaf algebras

Construction; candidate identification

Rains constructs elliptic Hecke and DAHA sheaf algebras, their spherical versions, and Fourier transforms. These provide geometric algebra realizations for comparison with the elliptic vertices. Matching the mirror-marked algebra and its parameters is a candidate identification.

[78]

Elliptic bispectrality and modular transforms

Trace and transform results

Quantum-affine traces satisfy MR, dual MR, qKZB, and dual qKZB equations. Felder–Varchenko develop the qKZB heat transform and its modular properties; elliptic-gamma identities supply the SL(3,Z) structure. These results give a connection layer alongside the elliptic-oper correspondence.

[69] [70] [85] [86]
Question for contributors: identify the elliptic-oper, trace-function, and sheaf-algebra descriptions under an explicit parameter dictionary. A public version of “Elliptic opers” [22] is welcome; its current link leads to the review’s announcement.
2024 handoutReview §2.6: statementsPermalink to vertex 2 Reflected vertex: eRS
Vertex 3 · eRS

Elliptic Ruijsenaars–Schneider

ℂp× × ℰx

Elliptic difference operators; the affine Laumon K-theory vertex.

Integrable System

Original construction. Ruijsenaars–Schneider (1986) introduce the relativistic Calogero–Moser family, including the elliptic model. [100]

Integrability. Ruijsenaars (1987) constructs mutually commuting quantum difference operators and proves Poisson commutativity of their classical counterparts using elliptic-function identities. This supplies the foundational type-A integrability proof. [101]

Opers
unknown
Baxter relation
unknown
Spin system
unknown
Finite-rank algebra
trig.el.DAHAℏ,c,p [6]
Large-rank algebra
eDIMℏ,q,p [65] [57] [58]
Bispectral equations
MR, dual MR, qKZB, and dual qKZB for normalized trace functions [69] [70]

Difference-elliptic operators

Construction

Cherednik’s difference-elliptic construction supplies the operator-theoretic source cited for this DAHA entry in §3.6.

[6]

Affine Laumon functions and eRS eigenfunctions

Conjectural spectral identification

Bullimore–Kim–Koroteev check the defect partition functions as eRS eigenfunctions in an instanton expansion. Shiraishi identifies a non-stationary Ruijsenaars series with affine Laumon Euler characteristics and conjectures the stationary eigenfunction limit.

[5] [62]

Elliptic DIM and commuting families

Result

Saito provides a free-field realization of the elliptic Macdonald operator and constructs commuting families using elliptic DIM and the elliptic Feigin–Odesskii algebra.

[57] [58]

The elliptic DIM reference in the 2024 handout

Result

Ghoneim–Kozçaz–Kurşun–Zenkevich relate elliptic DIM to four-dimensional gauge-theory network constructions and exhibit its decomposition into trigonometric DIM and an additional Heisenberg algebra.

[65]

Eigenvalue formula in the review

Review attribution

The review states the eRS eigenvalue formula as Theorem 3.10 and attributes it to Double Inozemtsev Limits. The formula is recorded here with that review attribution; the non-stationary Laumon and trace-function results below give their own precise spectral statements.

[19]

Affine Laumon partition functions and qKZ

Result in the stated parameter setup

Awata–Hasegawa–Kanno–Ohkawa–Shakirov–Shiraishi–Yamada prove the non-stationary difference equation for the relevant affine Laumon partition functions. With the stated mass tuning and truncation, it gives the quantum affine sl₂ qKZ equation with generic spins and Jackson integral solutions.

[73]

Quantum-affine traces and affine Macdonald functions

Result

Etingof–Schiffmann–Varchenko construct normalized traces satisfying four compatible difference systems; Sun supplies full proofs in specified conventions. In affine sl₂ with the three-dimensional evaluation representation, Sun identifies traces with Felder–Varchenko functions. Rains–Sun–Varchenko prove special-value results and initial affine-Macdonald cases.

[69] [70] [71] [72]

The normalized affine-Macdonald comparison

Open identification

The Etingof–Kirillov affine Macdonald construction and Shiraishi’s affine Laumon series give two descriptions to compare. Shiraishi proposes their equality up to normalization. The open task is an explicit parameter and grading dictionary, including the gl₁ factor, followed by a controlled stationary limit.

[87] [62] [71] [72]

Elliptic algebra and modular benchmarks

Results in the cited settings

Rains supplies elliptic DAHA sheaf algebras. Arthamonov–Shakirov give an explicit elliptic A₁ spherical-DAHA model at K=2, t=−q^(−K/2), with a PSL(2,Z) action. Shakirov’s 2026 elliptic CMM identities are proved to first elliptic order; all-orders identities and precise SL(3,Z) operator formulas are further targets.

[78] [82] [84] [85] [86]
Question for contributors: can the normalized affine Laumon function be identified with the quantum-affine trace under an explicit parameter dictionary? XYZ is the expected electric-frame partner; the magnetic-frame spin construction remains unknown.
2024 handoutReview §3.3Permalink to vertex 3 Reflected vertex: dual eRS
Vertex 4 · dual eCM

Dual elliptic Calogero–Moser

ℰp × ℂx

The elliptic differential-oper vertex in the November 2024 scheme.

Integrable System

Original construction. Braden–Marshakov–Mironov–Morozov (1999/2000), §3, construct the classical two-particle dual of elliptic Calogero–Moser, with elliptic momentum dependence and rational position dependence. [102]

Integrability. Marshakov (1999/2000), §3.2, proves Poisson commutativity of the many-body dual theta-function Hamiltonians: their ratios depend only on the original particle positions. Equations (28)–(30) give the determinant formula and the reduced symplectic structure. [103]

Opers
eG-opers [22]
Baxter relation
unknown
Spin system
Elliptic Gaudin (eGaudin) [22]
Finite-rank algebra
unknown
Large-rank algebra
unknown

Elliptic opers and the proposed correspondence

Announced programme

The review assigns elliptic G-opers and eGaudin to this vertex and attributes their development to the ongoing elliptic-opers project. It does not provide a separate public proof for this vertex.

[22]

An elliptic complex-rank algebra

Candidate

Global Cherednik algebras exist for smooth curves, with a Hamiltonian-reduction construction. Interpolation of the particle number in Deligne categories suggests an elliptic large-rank algebra. A filtered presentation, its central parameters, and the relation to the reflected eCM realization remain to be constructed.

[75] [76] [77]

Elliptic KZB and Casimir connections

Connection-level result

Toledano Laredo–Yang construct a DDCA-valued elliptic Casimir connection. Theorem 6.9 identifies it with KZB under type-A rank duality, up to a specified closed one-form. This is a connection-level comparison relevant to an extension of the Diamond; identifying it with the elliptic-oper entry is a further task.

[81]
Question for contributors: give an explicit elliptic algebra and degeneration maps compatible with the dual eCM polarization. The Baxter and algebra entries in the handout remain unknown.
2024 handoutReview §2.6Permalink to vertex 4 Reflected vertex: eCM
Vertex 5 · tRS

Trigonometric Ruijsenaars–Schneider

ℂp× × ℂx×

The central vertex: q-opers, XXZ Bethe equations, quantum K-theory, and DAHA.

Integrable System

Original construction. Ruijsenaars–Schneider (1986) introduce the trigonometric/hyperbolic relativistic many-body system as part of their Calogero–Moser generalization. [100]

Integrability. Ruijsenaars (1987) proves commutativity of the quantum difference Hamiltonians and Poisson commutativity of the classical integrals. The trigonometric model is a specialization of this construction. [101]

Opers
(G, q)-opers [14] [29]
Baxter relation
QQ-system [14] [29]
Spin system
XXZ [14] [29] [48]
Finite-rank algebra
DAHAℏ,q(X, Y) [8] [45] [46]
Large-rank algebra
Quantum toroidal gl₁ / DIM [54] [55] [56] [38]
Bispectral equations
Bispectral quantum KZ and tRS eigenvalue equations [68]

Opers and quantum/classical duality

Result

For the specified nondegenerate, twisted opers, QQ equations encode Bethe solutions. In type A, the canonical oper description gives a tRS Hamiltonian level set. The exact rank, singularity, and nondegeneracy assumptions are in the linked papers.

[14] [29] [34]

Quantum K-theory and Baxter operators

Result

Quantum multiplication for cotangent flag varieties is described by tRS relations. For cotangent Grassmannians, quantum multiplication by the exterior-algebra tautological class gives the XXZ Baxter operator. Vertex functions also connect qKZ and tRS.

[30] [48] [32] [37]

Stable spherical DAHA and Hilbert schemes

Result

Schiffmann–Vasserot identify the stable spherical DAHA with the elliptic Hall algebra and realize its action on equivariant K-theory of Hilbert schemes. Ding–Iohara supplies the current-algebra construction, with quantum toroidal conventions described in the survey.

[54] [55] [56] [38]

Mirror symmetry and branes

Scope: as in the papers

The oper and quiver descriptions support the bispectral/mirror correspondence. The brane/DAHA monograph treats SL(2); its category statement should not be read as an arbitrary-rank theorem.

[33] [34] [23]

DAHA and bispectral quantum KZ

Result

van Meer–Stokman construct bispectral qKZ equations from type-GL_N DAHA and a correspondence with joint eigenfunctions of trigonometric Ruijsenaars operators. Suitable specializations recover symmetric Macdonald polynomials.

[68]

Dynamical equations and the Toda direction

Result

Etingof–Varchenko construct dynamical Weyl groups and compatible dynamical difference equations for KZ and qKZ. In type A₁, Cherednik–Orr develop nil-DAHA, q-Whittaker functions, and q-Toda equations. Compatibility of a chosen bispectral transform with a Toda limit is a concrete extension to study.

[89] [83]

Elliptic modular deformations

Specified A₁ settings

The elliptic A₁ spherical-DAHA model at K=2 has an explicit PSL(2,Z) action. The elliptic CMM identities provide a related integral-transform problem, with a first-order proof in the elliptic parameter.

[82] [84]
Contributions comparing the bispectral transform with the nil/Toda limit are welcome.
2024 handoutReview §§2.1–2.4Permalink to vertex 5
Vertex 6 · eCM

Elliptic Calogero–Moser

ℂp × ℰx

Elliptic differential operators; the cohomological affine Laumon vertex.

Integrable System

Original construction. Calogero (1975) introduces the elliptic pair-potential extension of the many-body system. [98]

Classical integrability. Perelomov (1977) proves involutivity of the classical integrals. Krichever (1980) gives a Lax representation with spectral parameter and an algebraic-geometric integration of the elliptic particle dynamics through the KP equation. [105] [99]

Quantum integrability. Cherednik constructs elliptic quantum many-body operators. Etingof–Felder–Ma–Veselov give a commuting-Hamiltonian construction using elliptic Dunkl operators, including the usual Weyl-group elliptic Calogero–Moser systems. [7] [74]

Opers
unknown
Baxter relation
unknown
Spin system
unknown
Finite-rank algebra
rat.el.DAHAℏ,c,p; global Cherednik/Dunkl realization [7] [74] [75]
Large-rank algebra
unknown
Eigenfunctions
Normalized affine Laumon generating functions for a non-stationary operator [53]

Elliptic many-body operators

Construction

Cherednik’s elliptic quantum many-body and double affine KZ paper is the operator-theoretic reference assigned to this entry in the review.

[7]

Affine Laumon eigenfunctions

Non-stationary result

Neguț’s Affine Laumon spaces and integrable systems, Theorem 1.5, identifies the Chern-polynomial generating function with an eigenfunction of a non-stationary affine Calogero–Moser operator, after an explicit normalization. His ordinary Laumon construction gives the Calogero–Sutherland eigenfunctions at vertex 8.

[53] [42]

The qq-character approach

Related approach

The BPS/CFT papers develop qq-characters and their BPZ/KZ differential equations, providing the gauge-theory approach cited in the review’s discussion of eCM.

[43] [44]

Elliptic Dunkl operators and the critical twist

Result

Etingof–Felder–Ma–Veselov construct commuting elliptic Calogero–Moser Hamiltonians using elliptic Dunkl operators. Their geometric realization uses spherical global sections of an elliptic Cherednik sheaf at the critical twist.

[74] [75]

Complex-rank interpolation

Candidate

Global Cherednik algebras for curves and Deligne-category interpolation provide a candidate for the large-rank elliptic algebra. The tasks are explicit relations, a PBW statement, and degeneration to the multiplicative and additive curve realizations, with central quotients specified.

[75] [76] [77] [64] [80]

Elliptic Casimir and KZ connections

Connection-level result

On zero-weight spaces of suitable small DDCA modules, Theorem 5.3 of Toledano Laredo–Yang gives a rational Cherednik action and compares elliptic Casimir and elliptic KZ connections up to a scalar one-form. The construction supplies a precise algebra-to-connection comparison with its module hypotheses.

[81]
Question for contributors: construct an explicit elliptic complex-rank presentation and identify its action on the affine Laumon functions. The oper, Baxter, magnetic-frame spin, and large-rank algebra entries remain unknown.
2024 handoutReview §3.4Permalink to vertex 6 Reflected vertex: dual eCM
Vertex 7 · rRS

Rational Ruijsenaars–Schneider

ℂp× × ℂx

The trigonometric-oper / rational-RS vertex in the magnetic frame.

Integrable System

Original construction. Ruijsenaars–Schneider (1986) introduce the rational relativistic many-body system within their family of Calogero–Moser generalizations. [100]

Integrability. Ruijsenaars (1987) proves quantum commutativity and classical Poisson commutativity; the rational specialization gives the rRS Hamiltonians and their commuting integrals. [101]

Opers
Trigonometrically twisted G-opers [34]
Baxter relation
qqt-system [34]
Spin system
Trigonometric Gaudin (tGaudin) [34]
Finite-rank algebra
tCAℏ,c(x, Y) [8]
Large-rank algebra
unknown

Oper coordinates and degenerations

Result

The Zoo of Opers and Dualities treats the lower four vertices in type A. Its trigonometric-oper description connects rRS with the trigonometric Gaudin model and fits into the degenerations of the tRS picture.

[34]

rRS/tCM duality

Specified real forms

Fehér and collaborators establish rational-Ruijsenaars/Sutherland dualities by symplectic reduction, for the real forms and parameter ranges specified there. These complement the complex oper description and bispectral dualities.

[11] [12] [40]

KZ and dynamical equations

Connection-level results

Tarasov–Varchenko show that KZ and dynamical equations exchange under (gl_k,gl_n) duality. Etingof–Varchenko’s dynamical Weyl groups produce compatible dynamical difference equations for trigonometric KZ. These give rank-duality and compatibility data alongside the oper description.

[88] [89]
tGaudin here is the magnetic-frame partner. The electric-frame partner of rRS is XXX. The large-rank algebra is unknown.
2024 handoutReview §2.5Permalink to vertex 7 Reflected vertex: tCM
Vertex 8 · tCM

Trigonometric Calogero–Moser

ℂp × ℂx×

Additive difference opers, XXX, and the affine Yangian.

Integrable System

Original construction. Sutherland (1971) introduces the quantum many-body problem with inverse-sine-squared interaction on a circle, the trigonometric Calogero–Moser (Calogero–Sutherland) system. [95]

Classical integrability. Moser (1975) gives the Lax and isospectral formulation; Perelomov (1977) proves involutivity of the classical integrals. [97] [105]

Quantum solution. Sutherland (1972) constructs the excited states and their exact energies. The commuting-integral approach through trigonometric Dunkl operators is described in Etingof’s lectures, §6.4. [96] [67]

Opers
(G, ε)-opers [34]
Baxter relation
Qq-system [34]
Spin system
XXX [34] [61] [66]
Finite-rank algebra
tCAℏ,c(X, y) [8]
Large-rank algebra
Affine Yangian Yℏ,ε(ĝl₁) [59]
Eigenfunctions
Ordinary Laumon generating functions for Calogero–Sutherland [42]

Additive opers and quantum/classical duality

Result

Replacing the multiplicative shift by an additive shift leads to the (G, ε)-oper description. The type-A duality identifies its two coordinate descriptions with the XXX and tCM data in the magnetic frame.

[34]

Quantum cohomology and Laumon eigenfunctions

Result

Gorbounov–Rimányi–Tarasov–Varchenko identify quantum cohomology of cotangent flag varieties with a Yangian Bethe algebra. Neguț proves the Calogero–Sutherland eigenfunction formula using ordinary Laumon spaces.

[61] [42]

Affine Yangian and additive degeneration

Result

Tsymbaliuk studies the affine Yangian of gl₁ and its relation to quantum toroidal gl₁ through additive degeneration. This is the large-rank algebra in the diagram.

[59]

Calogero–Moser and Bethe-ansatz foundations

Handout references

The handout links tCM to Etingof’s lectures on Calogero–Moser geometry and representation theory, and XXX to Bethe’s 1931 spin-chain paper. These provide background for the modern oper and quantum-cohomology correspondences above.

[67] [66]

Curve realization and central parameters

Comparison problem

For the multiplicative curve, the global Cherednik construction gives the type-A trigonometric Cherednik algebra. Comparing a complex-rank realization to the affine Yangian requires an explicit choice of central extension or quotient, filtration, and normalization; these are additional data to the affine Yangian convention used here.

[75] [76] [77] [59]

Rational qKZ and dynamical differential equations

Connection-level result

Tarasov–Varchenko construct dynamical differential operators compatible with rational qKZ for gl_N. This supplies a connection partner for the rational qKZ system; identifying the trigonometric Casimir convention requires its gauge and normalization.

[90]
The Yangian governing the finite-rank XXX Bethe algebra and the affine Yangian of gl₁ in the large-rank entry are distinct roles; the two citations keep them separate.
2024 handoutReview §2.5Permalink to vertex 8 Reflected vertex: rRS
Vertex 9 · rCM

Rational Calogero–Moser

ℂp × ℂx

The rational differential endpoint: rational opers, Gaudin equations, rational Cherednik algebras, and DDCA.

Integrable System

Original construction. Calogero (1971) solves the quantum N-body problem with inverse-square pair interactions, with an optional quadratic confining term. This vertex uses the unconfined rational model. [94]

Classical integrability. Moser (1975) proves integrability of the rational system through isospectral evolution and commuting first integrals. [97]

Quantum integrability. Etingof’s lectures, §6, give the Dunkl-operator proof: invariant polynomials in commuting Dunkl operators yield a full commuting family of quantum Hamiltonians (Corollary 6.10 and the following gauge transformation). [67]

Opers
Rationally twisted G-opers [4] [34]
Baxter relation
qq-system [4] [34]
Spin system
Rational Gaudin (rGaudin) [4] [34] [41]
Finite-rank algebra
rCAℏ,c(x, y) [60] [8]
Large-rank algebra
DDCA — deformed double current algebra [63] [64] [80]

Differential opers and Gaudin Bethe equations

Result

Brinson–Sage–Zeitlin relate twisted Miura–Plücker opers, polynomial qq-solutions, and inhomogeneous Gaudin Bethe solutions, with explicit conditions for upgrading to Miura opers. The Zoo paper places the type-A rational-CM description in the degeneration diagram.

[4] [34]

Rational Cherednik algebra and Calogero–Moser space

Result

Etingof–Ginzburg provide a foundational construction linking symplectic reflection algebras, spherical rational Cherednik algebras, and Calogero–Moser geometry in type A.

[60]

DDCA, PBW, and Schur–Weyl duality

Type A; ranks as in the paper

Guay proves PBW results for affine Yangians and deformed double current algebras, describes a degeneration between them, and constructs a Schur–Weyl functor with rational Cherednik algebras in the specified type-A ranks.

[63]

A precise large-rank realization

Result

Etingof–Kalinov–Rains construct DDCA using spherical symplectic-reflection algebras in Deligne categories, equivalently filtered ultraproducts of finite-rank spherical algebras. Their gl₁ construction supplies the large-rank interpretation alongside Guay’s type-A theory.

[64]

The rational Gaudin Bethe algebra

Result

Mukhin–Tarasov–Varchenko prove that the Gaudin Hamiltonians generate the Bethe algebra for tensor powers of the vector representation of gl_N.

[41]

DDCA with matrix rank

Result

Kalinov realizes matrix-rank deformed double current algebras through endomorphism algebras in Deligne categories. This extends the interpolation picture alongside the gl₁ realization of Etingof–Kalinov–Rains. Matrix rank, particle number, and interpolation parameter have distinct roles.

[80] [64]

DDCA and the connection layer

Results in specified ranks

Rational KZ and dynamical equations exchange under type-A rank duality. In the elliptic setting, Toledano Laredo–Yang construct a DDCA-valued Casimir connection and compare it with KZB in Theorem 6.9. The latter uses the specified Lie algebras and modules; extending that identification to the stable gl₁ realization is a further question.

[88] [81]
Question for contributors: make the connection-level and interpolation realizations compatible under explicit degeneration maps. The DDCA sources include Guay’s type-A construction, the gl₁ realization, and Kalinov’s matrix-rank extension.
2024 handoutReview §2.5: the oper correspondencePermalink to vertex 9

Arrows & dualities

Each arrow records a limit of a coordinate or momentum type. The references distinguish established lower-vertex descriptions from the announced elliptic-oper programme.

The twelve degeneration arrows and their references
ArrowMeaning and scopeSources
DELL → dual eRS
1 → 2		One elliptic direction degenerates; the dual elliptic-RS branch.[31] [20] [39]
DELL → eRS
1 → 3		The momentum elliptic parameter tends to zero, giving eRS (review §3.3).[31] [19]
dual eRS → dual eCM
2 → 4		The coordinate dependence degenerates from trigonometric to rational. The elliptic-oper interpretation is announced in [22].[22]
dual eRS → tRS
2 → 5		The momentum elliptic curve degenerates to the multiplicative line. The source links the elliptic and ordinary q-oper vertices.[22] [14]
eRS → tRS
3 → 5		Trigonometric degeneration of elliptic difference operators.[6] [8]
eRS → eCM
3 → 6		Differential/nonrelativistic limit of elliptic difference operators.[6] [7]
dual eCM → rRS
4 → 7		Elliptic-to-trigonometric degeneration on the momentum side; the upper oper vertex is part of the announced programme.[22] [34]
tRS → rRS
5 → 7		Rational limit in the coordinate direction of tRS.[34]
tRS → tCM
5 → 8		Additive/differential limit in the momentum direction of tRS.[34]
eCM → tCM
6 → 8		Trigonometric degeneration in the coordinate direction of eCM.[7] [8]
rRS → rCM
7 → 9		Differential limit in the momentum direction of rRS.[34]
tCM → rCM
8 → 9		Rational limit in the coordinate direction of tCM. For the accompanying affine-Yangian/DDCA degeneration, see the algebra references.[34] [63] [64]

Reflection across the vertical axis. The pairs dual eRS/eRS, dual eCM/eCM, and rRS/tCM exchange coordinate and momentum roles in the scheme. For the lower vertices, see the oper and bispectral dualities [33] [34] [40]; the Sutherland/Ruijsenaars reductions [11] [12] [13] give precise results for specified real forms. The reflection is not an assertion that every elliptic correspondence or DELL self-duality has been proved.

Magnetic versus electric. The figure uses rRS/tGaudin and tCM/XXX. In the electric frame these become rRS/XXX and tCM/tGaudin, as explained in the review’s introductory conventions. [34]

Bispectrality & compatible equations

KZ and dynamical equations supply a connection layer alongside the Diamond. Each row specifies its relationship: compatibility, rank duality, or common eigenfunctions. An extension to adjacent faces requires explicit representation maps, parameter substitutions, and degeneration limits.

Spectral equations, their dynamical partners, and references
Spectral equationDynamical or dual partnerRelationship and sources
Rational KZRational dynamical / Casimir equationsExchange under (gl_k,gl_n) rank duality. [88]
Trigonometric KZDynamical difference equationsCompatibility via affine dynamical Weyl groups. [89]
Rational qKZDynamical differential equationsCompatibility for gl_N; the trigonometric Casimir normalization is additional data. [90]
Trigonometric qKZQuantum dynamical difference equationsQuantum dynamical-Weyl-group compatibility. [89]
Elliptic KZBDDCA-valued elliptic Casimir connectionType-A rank-duality identification up to an explicit closed one-form (Theorem 6.9). [81]
qKZB and dual qKZBMR and dual MR equationsFour compatible systems for normalized quantum-affine trace functions. [69] [70]

A question for contributors. Can a bispectral transform and a degeneration be made into a commuting square with explicit gauge factors? The type-GL_N DAHA/qKZ correspondence and the A₁ nil-DAHA limit give concrete starting points. [68] [83]

References

Click a reference number to return to its vertex in the Diamond; click a paper title to open the paper. “In the Diamond” lists every associated vertex. Citations in the vertex panels bring you back here. Two unpublished entries have an announcement link instead.

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Reference notes

The diagram follows the November 2024 handout, including its DDCA entry. The mathematical discussion also draws on the 2023 review. References were checked on 6–7 September 2026. Status labels describe the cited source’s claims.

DDCA at vertex 9

The handout identifies the large-rank algebra as DDCA. Its hyperlink points to Guay’s type-A paper [63]. The complementary gl₁ construction of Etingof–Kalinov–Rains [64] gives a precise interpretation using filtered ultraproducts of spherical algebras. [63] [64]

Elliptic DIM and many-body foundations

Reference [65] describes elliptic DIM at vertex 3. Bethe’s original paper [66] and Etingof’s lectures [67] provide foundations for the spin and many-body systems. [65] [66] [67]

A corrected q-Langlands link

Reference [29] prints the correct identifier, 1811.09937, but its external “Link” in the review targets 1508.02690. The links here use the correct paper. [29]

Laumon constructions

The affine eigenfunction formula is incorporated at vertex 6 with [53], Theorem 1.5, including its non-stationary operator and normalization. The ordinary Laumon / Calogero–Sutherland result [42] appears at vertex 8. Vertex 3 includes the 2024 affine Laumon / qKZ theorem in its stated setup. [42] [53] [73]

Algebra references

[54]–[56] supply the stable spherical DAHA / elliptic Hall / Ding–Iohara background; [57]–[58] cover elliptic DIM and elliptic Macdonald operators; [59] covers the affine Yangian; [60] covers rational Cherednik algebras. These citations support the stated constructions, with their original hypotheses.

Bispectral and geometric references

[68]–[93] cover connections, trace functions, complex rank, and surface quantization. Their results and scope are incorporated into the relevant vertex descriptions. Candidate elliptic and DELL identifications appear beside the known constructions.

Question marks and contributions

“?” marks an unresolved entry in the picture; the corresponding field is “unknown” in the text. References, constructions, and corrections are welcome. “Elliptic opers” [22] and the untitled [36] have no public identifier in the review; links to available versions are also welcome.