Spin Ruijsenaars-Schneider models are Coulomb branches

Gleb Arutyunov; Lukas Hardi

REVIEW 3 major objections 4 minor 8 references

Spin Ruijsenaars-Schneider models are Coulomb branches

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The rational and hyperbolic spin Ruijsenaars–Schneider models are the Poisson algebras of necklace-quiver Coulomb branches, with the coupling constant built in as a γ- or t-deformation.

desk verdict Solid, significant bridge between spin RS models and Coulomb branches; the main L-operator bracket is asserted without proof, but the paper deserves refereeing. read the letter →

arxiv 2603.03048 v3 pith:53UII7FZ submitted 2026-03-03 hep-th nlin.SI

Gleb Arutyunov , Lukas Hardi This is my paper · ORCID

classification hep-thnlin.SI

keywords Ruijsenaars-SchneidermodelCoulombbranchnecklacequivermonopoleoperatorsaffineYangianquantumtoroidalalgebraL-operatorsuperintegrability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to prove that two classical integrable many-body systems — the rational and hyperbolic spin Ruijsenaars–Schneider models — arise, without extra input, from the Coulomb branch geometry of 3d N=4 necklace quiver gauge theories. It shows that in the separated-variable (GKLO) realization of the abelianized Coulomb branch algebra, monopole operators assemble into L-operators whose Poisson brackets reproduce the Lax structure of the spin RS model. The trace Hamiltonians Tr L^n then generate exactly the known equations of motion, with the RS coupling constant encoded as a γ-deformation (rational case) or t-deformation (hyperbolic case). Because the same loop-algebra symmetries (affine Yangian and quantum toroidal) act as centralizers of the Hamiltonians, the model is superintegrable by construction. If correct, the result identifies the spin RS phase space with a gauge-theory object and suggests an elliptic analogue via elliptic Coulomb branches.

What carries the argument

The central object is the one-site L-operator, defined as L^{α±}_{ij}=u^{α+1,±}_j/(q^{α+1}_j−q^α_i) in the rational case and L^α_{ij}=u^{α+1,+}_j/(1−Q^{α+1}_j/Q^α_i) in the hyperbolic case, where u^{α±}_i are fundamental monopole operators of the quiver. These L-operators satisfy a Poisson bracket with r-, r̄-, and r-matrices; the total L-operator inherits the Lax bracket, and the traces Tr L^n are the commuting Hamiltonians. The work is carried out in the GKLO (separated canonical variables) representation of the Coulomb branch algebra, which makes the brackets explicit and exposes the affine Yangian / quantum toroidal loop symmetry that supplies superintegrability.

What would settle it

Compute both sides of equation (3.21) directly in the γ-deformed separated-variable algebra for ℓ=2 and ℓ=1, for all α, β; the paper gives the bracket only for general ℓ and leaves these cases unaudited. A mismatch for any pair would break Theorem 3.13. An independent check is to integrate the rational equations for N=2, ℓ=2 from H=γ Tr L and compare against the spin RS equations of motion.

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Extended reading notes

Core claim

On the abelianized cohomological Coulomb branch of the necklace quiver, the paper constructs one-site L-operators from monopole operators divided by coordinate differences; their total L-operator satisfies the Poisson bracket of the rational spin RS Lax matrix. The Hamiltonian H=γ Tr L generates the rational equations of motion with particle positions x_i=q_i^0 and potential V(z)=1/z−1/(z+γ). The same construction on the K-theoretic Coulomb branch, with x_i=log Q_i^0 and γ=−log t, generates the hyperbolic equations with potential 1/2 coth(z/2)−1/2 coth((z+γ)/2). In both settings the spin vectors satisfy a^1_i=1 and, after rescaling, reproduce the Poisson brackets found earlier by Hamiltonian

Load-bearing premise

The load-bearing premise is the one-site L-operator Poisson bracket: the paper states it with reference r-matrices but does not show the direct computation for ℓ=1 or 2, and the equations of motion and spin brackets follow from nothing else.

Editorial extensions

If this is right

The rational spin Ruijsenaars–Schneider equations of motion are derived from the Hamiltonian H=γ Tr L on the cohomological Coulomb branch algebra.
The hyperbolic spin RS equations are derived from the K-theoretic Coulomb branch with H=(t−1)Tr L, positions x_i=log Q_i^0, and coupling γ=−log t.
The Hamiltonians Tr L^n commute with each other and with the affine Yangian (rational) or quantum toroidal (hyperbolic) generators, so the model is superintegrable.
The rescaled spin variables reproduce the Poisson brackets of the earlier Hamiltonian-reduction description, so the Coulomb branch is the underlying phase space, not just a look-alike.
The same L-operator pattern is conjectured to extend to elliptic Coulomb branches, yielding the elliptic spin RS model and its quantization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the identification is the right one, quantization of the spin RS model can proceed from the quantized Coulomb branch algebra (affine Yangian of gl_ℓ), inheriting its R-matrix structure rather than starting from the classical Hamiltonian reduction.
The K-theoretic match is presented as an instance of mirror symmetry: the hyperbolic model should be equivalent to a multiplicative quiver variety, which may connect spin RS dynamics to cluster structure and Poisson-Lie geometry.
The superintegrability data suggest a complete set of Nℓ action variables given by the traces tr J[n]^k; finding explicit action-angle coordinates would be a concrete test and could yield separation of variables.
A cheap test of the elliptic conjecture would be to check whether an elliptic L-operator algebra with the elliptic r-matrix satisfies the same telescoping identity; if so, the elliptic equations of motion should follow by the paper's argument.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear —,

The short version: this paper genuinely connects the spin RS models to Coulomb branches, and the main identification is likely right. The missing piece is that the load-bearing L-operator bracket is asserted, not derived, and a few special cases are unexamined. I'd send it to a good referee, with instructions to push on that computation.

What's new: the authors give a GKLO representation of the necklace-quiver Coulomb branch and show that traces of the resulting total L-operator generate exactly the rational and hyperbolic spin RS equations of motion. That's a real bridge between two literatures. The EOM derivations in Thm 3.13 and 4.11 are explicit, with telescoping lemmas, and the affine Yangian/quantum toroidal symmetry comes out naturally rather than being bolted on. The Poisson brackets for the spin variables are also written in full and, after the stated rescaling, match the known Hamiltonian-reduction results of Arutyunov–Frolov and others. So the paper earns its main claim.

The soft spots are concentrated where the proofs should be. Proposition 3.6 (and its hyperbolic twin 4.6) states the one-site L-operator bracket with no derivation. Everything downstream — EOM, spin brackets, superintegrability — uses that bracket. Corollary 3.7 relies on an identity r^α + \bar{r}^α_21 − \bar{r}^α − r^α = 0 that is simply asserted; for ℓ ≥ 2 it is not obvious, and the cyclic cases ℓ=1,2 are explicitly not audited. The proof of Prop 3.14 uses several r-matrix identities (r^α e1e2=0, etc.) with no derivation. These are probably all true — the structure is consistent with the known reductions — but a referee cannot verify the central claim without seeing the computation. That's a completeness gap, not a sign of error. The paper also works entirely in the abelianized algebra A_{N,ℓ} and connects to the physical phase space by rescaling and comparison to known brackets, rather than by a structural theorem about reduction; that's worth tightening but is not fatal.

Who should read it: anyone interested in RS models, 3d N=4 Coulomb branches, or affine Yangian actions on integrable systems. The elliptic conjecture at the end is a natural next step.

My recommendation: send it to peer review, but ask the authors to supply the derivation of Prop 3.6/4.6, prove the r-matrix identity in Cor 3.7/4.7, and check ℓ=1,2. That's referee work, not desk-reject work.

Best,

Referee Report

3 major / 4 minor

Summary. The paper claims that the Poisson algebras of cohomological and K-theoretic Coulomb branches of the 3d N=4 necklace quiver reproduce, respectively, the rational and hyperbolic spin Ruijsenaars–Schneider models. The construction proceeds through a gamma-deformed GKLO realization of the abelianized Coulomb branch algebra, in which the monopole operators are expressed in terms of canonical variables (q_i^alpha, P_i^alpha) and (Q_i^alpha, P_i^alpha). The authors define one-site L-operators L^alpha (or L^{alpha pm} in the rational case) and a total L-operator, assert an r-matrix Poisson bracket for them, and derive the Krichever–Zabrodin equations of motion (1.1) from the first Hamiltonian H[1]=Tr L. They also derive Poisson brackets for the spin variables, show that a rescaling reproduces the brackets of [AF98] (rational) and [AO19, Fai26] (hyperbolic), exhibit commuting Hamiltonians and superintegrability via affine Yangian / quantum toroidal symmetries, and conjecture an elliptic analogue.

Significance. If the central construction is correct, the paper gives a substantial new structural identification: spin RS models appear as the dynamical system associated with the Coulomb branch of a familiar 3d N=4 quiver, with superintegrability explained by the (quantum) loop symmetry of the Coulomb branch. The derivation is constructive and does not fit parameters: the GKLO representation, L-operators, Hamiltonians, and equations of motion are all computed from the Coulomb branch algebra. The telescoping lemmas (Lemmas 3.12 and 4.10) that express the total L-operator in terms of spin variables are explicit and clear. However, the load-bearing L-operator brackets are asserted rather than proved, and the passage from the abelianized algebra to the physical spin RS phase space is not fully justified. These issues must be addressed before the central claim can be considered established.

major comments (3)

[Sec. 3.3, Prop. 3.6 / Eq. (3.21); Cor. 3.7] The one-site L-operator bracket (3.21) is the engine of the rational construction: Theorem 3.13, Proposition 3.14, Corollary 3.15, and the Hamiltonian hierarchy all depend on it. Yet Proposition 3.6 is stated without proof. The L-operators are defined in (3.19) from the gamma-deformed monopole operators (3.5), but no computation from the underlying brackets (2.4)-(2.7) or (3.3) is shown. The special cases ell=1,2, where the cyclic identification q^{alpha+ell}=q^alpha-gamma changes the structure, are not treated. Corollary 3.7 then relies on the identity r^alpha + \bar r^alpha_{21} - \bar r^alpha - r^alpha = 0, whose proof is not supplied and which is non-obvious for ell>=2. If (3.21) or this identity is incorrect, the derived equations of motion, spin brackets, and commuting Hamiltonians all fail. A full derivation, at least for general ell, is required.
[Sec. 4.3, Prop. 4.6 / Eq. (4.20); Cor. 4.7] The same issue appears in the K-theoretic case. Proposition 4.6 states the L-operator bracket (4.20) with r-matrices from [AKO19] but no derivation is given. Corollary 4.7 again invokes the identity r^alpha + \bar r^alpha_{21} - \bar r^alpha - r^alpha = 0 without proof. Moreover, the proof of Proposition 4.9 uses the identity (r^alpha + \bar r^alpha)e_2 = -1/2 e_2, which is stated without verification; this identity is needed for the centrality of H[n] and for the superintegrability claim. The hyperbolic equations of motion (Theorem 4.11) rest on Proposition 4.6, so the gap is load-bearing.
[Sec. 3.1/4.1 and Cor. 3.15/4.13] The paper works in the abelianized algebras A_{N,ell} and A^K_{N,ell} (Definitions 3.1 and 4.1), not directly in the full Coulomb branch. The homomorphisms psi and psi^K (Propositions 3.2 and 4.2) are asserted to be injective, but injectivity is not proved. More importantly, the relation between these abelianized coordinates and the physical spin RS phase space is not established: no Weyl/gauge invariance is imposed, and no explicit symplectic reduction or covering argument is given. Corollaries 3.15 and 4.13 show that after a rescaling the spin brackets coincide with those of [AF98] and [AO19,Fai26], but this bracket coincidence does not by itself prove that the spin RS phase space is a symplectic leaf of the Coulomb branch. The title claim 'are Coulomb branches' is stronger than what is demonstrated unless this identification is made precise.

minor comments (4)

[Eq. (3.23)] The definitions of \bar r^alpha and r^alpha in (3.23)-(3.24) are missing explicit summation ranges; e.g. (3.23) writes 1/(q_i^alpha-q_j^alpha)(e_ii-e_ij)\otimes e_jj without a sum over i,j. This should be clarified to avoid ambiguity.
[Sec. 4.4, proof of Thm. 4.11] In the displayed formula for \ddot x_i, the indices alpha,beta appear in Q^alpha_i+Q^beta_j over Q^alpha_i-Q^beta_j, but the preceding bracket is taken with Q^0_i. This is presumably a typo for Q^0_i, Q^0_j; please correct.
[Sec. 4.2, Cor. 4.4] In the displayed formula for K_alpha, lowercase q_i^alpha is used instead of uppercase Q_i^alpha, inconsistent with the notation of this section.
[Introduction, Eqs. (1.5)-(1.7)] The statement 'One can check that the Jacobi identity for these brackets is satisfied' is not backed by a proof or a reference. Since these brackets are used as the spin Poisson structure, a short verification or a reference to a computation would be appropriate.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central derivation is a constructive identification, though key one-site L-operator brackets are asserted rather than derived from the Coulomb branch algebra.

full rationale

The paper's claimed derivation chain is not circular. The cohomological and K-theoretic Coulomb branch Poisson algebras are taken from [BDG15, BFN18] as inputs, and the GKLO representations are explicit homomorphisms into algebras of difference operators (Definitions 3.1 and 4.1, Propositions 3.2 and 4.2). The one-site L-operators are then defined as concrete rational functions of the monopole operators (3.19) and (4.18), and the total L-operator is their sum/product. The equations of motion in Theorems 3.13 and 4.11 are computed by taking Poisson brackets with H = γ Tr L or H = (t−1) Tr L; no parameter is fitted to force the spin RS equations, and the potentials V(z) arise from algebraic identities such as γ/(q_i^0−q_j^0)(q_i^0−q_j^ℓ) = V(x_i−x_j). The r-matrices in Proposition 3.6 and 4.6 are imported from the earlier constructions [AF96] and [AKO19], and Corollaries 3.7 and 4.7 identify the resulting total-L bracket with the known Lax brackets from [AF98] and [AKO19]. This is an identification between a newly constructed L-operator algebra and a known one, not a restatement of the target equations by definition. The main genuine weakness is that Proposition 3.6 (and its hyperbolic analogue) is stated without proof: the paper does not show the computation of the one-site L-operator brackets from the underlying Coulomb branch brackets (2.4)–(2.7), and the special cases ℓ=1,2 for the r-matrix identities are not audited. Similarly, the identity r^α + r̄^α_{21} − r̄^α − r^α = 0 in Corollary 3.7 is invoked without proof. These are serious proof gaps and correctness risks, but they are not circularity: the claimed result does not reduce to its own inputs by construction. The relevant self-citations ([AF96], [AF98], [AKO19]) are published, parameter-free constructions that are external to the present identification; [AH25] appears only as motivation/conjecture and is not load-bearing. Accordingly, the honest finding is low circularity, with the score reflecting the unproved but load-bearing L-operator bracket assertion rather than any circular reduction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted in the paper: γ (cohomological) and t (K-theoretic) are the bifundamental mass/deformation parameters of the gauge theory and play the role of the RS coupling; N and ℓ are fixed integers. The axioms are the standard presentations of Coulomb branch Poisson algebras and the asserted injectivity of the GKLO map; the r-matrix identities are used but not proven.

assumptions (3)

domain assumption The abelianized Coulomb branch Poisson algebras C_{N,ℓ} and C^K_{N,ℓ} are as presented in Def 2.1 and 2.2, taken from [BDG15] and [FKRD18].
The paper builds all constructions on these presentations without re-deriving them from the 3d N=4 gauge theory.
domain assumption The GKLO homomorphisms ψ and ψ^K are injective (Prop 3.2 and 4.2), so working in A_{N,ℓ} is equivalent to working in the Coulomb branch algebra.
Injectivity is asserted after a bracket computation; the proof shown only verifies the Poisson brackets, not injectivity directly.
standard math The r-matrix identities (e.g., rα + r̄α_21 − r̄α − rα = 0, rα e1 e2 = 0, etc.) used in Cor. 3.7 and Prop. 3.14 hold for all ℓ including ℓ=1,2.
These identities are stated without proof; the special cases ℓ=1,2 receive separate treatment in Prop 3.2 but the r-matrix identities are not checked there.

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Pith. "Pith review of Spin Ruijsenaars-Schneider models are Coulomb branches." pith.science (2026). https://pith.science/paper/53UII7FZ

@misc{pith2026260303048,
  author       = {Pith},
  title        = {Pith review of: Spin Ruijsenaars-Schneider models are Coulomb branches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53UII7FZ}},
  note         = {Machine review of arXiv:2603.03048}
}

abstract

In this paper, we show that the Poisson algebras of homological and $K$-theoretic Coulomb branches of 3d $\mathcal{N}=4$ necklace quiver gauge theories provide Poisson structures and Hamiltonians that reproduce the equations of motion of the rational and hyperbolic spin Ruijsenaars-Schneider models, respectively. The construction is carried out in terms of monopole operators in the GKLO representation, also making the affine Yangian (and, in $K$-theory, quantum toroidal) superintegrability structure manifest. We conjecture that the Poisson algebras of elliptic Coulomb branches similarly reproduce the elliptic spin Ruijsenaars-Schneider model.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages

[1]

Arutyunov, L

[ACF97] G. Arutyunov, L. Chekhov, and S. Frolov. R-Matrix quantization of the elliptic Ruijsenaars-Schneider model.Theoretical and Mathematical Physics, 111(2):536–562, May 1997.doi:10.1007/bf02634266. [AF96] G. Arutyunov and S. Frolov. Quantum Dynamical R-matrices and Quantum Frobenius Group, 1996.arXiv:q-alg/9610009. [AF98] G. Arutyunov and S. Frolov. O...

arXiv 1997
[1996]

Chalykh and M

[CF20] O. Chalykh and M. Fairon. On the Hamiltonian formulation of the trigonometric spin Ruijsenaars–Schneider system.Letters in Mathematical Physics, 110(11):2893–2940, August 2020.doi:10.1007/s11005-020-01320-x. [Fai26] M. Fairon. Compatible Poisson structures on multiplicative quiver varieties.Revista Matem´ atica Iberoamericana,

work page doi:10.1007/s11005-020-01320-x 2020
[2005]

Krichever and A

[KZ95] I. Krichever and A. Zabrodin. Spin generalization of the Ruijsenaars-Schneider model, nonAbelian 2-d Toda chain and representations of Sklyanin algebra. Russ. Math. Surveys, 50:1101, 1995.arXiv:hep-th/9505039,doi:10.1070/ RM1995v050n06ABEH002632. [Res16] N. Reshetikhin. Degenerately integrable systems.Journal of Mathematical Sciences, 213(5):769–785,

arXiv 1995
[2008]

[Tsy23] A

URL:https://arxiv.org/abs/0808.3875,arXiv:0808.3875. [Tsy23] A. Tsymbaliuk. Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems.Letters in Mathematical Physics, 113(1):22,

arXiv
[2016]

Reshetikhin and M

[RS90] N. Reshetikhin and M. Semenov-Tian-Shansky. Central extensions of quantum current groups.Lett. Math. Phys., 19:133–142, 1990.doi:10.1007/BF01045884. [Sol08] F. Soloviev. On a Hamiltonian form of an elliptic spin Ruijsenaars-Schneider system,

doi:10.1007/bf01045884 1990
[2018]

Finkelberg, M

24 [FMP20] M. Finkelberg, M. Matviichuk, and A. Polishchuk. Elliptic zastava.arXiv preprint arXiv:2011.11220,

arXiv 2011
[2025]

Arutyunov, R

[AKO19] G. Arutyunov, R. Klabbers, and E. Olivucci. Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model.Journal of High Energy Physics, 2019(5), May 2019.doi:10.1007/jhep05(2019)069. [AO19] G. Arutyunov and E. Olivucci. Hyperbolic spin Ruijsenaars-Schneider model from Poisson reduction, 2019.arXiv:1906.02619. [BDG15] M. ...

arXiv 2019
[2026]

Fairon, L

[FFM21] M. Fairon, L. Feh´ er, and I. Marshall. Trigonometric real form of the spin RS model of Krichever and Zabrodin. InAnnales Henri Poincar´ e, volume 22, pages 615–675. Springer, 2021.doi:10.1007/s00023-020-00976-4. [FKRD18] M. Finkelberg, A. Kuznetsov, L. Rybnikov, and G. Dobrovolska. Towards a cluster structure on trigonometric zastava.Selecta Math...

work page doi:10.1007/s00023-020-00976-4 2021

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Arutyunov, L

[ACF97] G. Arutyunov, L. Chekhov, and S. Frolov. R-Matrix quantization of the elliptic Ruijsenaars-Schneider model.Theoretical and Mathematical Physics, 111(2):536–562, May 1997.doi:10.1007/bf02634266. [AF96] G. Arutyunov and S. Frolov. Quantum Dynamical R-matrices and Quantum Frobenius Group, 1996.arXiv:q-alg/9610009. [AF98] G. Arutyunov and S. Frolov. O...

arXiv 1997

[2] [1996]

Chalykh and M

[CF20] O. Chalykh and M. Fairon. On the Hamiltonian formulation of the trigonometric spin Ruijsenaars–Schneider system.Letters in Mathematical Physics, 110(11):2893–2940, August 2020.doi:10.1007/s11005-020-01320-x. [Fai26] M. Fairon. Compatible Poisson structures on multiplicative quiver varieties.Revista Matem´ atica Iberoamericana,

work page doi:10.1007/s11005-020-01320-x 2020

[3] [2005]

Krichever and A

[KZ95] I. Krichever and A. Zabrodin. Spin generalization of the Ruijsenaars-Schneider model, nonAbelian 2-d Toda chain and representations of Sklyanin algebra. Russ. Math. Surveys, 50:1101, 1995.arXiv:hep-th/9505039,doi:10.1070/ RM1995v050n06ABEH002632. [Res16] N. Reshetikhin. Degenerately integrable systems.Journal of Mathematical Sciences, 213(5):769–785,

arXiv 1995

[4] [2008]

[Tsy23] A

URL:https://arxiv.org/abs/0808.3875,arXiv:0808.3875. [Tsy23] A. Tsymbaliuk. Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems.Letters in Mathematical Physics, 113(1):22,

arXiv

[5] [2016]

Reshetikhin and M

[RS90] N. Reshetikhin and M. Semenov-Tian-Shansky. Central extensions of quantum current groups.Lett. Math. Phys., 19:133–142, 1990.doi:10.1007/BF01045884. [Sol08] F. Soloviev. On a Hamiltonian form of an elliptic spin Ruijsenaars-Schneider system,

doi:10.1007/bf01045884 1990

[6] [2018]

Finkelberg, M

24 [FMP20] M. Finkelberg, M. Matviichuk, and A. Polishchuk. Elliptic zastava.arXiv preprint arXiv:2011.11220,

arXiv 2011

[7] [2025]

Arutyunov, R

[AKO19] G. Arutyunov, R. Klabbers, and E. Olivucci. Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model.Journal of High Energy Physics, 2019(5), May 2019.doi:10.1007/jhep05(2019)069. [AO19] G. Arutyunov and E. Olivucci. Hyperbolic spin Ruijsenaars-Schneider model from Poisson reduction, 2019.arXiv:1906.02619. [BDG15] M. ...

arXiv 2019

[8] [2026]

Fairon, L

[FFM21] M. Fairon, L. Feh´ er, and I. Marshall. Trigonometric real form of the spin RS model of Krichever and Zabrodin. InAnnales Henri Poincar´ e, volume 22, pages 615–675. Springer, 2021.doi:10.1007/s00023-020-00976-4. [FKRD18] M. Finkelberg, A. Kuznetsov, L. Rybnikov, and G. Dobrovolska. Towards a cluster structure on trigonometric zastava.Selecta Math...

work page doi:10.1007/s00023-020-00976-4 2021