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REVIEW 5 major objections 4 minor 28 references

Spin Chain Integrability as a Supersymmetric Gauge Duality

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims to construct the first exact duality between 4D BCD-type supersymmetric gauge theories on $D^2\times T^2$ and open XYZ spin chains, with gauge vacuum equations matching the elliptic Bethe ansatz equations under explicit…

desk verdict A workmanlike extension of the Bethe/gauge dictionary with one genuinely new entry (C-type 4D) and an honest limitation: the 4D vacuum equation is imported from 3D without proof, so the headline 'first exact' claim is too strong. read the letter →

arxiv 2506.18662 v1 pith:UNMIQM5P submitted 2025-06-23 hep-th math-phmath.MP

Xiang-Mao Ding , Ting Zhang This is my paper · ORCID
classification hep-thmath-phmath.MP MSC 81T6082B2381R1281T13
keywords supersymmetricgaugetheoryspinchainBetheansatzXYZBethe/gaugecorrespondenceellipticR-matrixOmega-backgroundlocalization
topics Yang-Mills Mass Gap
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

This paper aims to establish an exact duality between four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories on $D^2\times T^2$ and the open XYZ spin chain, the fully anisotropic integrable spin chain built from the elliptic eight-vertex $R$-matrix. The paper derives effective superpotentials for gauge groups of type A, B, C, and D and shows that the vacuum equations $\exp(\sigma\,\partial W_{\rm eff}/\partial\Phi_i)=1$ coincide, term by term, with the Bethe ansatz equations of the XYZ chain: closed-boundary equations for A-type groups, and open-boundary equations with fixed boundary parameters for B/C/D-type groups. The claimed pay-off is a universal dimensional ladder — 2D $\mathcal{N}=(2,2)$ theories map to XXX chains, 3D $\mathcal{N}=2$ theories to XXZ chains, and 4D $\mathcal{N}=1$ theories to XYZ chains — with the $\Omega$-deformation parameter interpolating between dimensions. If correct, the result gives the first 4D realization of the Bethe/gauge correspondence for general-boundary XYZ chains and makes the non-perturbative vacuum structure of these gauge theories solvable through the integrable system.

What carries the argument

The central object is the elliptic function $\sigma(u)=\theta_{1/2,1/2}(u,\tau)$, together with the effective superpotential $W_{\rm eff}^{4d,R}(\Phi)$ obtained from the $D^2\times T^2$ one-loop determinants. The identity that carries the argument is the vacuum equation $\exp(\sigma\,\partial W_{\rm eff}/\partial\Phi_i)=1$: it is the bridge that converts the gauge-theory partition function into the Bethe ansatz equations of the spin chain. The second mechanism is the root normalization $2/\alpha^2$ inserted into the vector-multiplet superpotential: it makes the root-system data of the Lie algebra appear in the vacuum products with the precise powers needed to reproduce the three $\sigma$-function factors of the open XYZ BAE, and it is the feature that makes the B/C/D root systems select open boundary conditions while the A-type root system selects periodic ones.

What would settle it

Compute the next correction to the effective superpotential of a small B-type theory (for instance $SO(5)$ with $N_f=2L+12$) directly from the exact $D^2\times T^2$ partition function and check whether the vacuum equation remains $\exp(\sigma\,\partial W_{\rm eff}/\partial\Phi)=1$; if any correction survives, the equality with BAE (24) fails. A cheaper check is to test the phase conditions $P_{hB}=2\pi i k$, $P_{hC}=2\pi i k$, $P_{hD}=2\pi i k$ at generic masses: if they can only be satisfied by fine-tuning the dictionaries (57), (64), (69), the correspondence holds only on a parameter locus rather than as a duality.

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Extended reading notes

Core claim

On the gauge side, the paper uses the localized one-loop partition function on $D^2\times T^2$, with Robin-like boundary conditions for matter, to write the effective superpotential $W_{\rm eff}^{4d}(\Phi)$ as sums of cubic polynomials and dilogarithm/trilogarithm towers. Differentiating through the vacuum equation $\exp(\sigma\,\partial W_{\rm eff}/\partial\Phi_i)=1$ turns those sums into products of elliptic $\sigma$-functions. The paper then shows that for B-type ($SO(2N+1)$), C-type ($Sp(2N)$), and D-type ($SO(2N)$) gauge groups these products are exactly the open XYZ Bethe ansatz equations (24), provided the boundary parameters are fixed to $\alpha^\pm_1=1/2$, $\alpha^\pm_2=\tau/2$, $\alpha^\pm_3=(1+\tau)/2$ and the sign choices (59); for A-type groups the same procedure gives the closed-chain Bethe ansatz (20). The identification is organized by explicit dictionaries: $\Phi_i\leftrightarrow i u_i$, $m_{\rm adj}\leftrightarrow\eta$, $N\leftrightarrow L$, and flavor masses shifted by $\eta$ and half-periods, with $N_f$ fixed to $2L+12$, $2L+5$, or $2L+4$ for B, C, D respectively.

Load-bearing premise

The whole comparison rests on the assumption that the one-loop vacuum equation $\exp(\sigma\,\partial W_{\rm eff}/\partial\Phi)=1$, imported from the three-dimensional analysis, is the complete and exact vacuum condition for the four-dimensional theory on $D^2\times T^2$; any higher-loop or non-perturbative correction would add extra factors and break the matching with the Bethe ansatz equations.

Editorial extensions

If this is right

  • For B-, C-, and D-type 4D $\mathcal{N}=1$ gauge theories with the specified matter content, the space of supersymmetric vacua is parameterized by the Bethe roots of the open XYZ spin chain, so exact results from the spin chain become statements about the gauge theory.
  • The parameter dictionaries fix the matter content of the gauge theory: for a chain of length $L$, the number of flavors must be $N_f=2L+12$ (B-type), $N_f=2L+5$ (C-type), or $N_f=2L+4$ (D-type), so the duality predicts which theories have integrable vacuum structure.
  • For A-type gauge groups the same construction reproduces the closed-boundary XYZ Bethe ansatz, so the 4D correspondence covers periodic chains as well as open ones, with the gauge group determining the boundary condition.
  • Taking $\tau\to i\infty$ in the dictionaries should reproduce the 3D XXZ correspondence with generic off-diagonal boundaries, and further reduction to the XXX case, making the dimensional ladder a single family.
  • The elliptic modulus $\tau$ is identified with the torus modulus of $T^2$, so varying the geometry of the gauge-theory spacetime changes the anisotropy of the spin chain continuously.

Reading between the lines

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

  • The boundary parameters in (59) are locked to the half-periods $1/2$, $\tau/2$, $(1+\tau)/2$, so the 'general boundary' realized here is the most general off-diagonal $K$-matrix at a discrete, measure-zero set of boundary couplings; varying the $\alpha$ parameters freely should break the equality, a point the paper does not discuss.
  • The same mechanism with the coefficient $2/\alpha^2$ suggests a uniform normalization that unifies the 3D ($4/\alpha^2$) and 4D ($2/\alpha^2$) computations; one testable extension is to check whether exceptional gauge groups, which have multiple root lengths, require a separate normalization per root orbit.
  • Since the conclusion asserts the method extends to exceptional groups and quiver gauge theories without showing the dictionaries, the natural next check is to derive the E-type analogues of (57), (64), (69) and verify that the same half-period boundary parameters appear.
  • If the correspondence is exact, it implies a numerical prediction: for small $L$, the number of solutions of the gauge vacuum equations should equal the number of Bethe roots counted by the XYZ off-diagonal Bethe ansatz completeness; counting solutions on both sides would test the duality without computing any correlation function.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper claims to establish an exact duality between 4D N=1 supersymmetric gauge theories on D^2×T^2 and open XYZ spin chains with general boundary conditions, for gauge groups of type B, C, and D (and A-type with closed boundaries). The spin-chain side uses Bethe ansatz equations (BAE) taken from prior work on the XYZ model with non-diagonal boundaries. The gauge side constructs an effective superpotential from one-loop determinants of the localized partition function, postulates a vacuum equation exp(σ δW_eff/δΦ)=1, and derives vacuum equations for each gauge group. The correspondence is then implemented by parameter dictionaries mapping gauge theory masses and ranks to spin-chain parameters, and by fixing the boundary parameters to special values. The paper also includes an appendix giving a similar correspondence for 3D XXZ chains with fixed boundary parameters.

Significance. If the claimed duality were established rigorously, it would be a notable step in the Bethe/gauge correspondence program, extending the known 2D/XXX and 3D/XXZ dualities to 4D/XYZ with non-diagonal boundaries. The paper engages with a technically active literature and proposes explicit dictionaries, which is useful. However, the central claim is far stronger than what the evidence supports: the vacuum equation is imported from 3D without derivation, the boundary parameters are not general, and the phase conditions used to match the equations are imposed on dynamical variables rather than derived. As it stands, the paper does not provide a rigorous construction of the advertised duality.

major comments (5)
  1. [Sec. 3.2, Eq. (37)] The vacuum equation exp(σ δW_eff/δΦ)=1 is stated as 'derived from the 3D N=2 gauge theory vacuum configuration on D^2×S^1', and the text says the method is applied to 4D 'parallelly'. No derivation from the D^2×T^2 partition function is supplied, and non-perturbative corrections are not assessed. Since every vacuum equation in Section 4 is obtained from this equation, the claimed exact duality is conditional on an unjustified assumption. This is a load-bearing point that needs to be either derived or explicitly flagged as a conjecture.
  2. [Sec. 4.2, Eq. (55); Sec. 4.3, Eq. (62); Sec. 4.4, Eq. (67)] The phase functions PhB, PhC, and PhD contain explicit dependence on the dynamical variables Φ_i (e.g., terms like 16πi Φ_i m_adj/τ and sums over j of 8πi Φ_i m_adj/τ). The paper 'imposes' Ph=2πik with k∈Z and then drops the phase factor to obtain the clean vacuum equations (56), (63), and (68). This is not a choice of free parameters; it is a set of additional equations that the Bethe roots must satisfy, which would over-constrain or trivialize the system unless the dictionaries make the coefficients of Φ_i vanish identically. No such identity is shown. The derivation of the final vacuum equations is therefore not justified.
  3. [Sec. 2.2, Eq. (24); Sec. 4.2, Eq. (56)] The advertised 'general-boundary' XYZ spin chain has six free boundary parameters α_l^± according to the Hamiltonian (10) and BAE (24). However, the correspondence constructed in Section 4 fixes these parameters to the special values (59): α_1^±=1/2, α_2^±=τ/2, α_3^±=(1+τ)/2. The same happens in Appendix A with (A3), (A6), and (A8). Thus the realized duality covers only a measure-zero subfamily of boundary conditions, contradicting the abstract's claim of 'general-boundary XYZ spin chains' and the conclusion's statement that the paper treats 'the most generic boundary conditions'.
  4. [Sec. 4.2, dictionaries (57)-(58) and BAE (24)] The dictionary Φ_i ↔ i u_i and m_adj ↔ η leads to an apparent sign mismatch between the gauge vacuum equations and the BAE. For example, the BAE (24) contains factors σ(i(u_j-u_l-ηi)) = σ(i(u_j-u_l)+η), while the B-type vacuum equation (56) becomes σ(i u_i ± i u_j - η) = σ(i(u_i ± u_j)-η) under the same dictionary. The signs of the η-shifts differ, and no elliptic identity is invoked to reconcile them. Without a consistent identification of spectral parameters, the claimed equivalence of (56) and (24) is not demonstrated.
  5. [Sec. 4, Eqs. (45)-(68)] The vacuum equations are presented after the phrase 'Using formula (37)' without showing the differentiation of the lengthy superpotentials (45), (52), (60), and (66). Because the central claim is an exact equality between gauge vacuum equations and BAE, the algebra should be checkable. The omission makes it impossible to verify that the vacuum equations are correctly derived, and the subsequent match is therefore not rigorous. A detailed appendix or at least the key intermediate steps are needed.
minor comments (4)
  1. [Sec. 3.2, Eq. (35)] There are typos in the Li3 arguments: 'p.q' should be 'p,q', and in the second line of (35) the notation 'Li3(pqe−2πi(...);p,q)' should probably be 'Li3(pq e^{−2πi(...)};p,q)'.
  2. [Sec. 4.2, Eq. (55)] The expression for PhB contains the term '4πiΦ' without a subscript; it should be '4πiΦ_i' to be consistent with the surrounding terms.
  3. [Sec. 2.2, Eq. (22)] The condition on the sign factors ε_j^γ is written as a product over γ=± and j=1..3; the notation is ambiguous about whether ε_j^γ are constants or can depend on the index j in the BAE. Please clarify.
  4. [Sec. 5, Conclusion] The sentence 'The effective superpotentials come from the partition function in supersymmetric gauge theories' appears to be a fragment and the following sentence about exceptional gauge groups is not developed; the conclusion would benefit from a clearer statement of what is actually proven versus conjectured.

Circularity Check

3 steps flagged · score 7.0 of 10

The claimed 4D BCD/XYZ duality is enforced by hand: phase factors are set to 2πik, boundary parameters are fixed to (59), and flavor masses are assigned by dictionaries, so the match is a constructed identification rather than a derived prediction.

  1. fitted input called prediction [Section 4.1, Eqs. (48)-(51)]
    "By judiciously selecting mass parameters, one can construct the phase factor PhA = 4πil1uj + 2iϕ ... the invariance requirement of (48) is Nf∑(ma +m′a) =−Nm adj, which exactly replicates the field content condition in 4D N=1 gauge theories [18]."

    The phase PhA is not derived from the localization computation; it is constructed by choosing mass parameters, and the additional mass-sum constraint (50) is imposed so that the phase matches the closed XYZ BAE (20). The dictionary (51) then renames Φi→ui, madj→η, etc. The equivalence between the gauge vacuum equation (48) and the BAE is therefore fitted by hand rather than predicted.

  2. fitted input called prediction [Section 4.2, after Eq. (55); similarly for C/D at (62) and PhD]
    "Imposing the condition PhB = 2πik(k∈Z), we precisely derive the desired vacuum equations. ... Actually, we can choose suitable mass parameters to let the factore F be zero."

    The phase PhB is an output of the gauge-side vacuum equation (37). Setting it to 2πik by hand, and then choosing masses so the exponential factor vanishes, removes the only gauge-theoretic obstruction between the vacuum equation and the BAE form. The same 'Imposing the condition' is repeated for C and D. Thus 'deriving' (56), (63), and (68) is conditional on imposing the equality one then claims to establish.

1 more flagged steps
  1. fitted input called prediction [Section 4.2, dictionary (57)-(59)]
    "Through rigorous computation, we determine that the boundary parameters of the XYZ spin chain should satisfy ... (59). We thus establish the pivotal result: the vacuum equation (56) satisfies the BAE (24)."

    The boundary parameters {αl±, ϵl±} are free parameters of the open XYZ BAE (24). The dictionary (57) assigns each flavor mass ma to specific combinations of η and τ, and (59) fixes every boundary parameter so that the product of σ-factors in (56) is term-by-term the boundary factor of (24). With Nf=2L+12, N=L, and Φi=iui, the two equations are made identical by construction. No free datum remains to test the correspondence, and (59) fixes all six boundary parameters, so the advertised 'general-boundary' duality is actually a restricted subfamily selected to match.

full rationale

The two sides of the claimed duality are not self-referential: the open XYZ BAE (24) is taken from [28] and the 4D one-loop determinants from [15], so this is not definitional circularity of the usual kind. However, the central correspondences are not predictions. For A-type, Section 4.1 constructs the phase PhA by 'judiciously selecting mass parameters' and imposes a mass-sum condition to force agreement with the closed BAE. For B, C, and D, the phases PhB, PhC, and PhD are set to 2πik by fiat, and then mass dictionaries (57), (64), (69) and boundary parameters (59) are chosen so the remaining σ-products coincide term by term. Every free boundary parameter and virtually every flavor mass is consumed in making the identification, leaving nothing on either side to test the duality. In addition, the paper explicitly states that the key 4D vacuum equation (37) 'is derived from the 3D N=2 gauge theory vacuum configuration on D2×S1' and says 'We apply to the method to 4D parallelly'; no independent 4D derivation is supplied, so the dictionaries in Section 4 inherit this unverified premise. That is a missing-derivation risk rather than a circular reduction, but it reinforces the concern. The advertised 'general-boundary' correspondence is also narrower than claimed, because (59) fixes all six α parameters and all ϵ signs. Overall, the match reduces, by construction, to the choices made to enforce it, giving a partially circular fitted correspondence rather than a derived exact duality.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are postulated. The correspondence dictionary maps existing parameters of known theories, so the invented-entity ledger is empty.

free parameters (3)
  • flavour masses m_a, m'_a = m_adj maps to eta, and m_a maps to specific combinations of eta, tau, and 1/2 as in dictionaries (57), (64), (69)
    The masses are assigned after the fact so that the gauge vacuum equations reproduce the Bethe ansatz equations. The text says 'by judiciously selecting mass parameters' in Section 4.1, which is fitting, not predicting.
  • boundary parameters alpha_l^+ and alpha_l^- and signs epsilon_k^+ and epsilon_k^- = alpha_1^pm = 1/2, alpha_2^pm = tau/2, alpha_3^pm = (1+tau)/2, with epsilon signs as in Eq. (59)
    The six boundary parameters and sign choices of the open XYZ chain are set to specific values in Eq. (59) to make the BAE match the vacuum equations; no independent derivation is given.
  • root normalization coefficient in the vector multiplet superpotential = 2/alpha^2 instead of the 3D value 4/alpha^2
    The paper states the choice of 2/alpha^2 over 4/alpha^2 is for convenience because it eliminates square-root operations. This normalization freedom is selected to make the 4D match work.
assumptions (4)
  • domain assumption The 1-loop determinant formulas (29)-(31) from [15] are the correct building blocks for the 4D N=1 partition function on D^2×T^2.
    The entire effective superpotential is built from these determinants, imported from Longhi-Nieri-Pittelli without re-derivation.
  • domain assumption The vacuum equation exp(sigma dW/dPhi) = 1 is the exact vacuum condition for the 4D theory on D^2×T^2.
    The paper says 'we apply the method to 4D parallelly' in Section 3.2 and provides no derivation; in 3D this equation is standard, but its 4D validity is assumed.
  • domain assumption The Bethe ansatz equations (24) from [28] are the complete correct equations for the open XYZ spin chain with generic integrable boundaries.
    The spin chain side is taken as given from the off-diagonal Bethe ansatz literature; the correspondence is a comparison to these equations.
  • standard math The limits tau goes to i infinity (XXZ) and eta to 0 (XXX) preserve the structure of the correspondence.
    The dimensional and anisotropy limits are asserted in Sections 2.2 and 3.3 with asymptotic formulas, not proven to preserve the dictionary.

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Cite this review

Pith. "Pith review of Spin Chain Integrability as a Supersymmetric Gauge Duality." pith.science (2026). https://pith.science/paper/UNMIQM5P

@misc{pith2026250618662,
  author       = {Pith},
  title        = {Pith review of: Spin Chain Integrability as a Supersymmetric Gauge Duality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNMIQM5P}},
  note         = {Machine review of arXiv:2506.18662}
}
abstract

We establish a novel correspondence between 4D $\mathcal N=1$ supersymmetric gauge theories on $D^2\times T^2$ and open XYZ spin chains with generalized boundary conditions, extending beyond previous 3D Bethe/gauge duality frameworks. Our primary contribution is the rigorous construction of the first exact duality between 4D BCD-type gauge theories and general-boundary XYZ spin chains governed by elliptic $R$-matrices. This framework provides a universal mechanism for resolving supersymmetric gauge theory/spin-chain duality across dimensional hierarchies: 2D $\mathcal{N}=(2,2)$ $\longleftrightarrow$ XXX spin chain, 3D $\mathcal{N}=2$ $\longleftrightarrow$ XXZ spin chain, 4D $\mathcal{N}=1$ $\longleftrightarrow$ XYZ spin chain, mediated through the $\Omega$-deformation parameter $\epsilon$.

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Works this paper leans on

28 extracted references · 21 canonical work pages

  1. [27]

    Wang, R.-D Zhu,Bethe/Gauge correspondence forA N spin chains with integrable boundaries,J

    Z. Wang, R.-D Zhu,Bethe/Gauge correspondence forA N spin chains with integrable boundaries,J. High Energ. Phys.2024, 112 (2024). https://doi.org/10.1007/JHEP04(2024)112

    work page doi:10.1007/jhep04(2024)112 2024

  2. [1]

    R. J. Baxter,Eight-Vertex Model in Lattice Statistics, Phys. Rev. Lett. 26 (1971) 832

    work page 1971

  3. [2]

    R. J. Baxter,One-Dimensional Anisotropic Heisenberg Chain, Phys. Rev. Lett. 26 (1971) 834

    work page 1971

  4. [3]

    R. J. Baxter,Exactly Solved Models in Statistical Mechan- ics, Dover books on physics, Dover Publications(2007). https://books.google.ie/books?id=G3owDULfBuEC

    work page 2007

  5. [4]

    Cao, W.-L

    J. Cao, W.-L. Yang, S. Cui, K.-J. Shi and Y. Wang,Spin- 1/2 XYZ model revisit: General solutions via off-diagonal Bethe ansatz,Nucl. Phys. B886 (2014) 185

    work page 2014

  6. [6]

    Bethe/Gauge Correspondence for linear quiver theories with ABCD gauge symmetry and spin chains

    X.-M Ding and T. Zhang,Bethe/gauge correspondence for linear quiver theories with ABCD gauge symmetry and spin chains,Nuclear Physics B 991(2023) 116222 [arXiv:2303.04575]

    work page Pith review arXiv 2023

  7. [7]

    Langlands Dualities through Bethe/Gauge Correspondence for 3d Gauge Theories

    X.-M Ding and T. Zhang,Langlands Dualities through Bethe/Gauge Correspondence for 3d Gauge Theories, [arXiv:2312.13080]

    work page Pith review arXiv

  8. [8]

    H. M. Farkas and I. Kra,Theta Constants, Riemann Surfaces and the Modular Group, Graduate Studies in Mathematics 37, American Mathematical Society 2001

    work page 2001

Show all 28 references
  1. [9]

    Fujimori, T

    T. Fujimori, T. Kimura, M. Nitta and K. Ohashi, 2d partition function inΩ-background and vortex/instanton correspondence, JHEP 12 (2015) 110 [arXiv:1509.08630] [INSPIRE]. 27

    2015 arXiv

  2. [10]

    http://dx.doi.org 10.1007/978-1-4612-6398-2

    J.Humphreys,Introduction to Lie Algebra and Representation Theory, Springer Verlag, New York. http://dx.doi.org 10.1007/978-1-4612-6398-2

    doi

  3. [11]

    Hori and M

    K. Hori and M. Romo, Exact Results In Two-Dimensional (2,2) Super- symmetric Gauge Theories With Boundary, arXiv:1308.2438 [INSPIRE]

    arXiv

  4. [12]

    Korepin, N.M

    V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scat- tering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1993)

    1993

  5. [13]

    Kimura and R.-D Zhu,Bethe/gauge correspondence for SO/Sp gauge theories and open spin chains,JHEP03(2021) 227 [arxiv:2012.14197]

    T. Kimura and R.-D Zhu,Bethe/gauge correspondence for SO/Sp gauge theories and open spin chains,JHEP03(2021) 227 [arxiv:2012.14197]

    2021 arXiv

  6. [14]

    Lee and N

    K. Lee and N. Lee,Defects and type D relativistic Toda lattice for some 5d gauge theories,J. High Energ. Phys.02(2025), 210

    2025

  7. [15]

    Longhi, F

    P. Longhi, F. Nieri and A. Pittelli,Localization of 4dN= 1theories on D2×T 2,J. High Energ. Phys.12(2019), 147

    2019

  8. [16]

    Maulik and A

    D. Maulik and A. Okounkov,Quantum groups and quantum cohomology, arXiv preprint arXiv:1211.1287 (2012)

    2012 arXiv

  9. [17]

    Nakajima,Quiver varieties and tensor products, ii, in Symmetries, integrable systems and representations, pp

    H. Nakajima,Quiver varieties and tensor products, ii, in Symmetries, integrable systems and representations, pp. 403–428. Springer, 2013

    2013

  10. [18]

    Shatashvili,Supersymmetric vacua and Bethe ansatz,Nucl

    N.Nekrasov and S.L. Shatashvili,Supersymmetric vacua and Bethe ansatz,Nucl. Phys. B Proc. Suppl.192–193(2009) 91 [arXiv:0901.4744] [INSPIRE]

    2009 arXiv

  11. [19]

    Nekrasov and S.L

    N. Nekrasov and S.L. Shatashvili,Quantum integrability and supersym- metric vacua,Prog. Theor. Phys. Suppl.177(2009) 105 [arXiv:0901.4748] [INSPIRE]

    2009 arXiv

  12. [20]

    Orlando and S

    D. Orlando and S. Reffert,The Gauge-Bethe Correspondence and Geometric Representation Theory,Lett. Math. Phys.,98(2011) 289 [arXiv:1011.6120]

    2011 arXiv

  13. [21]

    Samaj and Z

    L. Samaj and Z. Bajnok,ˇIntroduction to the statistical physics of integrable many-body systems, Cambridge University Press, Cambridge (2013)

    2013

  14. [22]

    Seiberg and E

    N. Seiberg and E. Witten,Monopoles, duality and chiral symmetry break- ing in N=2 supersymmetric QCD,Nucl. Phys. B431 (1994) 484–550, arXiv:hep-th/9408099

    1994 arXiv

  15. [23]

    Seiberg and E

    N. Seiberg and E. Witten,Electric - magnetic duality, monopole conden- sation, and confinement in N=2 supersymmetric Yang-Mills theory,Nucl. Phys. B426 (1994) 19–52, arXiv:hep-th/9407087. [Erratum: Nucl.Phys.B 430, 485–486 (1994)]. 28

    1994 arXiv

  16. [24]

    Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press (1999)

    M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press (1999)

    1999

  17. [25]

    Takhtajan and L.D

    L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11

    1979

  18. [26]

    Y. Wang, W. L. Yang, J. Cho et al,Off-diagonal Bethe Ansatz for exactly solvable models[M], Springer Berlin Heidelberg(2015)

    2015

  19. [28]

    Z. Xin, J. Cao, W.-L Yang and Y. Wang,Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields, JHEP10(2024) 096 [arXiv:2303.03102]

    2024 arXiv

  20. [29]

    Yoshida and K

    Y. Yoshida and K. Sugiyama,Localization of three-dimensional N = 2 supersymmetric theories onS 1×D 2, Prog. Theor. Exp. Phys. 2020, 113B02. 29

    2020

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