Rigorous QQ-system foundation for $\mathrm{SO}(2r)$ spin chains: operator-level proofs for general representations, Wronskian Bethe completeness, and a $D_r$ Baxter $TQ$ analogue

Integrable spin chains with $\mathrm{SO}(2r)$ ($D_r$) Yangian symmetry still lack the complete Baxter Q-function machinery that exists for $A$-type chains. Ferrando, Frassek and Kazakov proposed the missing QQ-system -- Pluecker and spinorial QQ-relations on Q-functions labelled by subsets of $\{1,\dots,2r\}$, Weyl-type and QQ'-type factorisation formulas for transfer matrices, and a single Wronskian Bethe relation on $r+1$ basic Q-functions -- but verified it only at small finite length, observed extra non-physical solutions of the Wronskian Bethe equations, and noted the absence of any $D_r$ analogue of Baxter's $TQ$ equation. This problem asks for the rigorous completion of that program: prove the proposed relations at operator level in full generality, beyond the currently known minuscule-node cases; prove a completeness and selection theorem for the Wronskian Bethe ansatz; and establish a compact quantum-spectral-determinant ($TQ$/quantum-spectral-curve-type) formulation of the $D_r$ chain spectrum. Follow-up work has proven the key factorisation assumption at the minuscule Dynkin nodes and corroborated the extended Q-system at character level, leaving the general operator-level proof, the completeness theorem, and the compact $TQ$ form open.

Background

An integrable spin chain is a quantum system -- here a periodic chain of $L$ sites, each carrying a finite-dimensional representation of a Lie algebra $\mathfrak{g}$ -- with enough commuting conserved quantities that its spectrum can, in principle, be computed exactly. For rational (XXX-type) chains the relevant symmetry is the Yangian $Y(\mathfrak{g})$, a Hopf algebra of non-local charges; the conserved quantities are generated by transfer matrices $T_\lambda(x)$, traces of monodromy matrices built from an $R$-matrix solving the Yang-Baxter equation, with the auxiliary space ranging over finite-dimensional $Y(\mathfrak{g})$-modules $\lambda$. A Lax matrix is a solution $T(x)$ of the Yangian $RTT$ relation $R(x-y)\,T(x)T(y)=T(y)T(x)\,R(x-y)$, which guarantees that the resulting transfer matrices commute; constructing Lax matrices for new auxiliary representations is the basic route to new $Q$- and $T$-operators. Baxter showed that the spectral problem is often best encoded in polynomial Q-functions of the spectral parameter $x$: eigenvalues of transfer matrices and the Bethe ansatz equations (BAE) that determine them are packed into functional relations among Q's. For $\mathfrak{g}=\mathfrak{sl}_n$ ($A$-type) this Q-function theory is essentially complete. There is a full QQ-system: a Hasse diagram (the graph of subsets ordered by inclusion) of Q-functions $Q_I$ labelled by subsets $I$ of Dynkin nodes, tied by bilinear Pluecker-type QQ-relations of the schematic form $Q_I Q_{I\cup\{i,j\}}-Q_{I\cup\{i\}}Q_{I\cup\{j\}}=$ (a shifted pair of $Q_{I\cup\{i\}}$ and $Q_{I\cup\{j\}}$), where a shift $Q^{[n]}$ means $Q(x+n)$. The BAE can moreover be rewritten as a single Wronskian (Casoratian, i.e. discrete Wronskian determinant) relation on $r+1$ basic Q-functions, a formulation whose solutions are in bijection with the physical eigenstates -- proven for rational $\mathfrak{gl}_{m|n}$ chains by Chernyak, Leurent and Volin.

The Kirillov-Reshetikhin modules are the standard family of finite-dimensional $Y(\mathfrak{g})$-modules labelled by a Dynkin node $a$ and an integer $s$ (the 'rectangular' representations); their transfer matrices satisfy the Hirota bilinear fusion relations. For the orthogonal series $D_r=\mathfrak{so}(2r)$, in contrast, the Yangian $R$-matrix and explicit Lax matrices are known only for the symmetric-traceless (node $1$, including the vector representation) and the spinorial (nodes $r-1$ and $r$) members of this family -- the three minuscule nodes of $D_r$ -- so that transfer matrices of the other representations are accessible only through the Hirota equation and character-level tableaux formulas; and no compact Baxter $TQ$ equation -- an equation expressing a transfer matrix through a single 'quantum spectral determinant' Q-function -- is known.

Such $TQ$ equations are the engine of the quantum spectral curve (QSC), the formalism solving the planar spectrum of $\mathcal{N}=4$ super Yang-Mills; a $D$-type analogue is the natural route to a QSC for fishnet-type conformal field theories in $d>4$ dimensions -- highly symmetric bi-scalar gauge theories whose planar Feynman graphs form regular square lattices -- whose conformal symmetry $SO(2,d)$ is genuinely $D$-type (for $d=4$ the accidental isomorphism $SO(2,4)\simeq SU(2,2)$ lets one use $A$-type methods, but no such isomorphism exists for $d>4$).

Ferrando, Frassek and Kazakov proposed the missing $D_r$ QQ-system: a Hasse diagram of Q-functions $Q_I$ labelled by 'acceptable' subsets $I\subset\{1,\dots,2r\}$, i.e. subsets containing at most one member of each pair $\{k,k'=2r-k+1\}$ (so $|I|\le r$, with $2^r$ maximal such sets), related along the tail of the Dynkin diagram by Pluecker-type QQ-relations, with the top two levels $|I|=r-1,r$ factorising into $2^r$ spinorial Q-functions $S_I$ obeying their own QQ-relations. They derived Weyl-type determinant formulas expressing transfer matrices of all symmetric and antisymmetric representations through $r+1$ basic Q-functions, QQ'-type factorisation formulas expressing them through the $2r$ single-index Q-functions, and rewrote the Bethe ansatz as a single Wronskian relation on spinorial Q-functions, one such equation for each choice of a maximal acceptable set $I_r$. All these relations were motivated by asymptotics, character factorisations and Lax-matrix factorisations, and verified against explicit oscillator Q-operators constructed by Frassek at small finite length -- but not proven; and the Wronskian Bethe equations were observed to admit extra solutions beyond the physical Bethe roots.

Problem Statement

The problem is to complete this program: to place the proposed $D_r$ QQ-system on rigorous operator-level footing and to establish it as a complete solution method for the spectrum of rational Yangian $\mathrm{SO}(2r)$ spin chains of arbitrary rank $r$ -- that is, to bring the $D$-type Q-function description of the spectrum to the same status as the $A$-type one. Concretely, for the rational $\mathrm{SO}(2r)$ chain: (1) Operator-level proof of the QQ-system. Prove, at operator level and in full generality (arbitrary rank $r$ and chain length $L$), the Pluecker QQ-relations, the spinorial QQ-relations, the QQ'-type factorisation relations and the Weyl-type transfer-matrix formulas proposed by Ferrando, Frassek and Kazakov (their equations (5.1), (5.8), (8.7), (8.14)). Explicit operator constructions (Lax matrices solving the Yangian $RTT$ relation and Q-operators) are currently available only at the minuscule nodes of $D_r$ -- the vector node $1$ and the two spinor nodes $r-1$ and $r$ -- and none is known for the rectangular Kirillov-Reshetikhin modules at the non-minuscule nodes $2\le a\le r-2$; overcoming this gap is a central open obstacle, but the proof is not required to proceed through any particular construction route. (2) Completeness of the Wronskian Bethe ansatz. Resolve the structure of the Wronskian Bethe relation (their equation (7.2)): characterise the extra solutions it admits beyond the physical Bethe roots, and prove a completeness and selection statement -- a rule specifying which of the $2^r$ choices of the maximal acceptable set $I_r$, or which combination of the corresponding Wronskian equations, isolates exactly the physical spectrum, i.e. places the polynomial solutions in the correct analytic class (asymptotics fixed by the global charges) in bijection with the eigenstates of the chain, with solution counts matching the Hilbert-space dimension. (3) $D_r$ analogue of Baxter's $TQ$ equation. Determine whether a compact quantum-spectral-determinant form exists for $D_r$ -- a minimal closed system of functional equations on the basic Q-functions, equivalent to the full QQ-system, from which all transfer-matrix eigenvalues and Q-functions follow -- and construct it; an admissible answer is likewise a proof that the QQ'-type relations already constitute such a minimal system, closing the question in the negative direction. These three deliverables are one objective in three linked steps: without (1) the proposed relations remain conjectural; without (2) the Wronskian Bethe ansatz is not a reliable solution method; and (3) is the compact form that would let the system be used the way $TQ$ equations and quantum spectral curves are used for $A$-type problems.

References

1. G. Ferrando, R. Frassek and V. Kazakov, QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains, J. High Energ. Phys. 02 (2021) 193, DOI 10.1007/JHEP02(2021)193, arXiv:2008.04336, https://arxiv.org/abs/2008.04336
2. R. Frassek, Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains, Nucl. Phys. B 956 (2020) 115063, DOI 10.1016/j.nuclphysb.2020.115063, arXiv:2001.06825, https://arxiv.org/abs/2001.06825
3. S. Ekhammar, H. Shu and D. Volin, Extended systems of Baxter Q-functions and fused flags I: simply-laced case, arXiv:2008.10597 (2020, preprint), https://arxiv.org/abs/2008.10597
4. R. Frassek, I. Karpov and A. Tsymbaliuk, Transfer matrices of rational spin chains via novel BGG-type resolutions, Commun. Math. Phys. 400 (2023) 1-82, DOI 10.1007/s00220-022-04620-6, arXiv:2112.12065, https://arxiv.org/abs/2112.12065
5. E. Frenkel and D. Hernandez, Extended Baxter Relations and QQ-Systems for Quantum Affine Algebras, Commun. Math. Phys. 405 (2024) 190, DOI 10.1007/s00220-024-05051-1, arXiv:2312.13256, https://arxiv.org/abs/2312.13256
6. P. Koroteev and A. M. Zeitlin, q-opers, QQ-systems, and Bethe Ansatz II: Generalized minors, J. Reine Angew. Math. (published online 2023), DOI 10.1515/crelle-2022-0084, arXiv:2108.04184, https://arxiv.org/abs/2108.04184
7. D. Chernyak, S. Leurent and D. Volin, Completeness of Wronskian Bethe Equations for Rational $\mathfrak{gl}_{m|n}$ Spin Chains, Commun. Math. Phys. 391 (2022) 969-1045, DOI 10.1007/s00220-021-04275-9, arXiv:2004.02865, https://arxiv.org/abs/2004.02865
8. Y. Jiang, Y.-C. Liu, Y. Miao and Z.-X. Tan, Rational Q-systems for integrable spin chains without $U(1)$ symmetry, SciPost Phys. 20 (2026) 175, DOI 10.21468/SciPostPhys.20.6.175, arXiv:2512.01551, https://arxiv.org/abs/2512.01551
9. R. Frassek and A. Tsymbaliuk, Rational Lax Matrices from Antidominantly Shifted Extended Yangians: BCD Types, Commun. Math. Phys. 392 (2022) 545-619, DOI 10.1007/s00220-022-04345-6, arXiv:2104.14518, https://arxiv.org/abs/2104.14518
10. R. Frassek and A. Tsymbaliuk, Orthosymplectic superoscillator Lax matrices, Lett. Math. Phys. 114 (2024) 49, DOI 10.1007/s11005-024-01789-w, arXiv:2309.14199, https://arxiv.org/abs/2309.14199
11. Z. Tsuboi, Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system, Nucl. Phys. B 972 (2021) 115563, DOI 10.1016/j.nuclphysb.2021.115563, arXiv:2106.08931, https://arxiv.org/abs/2106.08931
12. Z. Tsuboi, Folding QQ-relations and transfer matrix eigenvalues: Towards a unified approach to Bethe ansatz for super spin chains, Nucl. Phys. B 1005 (2024) 116607, DOI 10.1016/j.nuclphysb.2024.116607, arXiv:2309.16660, https://arxiv.org/abs/2309.16660
13. Y. Boujakhrout and E. H. Saidi, Minuscule ABCDE Lax operators from 4D Chern-Simons theory, Nucl. Phys. B 981 (2022) 115859, DOI 10.1016/j.nuclphysb.2022.115859, arXiv:2207.14777, https://arxiv.org/abs/2207.14777
14. K. Costello, D. Gaiotto and J. Yagi, Q-operators are 't Hooft lines, J. High Energ. Phys. 11 (2024) 003, DOI 10.1007/JHEP11(2024)003, arXiv:2103.01835, https://arxiv.org/abs/2103.01835
15. S. Ekhammar and D. Volin, Bethe algebra using pure spinors, Lett. Math. Phys. 115 (2025) 20, DOI 10.1007/s11005-024-01894-w, arXiv:2104.04539, https://arxiv.org/abs/2104.04539
16. F. Chernikov, S. Ekhammar, N. Gromov and B. Smith, Gluing Quantum Spectral Curves: a two-copy $\mathfrak{osp}(4|2)$ construction, arXiv:2511.09654 (2025, preprint), https://arxiv.org/abs/2511.09654
17. A. Cavaglià, R. Frassek, N. Primi and R. Tateo, On the quantum spectral curve for $\mathrm{AdS}_3\times\mathrm{S}^3\times\mathrm{S}^3\times\mathrm{S}^1$ strings and the $\mathfrak{d}(2,1;\alpha)$ Q-system, arXiv:2511.09635 (2025, preprint), https://arxiv.org/abs/2511.09635