Completeness of the Wronskian Bethe ansatz for rational gl(m|n) spin chains in arbitrary highest-weight representations and the exceptional specialisation locus

The theorem of Chernyak, Leurent and Volin (Commun. Math. Phys. 391 (2022) 969) proves, for rational gl(m|n) supersymmetric spin chains whose sites all carry the defining representation, that the algebra of commuting charges (the Bethe algebra) is isomorphic to the polynomial ring defined by Wronskian Bethe equations (WBE) among Baxter Q-functions. This establishes completeness and faithfulness of the Bethe ansatz for every value of the inhomogeneity invariants, with the isomorphism after specialisation to numerical inhomogeneities guaranteed by the sufficient condition that no two inhomogeneities differ by the quantum spacing $\hbar$. The source itself leaves two coupled gaps that constitute this problem: extend the theorem to chains whose sites carry arbitrary finite-dimensional highest-weight representations of gl(m|n), which requires identifying refined quantisation conditions beyond the WBE (known to be insufficient in that generality), and characterise exactly the exceptional locus of inhomogeneity tuples where the specialised isomorphism fails on nilpotent parts. Both gaps remain open as of 2026: substantial partial progress exists only for bosonic gl(n) chains (separation of variables in any compact representation, and rigorous Q-system counting for spin-s and A-type chains) and for supersymmetric chains in the defining or fundamental representation; no completeness theorem exists for general highest-weight supersymmetric representations, and the exceptional locus is still located only up to containment in the critical locus of the master map.

Background

An XXX-type (rational) spin chain with $\mathfrak{gl}_{m|n}$ symmetry is a quantum system whose Hilbert space is a tensor product of $L$ local modules of the Lie superalgebra $\mathfrak{gl}_{m|n}$ — the graded algebra of endomorphisms of $\mathbb{C}^{m|n}$, with $m$ bosonic and $n$ fermionic directions. Its conserved charges are generated by a transfer matrix depending on a spectral parameter $u$, and the full quantum symmetry is the Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$, a Hopf-algebra deformation of the enveloping algebra of the current algebra $\mathfrak{gl}_{m|n}[u]$; the $L$ sites carry its evaluation representations, shifted by site-dependent complex inhomogeneities $\theta_\ell$, and a diagonal boundary matrix $G$ (the twist) encodes quasi-periodic boundary conditions. Coefficients of the twisted transfer matrix and of its fused descendants generate a commutative algebra of operators, the Bethe algebra; diagonalising it solves the spectral problem.

In the analytic Bethe ansatz the joint eigenvalues are encoded in Baxter Q-functions, polynomials in $u$ obeying functional relations. In nested form these are the familiar nested Bethe equations, which suffer from singular and non-physical (spurious) solutions. Chernyak, Leurent and Volin instead use Wronskian Bethe equations (WBE): relations expressing that finite-difference Wronskians — determinants built from Q-functions and their shifts $u\mapsto u\pm\hbar$ — equal known twist polynomials, which by construction admit only physical solutions. For chains whose every site carries the defining representation $\mathbb{C}^{m|n}$ (both with generic twist and twist-less), they prove that the Bethe algebra restricted to a weight subspace of weight $\Lambda$ is isomorphic, over the polynomial ring $\mathbb{C}[\chi]$ in the elementary symmetric polynomials $\chi_\ell$ of the inhomogeneities, to the Wronskian algebra $\mathcal{W}_\Lambda=\mathbb{C}[\chi][c_1,\ldots,c_L]/\mathcal{I}_\Lambda$ defined by the WBE. This yields completeness — the algebraic number of WBE solutions equals $d_\Lambda$, the dimension of the highest-weight subspace, which in the twist-less case is the number of standard Young tableaux of the shape labelling the irreducible representations occurring in the chain via supersymmetric Schur–Weyl duality — and faithfulness, hence maximality of the Bethe algebra as a commutative subalgebra. The bosonic $\mathfrak{gl}_n$ counterpart for the defining representation was proven earlier by Mukhin, Tarasov and Varchenko through the Wronski map.

Two restrictions of this theorem are essential. First, the proof covers only defining-representation sites. For chains in arbitrary finite-dimensional highest-weight local representations the authors expect an extension, but warn that "an analog of the quantisation condition will not be sufficient": as made explicit in the bosonic case by Ryan and Volin, beyond (anti)fundamental representations the single Wronskian quantisation condition admits more solutions than $d_\Lambda$, so extra conditions — expected to be polynomiality requirements on the full Q-system attached to a Young diagram — must be identified and proved to cut the count down exactly to $d_\Lambda$.

Second, the specialisation of the isomorphism to numerical inhomogeneities $\bar\theta$ is proved under the sufficient condition $\bar\theta_\ell+\hbar\neq\bar\theta_{\ell'}$ for $\ell<\ell'$. The paper confines the image $\chi(\Theta_{\rm not})$ of the exceptional set $\Theta_{\rm not}$ — the inhomogeneity tuples $\bar\theta$ where the specialised map $\varphi_{\bar\theta}$ fails to be an algebra isomorphism — inside the critical locus $\chi(\mathrm{Se}_{\rm crit})$ of the master map $c\mapsto\chi$ at which WBE solutions degenerate; away from it distinct solutions label one-dimensional eigenspaces, while at critical points the Bethe algebra may become non-diagonalisable and only the nilpotent (Jordan-block) part of the isomorphism can fail. An explicit $\mathfrak{gl}_2$, $L=3$ example with all inhomogeneities equal shows that the sufficient condition is not necessary. The precise location of $\Theta_{\rm not}$ — necessary and sufficient conditions on $\bar\theta$ — is left open, the authors noting it "would probably require exploration of Yangian representation theory beyond techniques developed in the paper".

Problem Statement

Complete the completeness-and-faithfulness theorem of Chernyak, Leurent and Volin for rational $\mathfrak{gl}_{m|n}$ spin chains along the two directions the source leaves open, which together constitute one objective — a complete completeness theorem for the rational supersymmetric Bethe ansatz:

1. Arbitrary local representations. For $\mathfrak{gl}_{m|n}$ chains whose sites carry arbitrary finite-dimensional highest-weight representations (both with generic twist and twist-less), identify the refined quantisation conditions that generalise the Wronskian Bethe equations — the conditions on the Q-system attached to the underlying Young diagram which select precisely the physical solutions — and prove, with these conditions, that the restricted Bethe algebra is isomorphic over $\mathbb{C}[\chi]$ to the corresponding generalised Wronskian algebra, that the algebraic number of solutions equals the relevant highest-weight subspace dimension for every value of $\chi$, and that the isomorphism specialises to numerical inhomogeneities under explicitly stated conditions.

2. Exceptional specialisation locus. Determine necessary and sufficient conditions on the inhomogeneities $\bar\theta$ (with $\bar\chi$ arbitrary) under which the specialised map $\varphi_{\bar\theta}$ is an isomorphism including nilpotent parts — equivalently, locate $\Theta_{\rm not}$ exactly. This is open already for sites in the defining representation: the known sufficient condition $\bar\theta_\ell+\hbar\neq\bar\theta_{\ell'}$ is provably not necessary, and the only structural information is the containment of $\chi(\Theta_{\rm not})$ inside the critical locus of the master map.

Both components are required for a complete answer, and neither discharges the other: component 1 requires the general-representation isomorphism with its specialisation to numerical inhomogeneities stated under explicit conditions, while component 2 requires the exact locus, a strictly finer demand on the inhomogeneities that no sufficient condition settles. No restriction on $m$, $n$, $L$, or the local representations beyond finite-dimensional highest weight is admitted; partial results confined to special representation classes (bosonic $\mathfrak{gl}_n$, spin-$s$ symmetric, or defining representations) reproduce known cases. The source's outlook also proposes extending the algebraic completeness to non-compact spin chains and the quantum spectral curve of planar $\mathcal{N}=4$ super-Yang-Mills, and extending the geometric Bethe/Gauge realisation to supersymmetric Yangians; those are separable research programs, both open, and not part of this problem's completion boundary.

References

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