Bethe ansatz equations for principal series modules of the trigonometric Cherednik algebra at critical level: classification of solutions

For any module $M$ of the extended affine Weyl group $W^a$, the trigonometric Cherednik (degenerate double affine Hecke) algebra at critical level produces commuting quantum Hamiltonians for root-system generalizations of quantum spin-particle systems on a circle with delta-function interactions, and the associated eigenfunctions are governed by matrix-valued Bethe ansatz equations $J^k_y(\lambda)m = e^{\lambda(y)}m$ ($y$ in the translation lattice $Y$), expressed through the algebra's normalized intertwiners $J^k_y(\lambda)$. Emsiz, Opdam, and Stokman (2009) derived these equations for arbitrary root systems but solved them only for the trivial module (scalar case) and the left-regular module $\mathrm{Fun}(W^a)$, and proved a single necessary condition (purely imaginary momenta for unitary modules in the repulsive regime). The classification of solutions for the principal series modules $M(t)$ — which for type $A$ and $t=0$ reduce to C. N. Yang's nested Bethe ansatz for the spin-$1/2$ delta gas — remains open.

Background

One-dimensional quantum many-body systems with delta-function interactions are among the fundamental exactly solvable models. Yang (1967) solved the spin-$1/2$ fermion gas with repulsive delta interactions by the nested Bethe ansatz, and Yang–Yang (1969) analyzed the corresponding scalar (spinless) Bethe equations. The Bethe ansatz starts from a plane-wave expansion on a Weyl chamber (a fundamental domain for the action of the finite reflection group) and imposes single-valuedness on the circle; the admissible momenta and internal-spin vectors are then constrained by algebraic Bethe ansatz equations.

Emsiz–Opdam–Stokman (2009) recast this construction for arbitrary affine root systems. Let $V$ be the Euclidean space of an irreducible reduced root system $R$, and let $W^a = W \ltimes Y$ be the extended affine Weyl group (it is a semidirect product of the finite Weyl group $W$ with the translation lattice $Y$, which contains the coroot lattice $Q^\vee$, the lattice spanned by the coroots dual to the roots); the affine Weyl group $W \ltimes Q^\vee$ is the normal subgroup of $W^a$ generated by the reflections in the affine hyperplanes orthogonal to the affine roots. The trigonometric Cherednik algebra $\hat{A}(k)$ (a degenerate, trigonometric version of Cherednik's double affine Hecke algebra) depends on multiplicity parameters $k_a$ attached to the affine roots. At the critical level — the parameter value at which the central constant $c=\sum_a k_a$ vanishes — the center of $\hat{A}(k)$ is large enough that, for every $W^a$-module $M$, it yields a commuting family of differential-reflection (Dunkl-type) operators that descends to a commuting family of quantum Hamiltonians $H^k_M$ on a physical space of $M$-valued functions: a root-system generalization of a quantum spin-particle system on a circle with delta-function interactions. The sign of the parameters $k_a$ distinguishes repulsive ($k_a<0$) from attractive delta interactions.

The spectral analysis is encoded by the normalized intertwiners $J^k_w(\lambda)$ of $\hat{A}(k)$: explicit matrix-valued rational functions of the spectral parameter (momentum) $\lambda$, indexed by $w\in W^a$ and satisfying a cocycle relation. A plane wave propagates to a global eigenfunction $\psi_{\lambda m}=\sum_{w\in W} e^{w\lambda}\otimes J^k_w(\lambda)m$, and Theorem 5.10 of the source shows that $\psi_{\lambda m}$ is $W^a$-invariant (physically: single-valued on the circle) if and only if the Bethe ansatz equations $J^k_y(\lambda)\,m = e^{\lambda(y)}\,m \quad \forall\, y\in Y$ hold (equation (5.7) of the source; $e^{\lambda(y)}$ denotes the exponentiated pairing). For the trivial module these reduce to scalar Bethe equations whose type-$A$ solutions were analyzed by Yang–Yang, and for the module $M=\mathrm{Fun}_\mathbb{C}(W^a)$ of complex functions on the extended affine Weyl group under the left-regular action, Corollary 5.13 of the source gives all solutions explicitly. The principal series modules are the $|W|$-dimensional modules $M(t)=\bigoplus_{w\in W}\mathbb{C}\,v_w(t)$ ($t\in V^*_\mathbb{C}$) on which translations act through the character $t$; $M(t)$ is unitarizable exactly when $t$ is purely imaginary, and for root system $A$ and $t=0$ one recovers the internal spin module of Yang's fermionic delta gas, whose Bethe ansatz equations are Yang's nested Bethe ansatz equations.

Beyond these cases almost nothing is known. The source's Proposition 5.15 is the only general result: if $M$ is unitary, all couplings are repulsive ($k_a<0$), and $m\neq 0$ solves (5.7), then $\lambda$ must be purely imaginary. The source states that describing the solutions of (5.7) for the principal series module $M(t)$ is a natural open problem (its Theorem 5.10 and Proposition 5.15 are stated for general modules, but the classification question is posed only for $M(t)$), and it does not address completeness of the Bethe ansatz (whether the Bethe eigenfunctions span the physical state space) or the associated spectral analysis. Later work settled only scalar or discrete analogs: Emsiz (2010) proved completeness of the Bethe ansatz for the Laplacian on a Weyl alcove with repulsive boundary conditions (the trivial-module case for arbitrary root systems); van Diejen–Emsiz (2017) proved orthogonality of the Bethe eigenfunctions for the hyperoctahedral (type $B/C$) Weyl alcove; van Diejen–Emsiz–Zurrián (2018) proved completeness for an open $q$-boson chain (a discrete $q$-analog); van Diejen–Emsiz–Zurrián (2024) constructed the basic representation of the double affine Hecke algebra at critical level for general affine root systems, providing structural groundwork but no Bethe-ansatz classification. The matrix-valued (module-valued) problem has remained untouched.

Problem Statement

Describe the solutions of the Bethe ansatz equations $J^k_y(\lambda)\,m = e^{\lambda(y)}\,m \quad \text{for all } y \in Y,$ of the trigonometric Cherednik algebra at critical level for the principal series modules $M(t)$ ($t \in V^*_\mathbb{C}$) of the extended affine Weyl group $W^a$. Here $J^k_y(\lambda)$ are the normalized intertwiners acting on the module, $\lambda \in \mathbb{C}^k_{\mathrm{reg}}$ is the spectral parameter (momentum), where $\mathbb{C}^k_{\mathrm{reg}}$ denotes the open subset of the complexified dual space $V^*_\mathbb{C}$ on which the intertwiner matrices $J^k_w(\lambda)$ have no poles, and $Y$ is the translation lattice; solutions are the admissible pairs $(\lambda, m)$, $0 \neq m \in M$, whose plane-wave expansions $\psi_{\lambda m}=\sum_{w\in W}e^{w\lambda}\otimes J^k_w(\lambda)m$ are the joint eigenfunctions of the commuting quantum Hamiltonians of the associated spin-particle system with delta-function interactions on the circle. The source question is to classify or parametrize the solution pairs $(\lambda, m)$ for all principal series modules $M(t)$ — beyond Yang's type-$A$ analysis for $t=0$ — in the repulsive regime ($k_a<0$, where the only known general fact is the necessary condition of Proposition 5.15 that $\lambda$ be purely imaginary for unitary $M$) as well as in the non-repulsive (attractive) regimes where no bound of this kind is available.

The settled cases — the trivial module (scalar Bethe equations, Corollary 5.11 of the source, with completeness for repulsive couplings due to Emsiz 2010) and the regular module $\mathrm{Fun}(W^a)$ (Corollary 5.13 of the source, all solutions explicit) — are excluded from what is asked; the problem is the module-valued (matrix) case, of which Yang's nested Bethe ansatz is the type-$A$, $t=0$ instance.

References

1. E. Emsiz, E. M. Opdam, J. V. Stokman, "Trigonometric Cherednik algebra at critical level and quantum many-body problems", Selecta Mathematica (N.S.) 14 (2009) 571–605. DOI: 10.1007/s00029-009-0516-y. https://arxiv.org/abs/0804.0046
2. C. N. Yang, "Some exact results for the many-body problem in one dimension with repulsive delta-function interaction", Physical Review Letters 19 (1967) 1312–1315. DOI: 10.1103/PhysRevLett.19.1312. https://doi.org/10.1103/PhysRevLett.19.1312
3. C. N. Yang, C. P. Yang, "Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction", Journal of Mathematical Physics 10 (1969) 1115–1122. DOI: 10.1063/1.1664947. https://doi.org/10.1063/1.1664947
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7. J. F. van Diejen, E. Emsiz, I. N. Zurrián, "Completeness of the Bethe Ansatz for an Open q-Boson System with Integrable Boundary Interactions", Annales Henri Poincaré 19 (2018) 1349–1384. DOI: 10.1007/s00023-018-0658-6. https://arxiv.org/abs/1611.05922
8. J. F. van Diejen, E. Emsiz, "Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials", Letters in Mathematical Physics 109 (2019) 89–112. DOI: 10.1007/s11005-018-1101-0. https://arxiv.org/abs/1903.00867
9. J. F. van Diejen, E. Emsiz, I. N. Zurrián, "On the basic representation of the double affine Hecke algebra at critical level", Journal of Algebra and Its Applications 23 (2024) 2450061. DOI: 10.1142/S0219498824500610. https://arxiv.org/abs/2412.09397