Bethe subspaces and wonderful models for toric arrangements

TL;DR

This work introduces Bethe subspaces in the trigonometric holonomy Lie algebra $t^{trig}_Φ$ for arbitrary root systems and treats them as quadratic components of Bethe subalgebras tied to integrable spin chains. It constructs a compact parameter space via the De Concini–Gaiffi wonderful model $X_Φ$ and proves that the Bethe subspaces extend regularly to $X_Φ$, forming a finite, generically bijective map to a Grassmannian; this extension is faithful for classical types and aligns with known results in type $A_n$. The Bethe subspaces assemble into a vector bundle $\mathcal{Q}$ over $X_Φ$, which is canonically identified with the logarithmic tangent bundle $T X_Φ(-\log D)$, and its restriction to the central fiber recovers the Gaudin subalgebra bundle. The paper develops several key representations of $t^{trig}_Φ$ into Yangians, trigonometric Gaudin algebras, graded affine Hecke algebras, and geometric contexts like hypertoric quantum cohomology, and it lays out conjectures extending the framework to Yangians and to cohomology of Springer resolutions. Overall, $X_Φ$ emerges as the intrinsic parameter space governing degenerations of Bethe-type commutative subalgebras across multiple representation-theoretic settings, with a rich geometric–algebraic structure linking Gaudin systems, quantum cohomology, and spectral theory.

Abstract

We study the family of commutative subspaces in the trigonometric holonomy Lie algebra $t^{\mathrm{trig}}_Φ$, introduced by Toledano Laredo, for an arbitrary root system $Φ$. We call these subspaces \emph{Bethe subspaces} because they can be regarded as quadratic components of \emph{Bethe subalgebras} in the Yangian corresponding to the root system $Φ$, that are responsible for integrals of the generalized XXX Heisenberg spin chain. Bethe subspaces are naturally parametrized by the complement of the corresponding toric arrangement . We prove that this family extends regularly to the minimal wonderful model $X_Φ$ of the toric arrangement described by De Concini and Gaiffi, thus giving a compactification of the parameter space for Bethe subspaces. For classical types $A_n, B_n, C_n, D_n$, we show that this extension is faithful. As a special case, when $Φ$ is of type $A_n$, our construction agrees with the main result of Aguirre--Felder--Veselov on the closure of the set of quadratic Gaudin subalgebras. Our work is also closely related to, and refines in this root system setting, a parallel compactification result of J. Peters obtained for more general toric arrangements arising from quantum multiplication on hypertoric varieties. Next, we show that the Bethe subspaces assemble into a vector bundle over $X_Φ$, which we identify with the logarithmic tangent bundle of $X_Φ$. Finally, we formulate conjectures extending these results to the setting of Bethe subalgebras in Yangians and to the quantum cohomology rings of Springer resolutions. We plan to address this in our next papers.

Theorems & Definitions (64)

• Remark
• Definition 2.3
• Remark
• Lemma 2.4
• proof
• Theorem 2.5
• Definition 2.7
• Proposition 2.8
• Lemma 2.9
• proof