Bethe subspaces and wonderful models for toric arrangements
Aleksei Ilin, Leonid Rybnikov

TL;DR
This paper studies Bethe subspaces related to toric arrangements and their compactifications, revealing their geometric structure and connections to integrable models, with extensions to quantum cohomology conjectured for future work.
Contribution
It introduces a geometric framework for Bethe subspaces via wonderful models, extending their parameter space and linking to integrable systems and quantum cohomology.
Findings
Bethe subspaces form a vector bundle over the minimal wonderful model.
Extension of Bethe subspaces to the minimal wonderful model is faithful for classical types.
Connections established with Gaudin subalgebras and hypertoric varieties.
Abstract
We study the family of commutative subspaces in the trigonometric holonomy Lie algebra , introduced by Toledano Laredo, for an arbitrary root system . 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 , 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 of the toric arrangement described by De Concini and Gaiffi, thus giving a compactification of the parameter space for Bethe subspaces. For classical types , we show that this extension is faithful. As a special case, when is of type…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology
