Cactus flower spaces and monodromy of Bethe vectors

TL;DR

This paper explores the relationship between cactus flower moduli spaces, operadic coverings, and monodromy of Bethe vectors, connecting geometric, algebraic, and combinatorial structures in Gaudin models and quantum cohomology.

Contribution

It establishes a correspondence between operadic coverings of cactus flower spaces and coboundary monoidal categories, and recovers Kashiwara crystals from Bethe eigenlines in Gaudin models.

Findings

Classifies operadic coverings via coboundary monoidal categories.

Recovers Kashiwara crystals from Bethe eigenlines.

Computes monodromy of Bethe eigenlines for trigonometric Gaudin models.

Abstract

We continue the study of cactus flower moduli spaces $\overline{F}_n$ and Gaudin models started in arXiv:2308.06880, arXiv:2407.06424. We show that isomorphism classes of operadic coverings of the real form $\overline{F}_n(\mathbb{R})$ are naturally one-to-one with equivalence classes of concrete coboundary monoidal categories (i.e. coboundary monoidal categories that admit a faithful monoidal functor to sets) with certain semisimplicity and finiteness conditions. Following the strategy of arXiv:1708.05105, for any complex semisimple Lie algebra $\mathfrak{g}$, we recover Kashiwara $\mathfrak{g}$-crystals, as a concrete coboundary category, from the coverings given by Bethe eigenlines for inhomogeneous Gaudin models. Using this, we compute the monodromy of Bethe eigenlines for trigonometric Gaudin models over two different real loci. In the particular case of minuscule highest weights, this can be regarded as combinatorial version of the wall-crossing conjecture of Bezrukavnikov and Okounkov for quantum cohomology of symplectic resolutions in the case of minuscule resolutions of slices in the affine Grassmannian.