Gaudin models and moduli space of flower curves

TL;DR

This paper introduces a family of commutative subalgebras in the universal enveloping algebra related to Gaudin models, explores their degenerations, and connects them to moduli spaces of flower curves, with applications to quantum integrable systems.

Contribution

It constructs and studies the family of trigonometric Gaudin subalgebras, linking them to moduli spaces of flower curves and quantum cohomology, and analyzes their degenerations and actions.

Findings

Trigonometric Gaudin subalgebras form a universal family parameterized by moduli spaces.

Degenerations of these subalgebras relate to inhomogeneous Gaudin models.

The subalgebras act on tensor products of finite-dimensional modules, inducing monodromy actions.

Abstract

We introduce and study the family of trigonometric Gaudin subalgebras in $U g^{\otimes n}$ for arbitrary simple Lie algebra $g$. This is the family of commutative subalgebras of maximal possible transcendence degree that serve as a universal source for higher integrals of the trigonometric Gaudin quantum spin chain attached to $g$. We study the parameter space that indexes all possible degenerations of subalgebras from this family. In particular, we show that (rational) inhomogeneous Gaudin subalgebras of $ U g^{\otimes n}$ previously studied in \cite{ffry} arise as certain limits of trigonometric Gaudin subalgebras. Moreover, we show that both families of commutative subalgebras glue together into the one parameterized by the space $\overline{\mathcal F}_n$, which is the total space of degeneration of the Deligne-Mumford space of stable rational curves $ \overline M_{n+2} $ to the moduli space of cactus flower curves $ \overline F_n $ recently introduced in \cite{iklpr}. As an application, we show that trigonometric Gaudin subalgebras act on tensor products of irreducible finite-dimensional $g$-modules without multiplicities, under some explicit assumptions on the parameters in terms of two different real forms of $\overline M_{n+2}$. This gives rise to a monodromy action of the affine cactus group on the set of eigenstates for the trigonometric Gaudin model. We also explain the relation between the trigonometric Gaudin model and the quantum cohomology of affine Grassmannians slices.