Gaudin models and moduli space of flower curves

TL;DR

The article develops a unified framework for three Gaudin models—rational homogeneous, rational inhomogeneous, and trig—by constructing their compactified parameter spaces via moduli spaces of curves, specifically $\overline M_{n+1}$, $\overline M_{n+2}$, and the cactus-flower space $\overline F_n$. It shows that trig algebras degenerately encompass inhomogeneous ones through a flat, operadic family over the degeneration space $\overline\mathcal F_n$, with a central fiber governed by cactus flower data, thereby linking integrable systems to geometric representation theory and the quantum cohomology of affine Grassmannian slices. The work develops universal Gaudin algebras, their reductions, and their actions on tensor products of finite-dimensional representations, and it studies spectra, monodromy, and real-form conditions to obtain simple spectra and crystal-structure interpretations. These constructions yield a coherent picture connecting Bethe ansatz, Howe duality, opers, and affine Grassmannian geometry, with conjectural ties to the Mau-Woodward compactification and crystal monodromies. The results provide robust tools for understanding degenerations among integrable systems and their geometric avatars, along with explicit descriptions of limit subalgebras and their Hilbert-Poincaré data.

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.

Figures (7)

• Figure 1: A point of $\overline M_{9+1}$ (a cactus curve), a point of $\overline \mathfrak{t}_9$ (a flower curve), and a point of $\overline F_9$ (a cactus flower curve).
• Figure 2: A planar binary forest $\tau$ and the corresponding cactus flower curve. On ${\mathcal{W}}_\tau$, we have the following regular functions: $\nu_{14}, \nu_{24}, \nu_{34},$$\delta_{12}, \delta_{13}, \delta_{23},$$\mu_{142}, \mu_{143}, \mu_{243}, \mu_{132}$.
• Figure 3: This is a curve $C \in \overline M_{7+1}$. We have $n = 7$, $m= 4$, and $\mathcal{B} = (\{1\}, \{2,3\}, \{4\}, \{5,6,7\})$. Each component contributes the algebra shown and the whole curve gives $\mathcal{A}(C) = \Delta^\mathcal{B}(\mathcal{A}(w_1,w_2,w_3,w_4)) \mathcal{A}(z_2,z_3) \mathcal{A}(z_5,z_6,z_7)$.
• Figure 4: This is a curve $C \in \overline M_{6+2}$. We have $n = 6$, $m= 4$, and $\mathcal{B} = (\{0\}, \{1,2\}, \{3\}, \{4,5,6\})$. Each component contributes the algebra shown and the whole curve gives $\mathcal{A}^{trig}_\theta(C) = \Delta^{\mathcal{B}'}(\mathcal{A}^{trig}_\theta(w_1, w_2, w_3)) \mathcal{A}(z_1, z_2) \mathcal{A}(z_4, z_6, z_6)$.
• Figure 5: This is a curve $C \in \overline M_{6+2}$ where $z_0$ does not lie on the same component as $z_{6+1}$. We have $n = 6$, $m= 4$, and $\mathcal{B} = (\{0,1,2,3\}, \{4\} , \{5\},\{6\})$. Each component contributes the algebra shown and the whole curve gives $\mathcal{A}^{trig}_\theta(C) = \psi_\theta(j_{\mathcal{B}}^{6+1} \mathcal{A}(0,z_1, z_2, z_3))) \psi_\theta( \Delta^\mathcal{B}( \mathcal{A}(0, z_4, z_5, z_6))$.

Theorems & Definitions (192)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Conjecture 1.6
• Theorem 1.7
• proof
• Conjecture 1.8
• Conjecture 1.9