Gaudin subalgebras and stable rational curves

TL;DR

This paper establishes a geometric correspondence between Gaudin subalgebras and the moduli space of stable genus zero curves with marked points, revealing new algebraic and geometric structures.

Contribution

It demonstrates that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves, providing new insights into their geometric and algebraic properties.

Findings

Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves.

The sheaf of Gaudin subalgebras is isomorphic to a sheaf of twisted differential operators.

Spectrum of associated sheaves forms coisotropic subschemes of a twisted cotangent bundle.

Abstract

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n. We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of the moduli space in a Grassmannian of (n-1)-planes in an n(n-1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over the moduli space is isomorphic to a sheaf of twisted first order differential operators. For each representation of the Kohno--Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of the moduli space.