Gaudin subalgebras and stable rational curves

Abstract

AbstractGaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra $\Xmathfrak {t}_{\hspace *{.3pt}n}$. We show that Gaudin subalgebras form a variety isomorphic to the moduli space $\bar M_{0,n+1}$ of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of $\bar M_{0,n+1}$ in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over $\bar M_{0,n+1}$ 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 $\bar M_{0,n+1}$.