Segal-Sugawara vectors for the Lie algebra of type $G_2$

TL;DR

This paper provides explicit formulas for Segal-Sugawara vectors for the Lie algebra of type G_2, proving the Feigin-Frenkel theorem, and applying these results to Vinberg's quantization problem and Gaudin model eigenvalues.

Contribution

It introduces explicit formulas for Segal-Sugawara vectors for G_2 and uses them to prove the Feigin-Frenkel theorem and solve Vinberg's quantization problem.

Findings

Explicit formulas for Segal-Sugawara vectors for G_2

Proof of the Feigin-Frenkel theorem for G_2

Formulas for generators of maximal commutative subalgebras

Abstract

Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg's quantization problem by providing formulas for generators of maximal commutative subalgebras of $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\mathfrak{g}$.