A Feigin-Frenkel theorem with n singularities

TL;DR

This paper extends the Feigin-Frenkel theorem to a setting with n singularities, describing the center of a generalized affine algebra and linking it to G^L-Opers on an n-pointed disk.

Contribution

It provides an algebraic and geometric description of the center for affine algebras with multiple singularities, generalizing the classical Feigin-Frenkel theorem.

Findings

Explicit generators for the center are constructed.

The center is characterized as functions on G^L-Opers over the n-pointed disk.

Factorization properties relate the new isomorphism to the classical one.

Abstract

For a simple Lie algebra g we consider an analogue of the affine algebra ^gk with n singularities, defined starting from the ring of functions on the n-pointed disk. We study the center of its completed enveloping algebra and prove an analogue of the Feigin-Frenkel theorem in this setting. In particular, we first give an algebraic description of the center by providing explicit topological generators; we then characterize the center geometrically as the ring of functions on the space of G^L-Opers over the n-pointed disk. Finally, we prove some factorization properties of our isomorphism, thus establishing a relation between our isomorphism and the usual isomorphism of Feigin-Frenkel.