Compactifications of moduli of $G$-bundles and conformal blocks

TL;DR

This paper proves finite generation of conformal blocks algebras over the moduli space of stable curves for certain Lie types and links this to a specific compactification of the moduli space of G-bundles.

Contribution

It establishes finite generation of conformal blocks algebras for genus g≥2 curves and connects these to Schmitt and Muñoz-Castañeda's G-bundle compactification.

Findings

Conformal blocks algebra is finitely generated for type A and C.

Explicit connection to a known G-bundle compactification.

Results hold for stable curves of genus g≥2.

Abstract

For a stable curve of genus $g\geq 2$ and simple Lie algebra of type A or C, we show that the conformal blocks algebra $\mathcal{A}$ on $\overline{\mathcal{M}}_g$ is finitely generated and establish an explicit connection to Schmitt and Mu\~{n}oz-Casta\~{n}eda's compactification of the moduli space of $G$-bundles.