Moduli spaces and the algebra of conformal blocks

TL;DR

This paper investigates the geometric properties of moduli spaces of parabolic G-bundles and proves finite generation of conformal blocks algebras for classical Lie groups, advancing understanding in algebraic geometry and representation theory.

Contribution

It establishes that certain moduli spaces are of Fano type and proves the finite generation of conformal blocks algebras for classical simple Lie groups.

Findings

Moduli spaces are of Fano type for genus g ≥ 3.

Conformal blocks algebras are finitely generated for classical Lie groups.

Results apply to complex smooth projective curves.

Abstract

For a classical simple and simply connected group $G$, let $\mathcal{M}_{G,\omega}$ be the moduli space of $\omega$-semistable parabolic $G$-bundles on a complex smooth projective curve of genus $g$. We prove two results in this article: (1) $\mathcal{M}_{G,\omega}$ is of Fano type when $g\geq 3$; (2) the algebra of conformal blocks on any $n$-pointed stable curve for a classical simple Lie algebra is finitely generated.