Conformal blocks in algebraic geometry

TL;DR

This paper surveys the algebraic geometric theory of conformal blocks, detailing their construction from vertex operator algebras, their properties, and their relation to line bundles on moduli spaces of principal bundles.

Contribution

It provides a comprehensive summary of conformal blocks' construction, properties, and their connection to line bundles on moduli spaces, emphasizing explicit rank and Chern class computations.

Findings

Conformal blocks define vector bundles on moduli spaces of curves.

Factorization and sewing properties are fundamental to conformal blocks.

Line bundles on moduli of principal bundles can be understood via conformal blocks.

Abstract

These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe their fundamental properties -- such as factorization and sewing -- which imply that conformal blocks define vector bundles on moduli spaces of curves whose ranks and Chern classes can be computed explicitly. The second aim is to describe how line bundles on moduli of principal bundles over a curve -- and their sections -- can be understood via (twisted) conformal blocks for (twisted) affine Lie algebras.