Vector bundles on curves and generalized theta functions: recent results and open problems

TL;DR

This survey reviews recent advances in understanding vector bundles on curves and their associated generalized theta functions, highlighting the proof of the Verlinde formula and open problems in the field.

Contribution

It summarizes recent progress and open questions regarding the dimensions and properties of spaces of sections of determinant line bundles on Riemann surfaces.

Findings

Proof of the Verlinde formula for these spaces

Advances in understanding the geometry of vector bundles on curves

Identification of key open problems in the area

Abstract

Riemann surface carries a natural line bundle, the determinant bundle. The space of sections of this line bundle (or its multiples) constitutes a natural non-abelian generalization of the spaces of theta functions on the Jacobian. There has been much progress in the last few years towards a better understanding of these spaces, including a rigorous proof of the celebrated Verlinde formula which gives their dimension. This survey paper tries to explain what is now known and what remains open.