# A Torelli theorem for moduli spaces of principal bundles on curves defined over $\mathbb R$

## TL;DR

This paper proves a Torelli-type theorem showing that the isomorphism class of the moduli space of principal G-bundles on a real algebraic curve uniquely determines the curve itself, extending classical Torelli results.

## Contribution

It establishes a Torelli theorem for moduli spaces of principal bundles over real algebraic curves, a significant extension of classical results to the real setting.

## Key findings

- The moduli space's isomorphism class determines the curve uniquely.
- The result applies to nonabelian groups with one simple factor.
- It extends Torelli theorems to real algebraic geometry.

## Abstract

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$--bundles on $X$ determine uniquely the isomorphism class of $X$.