# Determinant bundle over the universal moduli space of principal bundles over the Teichm\"uller space

## TL;DR

This paper investigates the curvature properties of the determinant bundle associated with the adjoint vector bundle over the moduli space of stable principal G-bundles as the complex structure of the underlying Riemann surface varies within Teichmüller space.

## Contribution

It provides a detailed analysis of the curvature of the determinant bundle over the universal moduli space as the conformal structure changes.

## Key findings

- Curvature formulas for the determinant bundle derived.
- Insights into the geometric structure of moduli spaces.
- Connections between Teichmüller theory and bundle curvature.

## Abstract

For a Riemann surface $X$ and the moduli of regularly stable $G$-bundles $M$, there is a naturally occuring "$adjoint$" vector bundle over $X \times M$. One can take the determinant of this vector bundle with respect to the projection map onto $M$. Our aim here is to study the curvature of the determinant bundle as the conformal structure on $X$ varies over the Teichm\"uller space.