Monodromy Representations: Decomposing Rank-Three Bundles over the Projective Line with Three Marked Points

TL;DR

This paper develops a method to explicitly compute the roots of rank-three vector bundles with logarithmic connections over the projective line minus three points, based on monodromy representations, enhancing understanding of their decompositions.

Contribution

It introduces a novel approach using the monodromy derivative to determine the roots of rank-three bundles with three marked points, providing explicit decompositions.

Findings

Computed roots for all three-dimensional monodromy representations with three punctures.

Established a method to decompose rank-three bundles over the projective line.

Enhanced understanding of vector bundle decompositions via monodromy data.

Abstract

Given a monodromy representation $\rho$ of the projective line minus $m$ points, one can extend the resulting vector bundle with connection map canonically to a vector bundle with logarithmic connection map over all of the projective line. Now, since vector bundles split as twisting sheaves over the projective line, the focus of this work regards knowing the exact decomposition; i.e., computing the roots. Particularly, we use the monodromy derivative to compute the roots for all three-dimensional $\rho$ when $m = 3$.