Computing the Roots of Twisting Sheaves over the Projective Line arising from Monodromy Representations

TL;DR

This paper develops methods to compute the roots of twisting sheaves over the projective line derived from monodromy representations, providing explicit results for specific cases of dimension and number of punctures.

Contribution

It introduces a systematic approach to determine the roots of twisting sheaves associated with monodromy representations, extending known results to new cases.

Findings

Computed roots for all finite-dimensional cases when m=2

Determined roots for all representations of dimension less than 3 when m=3

Provided explicit decomposition of vector bundles with logarithmic connections

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 compute the roots for all finite-dimensional $\rho$ when $m = 2$ and for all $\rho$ of dimension less than $3$ when $m = 3$.