Parallel Transports in Webs

TL;DR

This paper proves that for connected reductive algebraic groups, webs are holonomically isolated and their parallel transports form a specific Lie subgroup, providing explicit calculations and criteria for holonomical independence.

Contribution

It explicitly calculates the Lie subgroup of parallel transports in webs and establishes criteria for their holonomical independence, independent of trivializations.

Findings

Webs are holonomically isolated for connected reductive groups.

Parallel transports in webs form a specific Lie subgroup.

Criteria for holonomical independence are derived.

Abstract

For connected reductive linear algebraic structure groups it is proven that every web is holonomically isolated. The possible tuples of parallel transports in a web form a Lie subgroup of the corresponding power of the structure group. This Lie subgroup is explicitly calculated and turns out to be independent of the chosen local trivializations. Moreover, explicit necessary and sufficient criteria for the holonomical independence of webs are derived. The results above can even be sharpened: Given an arbitrary neighbourhood of the base points of a web, then this neighbourhood contains some segments of the web whose parameter intervals coincide, but do not include 0 (that corresponds to the base points of the web), and whose parallel transports already form the same Lie subgroup as those of the full web do.