Approximating the parallel transport of an induced connection

TL;DR

This paper introduces efficient, order 4 accurate numerical methods for approximating parallel transport of induced connections on sub-bundles, which are simpler and insensitive to trivialisation choices, with applications in computational geometry.

Contribution

The paper presents novel numerical algorithms for parallel transport that outperform naive methods in simplicity and robustness, applicable beyond skyrmion computations.

Findings

Methods achieve accuracy up to order 4.

Algorithms are insensitive to trivialisation choices.

Applicable to a broad range of computational geometry problems.

Abstract

Efficient numerical methods to approximate the parallel transport operators of the induced connection on a sub-bundle of a vector bundle are presented. These methods are simpler than naive applications of a Runge--Kutta algorithm, and have accuracy up to order 4. They have the desirable property of being insensitive to choices of trivialisation of the sub-bundle. The methods were developed in order to solve a problem of computing skyrmions using the Atiyah--Manton--Sutcliffe and Atiyah--Drinfeld--Hitchin--Manin constructions, but are applicable to a broader range of problems in computational geometry.