Characteristic Classes in Computer Algebra

TL;DR

This paper presents a framework within SageMath for computing characteristic classes of vector bundles using symbolic calculus and the Chern-Weil approach, including implementation details and application examples.

Contribution

It introduces a novel SageMath-based framework for symbolic computation of characteristic classes via connections, with practical implementation and application demonstrations.

Findings

Successful implementation of characteristic classes computation in SageMath

Demonstrated applications in Jupyter Notebook

Discussion on computational cost and efficiency

Abstract

We develop a framework to compute characteristic classes and their forms in the computer algebra system SageMath using symbolic calculus. In order to do this, we make use of the Chern-Weil approach in which characteristic classes of vector bundles in the de Rham cohomology are obtained by arbitrary connections. Along the way, we implement the notion of vector bundles, their sections and connections as well as mixed differential forms in SageMath. We conclude by discussing some application examples exposed in Jupyter Notebook and eventually address the issue of computational cost.