Gauge field theories: various mathematical approaches

TL;DR

This paper reviews modern mathematical frameworks like differential geometry, noncommutative geometry, and Lie algebroids for formulating gauge field theories, highlighting their similarities and differences.

Contribution

It compares various advanced mathematical approaches to gauge theories, emphasizing their common patterns and improvements over traditional paradigms.

Findings

Different mathematical frameworks provide rigorous descriptions of gauge theories.

A common mathematical pattern underlies these diverse approaches.

Each approach offers unique insights into gauge field formulations.

Abstract

This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common mathematical pattern on which the paper concludes.