Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics

Tue, 23 Nov 2021 09:00:00 +0000

Throughout this series, we have built differential geometry from the ground up: manifolds, tangent spaces, differential forms, integration, Riemannian metrics, connections, and curvature. A recurring theme has been the tangent bundle $$TM$$ — the collection of all tangent spaces glued together into a single geometric object. The Levi-Civita connection is a rule for differentiating sections of $$TM$$ (i.e., vector fields), and the Riemann curvature tensor measures the non-commutativity of this differentiation.

But the tangent bundle is just one example of a much more general construction: a fiber bundle. And the Levi-Civita connection is just one example of a connection on a vector bundle. This generalization is not merely aesthetic — it is the mathematical language of gauge theory, the framework underlying all of modern particle physics. Electromagnetism, the weak force, the strong force, and even gravity can all be described as connections on appropriate bundles, with their dynamics governed by curvature.

Gauge-Theory on Chen Kai Blog

Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics