Exploring Topological Defects through Differential Geometry and Homotopy Theory

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Natuurkunde (BSc)

Strubbe, Nick (2024)

Exploring Topological Defects through Differential Geometry and Homotopy Theory

Bachelor thesis | Natuurkunde (BSc)

This thesis investigates the mathematical and physical foundations of topological defects. We first introduce the mathematical background, which consists of the theory of Lie groups and their corresponding Lie algebras, and fibre bundles, principal bundles and connections on principal bundles. We also give an introduction to classical field theory, and present the Lagrangian formalism for fields and Yang-Mills theory. We cover spontaneous symmetry breaking, and we explain how this can lead to topological defects using the Kibble mechanism. Finally, we classify topological defects using homotopy groups, for which we develop the underlying framework.

Topological defect

spontaneous symmetry breaking

Kibble mechanism

gauge theory

classical field theory

Lagrangian formalism

vacuum manifold

Yang-Mills theory

differential geometry

Lie group

Lie algebra

fibre bundle