Brongers, Bram (2021) An Exploration of Gauge Theory and the Atiyah-Singer Index Theorem. Bachelor's Thesis, Mathematics.
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Abstract
This thesis is concerned with the mathematics that underlies gauge theories in physics, which will serve as a prelude to the Atiyah-Singer index theorem (index theorem for short). What this means is that we will study connections and their curvature on vector bundles and principal bundles. In the process of introducing these concepts, we will also acquaint ourselves with Chern-Weil theory, which provides a powerful tool to study vector bundles, and is one of the key ingredients in our exposition of the index theorem. With the mathematical preliminaries covered, we will give physical examples of gauge theories (Yang-Mills, Chern-Simons). After doing so, we return to the abstract realm of mathematics to familiarise ourselves with Clifford algebras and spinor bundles, to be used in an application of the index theorem later on, and to complete our mathematical framing of the standard model of particle physics. We provide the Lagrangian of the standard model of particle physics with this knowledge, completing our notion of a gauge theory. Then we turn our attention to the index theorem, explaining the various terms involved in the statement of the theorem, and sketching the K-theory proof. Finally, we perform a calculation with the index theorem that is relevant to gauge theory, namely computing the dimension of the moduli space of irreducible self-dual Yang-Mills connections.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Seri, M. and Veen, R.I. van der |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 19 Jul 2021 13:37 |
| Last Modified: | 19 Jul 2021 13:37 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/25344 |
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