Differential geometry and gauge theory
The goal of these lectures is to define mathematical gauge theory and relate it to gauge theory (from physics).
To achieve this, we will build a mathematical background: a lot of differential geometry, some algebraic topology, a bit of Riemannian geometry, and an ounce of category theory. We will also see how to use these effectively in physics.
1 . Basics of differential geometry: Manifolds and vector fields, differential forms, Hodge theory, de Rham cohomology. We will follow [Paulin] (in French) / [Lang] (in English), and [Warner] for the part on Hodge theory.
2 . First peeks at gauge theory: Maxwell’s equations, the Schrödinger equation,Yang-Mills theory. This will be based on [Sontz].
3 . More differential geometry (and a bit of Riemannian geometry): Fiber and principal bundles, connections and curvature. Following [Husemoller], and [GallotHulinLafontaine].
4 . Finally: Mathematical gauge theory, based on [Sontz] and [Husemoller].
A tiny bit of differential calculus is required (you will need to recall how to compute gradients, divergences and curls).
Oral exam + short report
- [GallotHulinLafontaine] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry. Universitext. Springer-Verlag, Berlin, third edition, 2004.
- [Husemoller] Husemöller, Dale Fibre Bundles. Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994.
- [Lang] Lang, Serge Fundamentals of differential geometry. Graduate Texts in Mathematics, 191. Springer-Verlag, New York, 1999.
- [Paulin] Paulin, Frédéric Variétés différentielles et formes différentielles. Lecture Notes.
- [Sontz] Sontz, Stephen Bruce Principal bundles. The classical case. Universitext, Springer, Cham, 2015.
- [Warner] Warner, Frank W. Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
See also https://www.imo.universite-paris-saclay.fr/~remi.leclercq/Enseignement/M2Phy/index.html for more details.