Differential geometry and gauge theory

Diploma(s)

Master ICFP

M2 ICFP

Spring semester

Level Master 2 3 ECTS - English

Cours en option pour le parcours de Physique théorique

Instructor(s) Rémi LECLERCQ ( Université Paris-Saclay )

Education office

enseignement@phys.ens.fr

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.

Additional info can be found https://www.imo.universite-paris-saclay.fr/~remi.leclercq/Enseignement/M2Phy/index.html

Syllabus

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].

Prerequisities

A tiny bit of differential calculus is required (you will need to recall how to compute gradients, divergences and curls).

Evaluation

Oral exam + short report

Bibliography

[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.