M2 Physique Théorique | Differential geometry and gauge theory

Table of Contents

1 Description

These lectures are part of a mathematical course oriented towards theoretical physics graduate students, see also this page at ENS for more context.

Factual info:

  • Lectures occur on Tuesdays 14:00-17:00
  • at Jussieu, in room 14-24-205

The subject is differential geometry and gauge theory and the main goal of these lectures is to define mathematical gauge theory and relate it to gauge theory (from physics). This topic is covered by numerous books or survey articles with (rather quite) different proportions of details in maths and physics respectively, see for instance [BaezMuniain94], [EguchiGilkeyHanson80], [RudolphSchmidt17, I & II] and [Sontz15]. The latter will guide our approach.

On our way to gauge theories, we will build a mathematical background (made of a lot of differential geometry, some algebraic topology, a bit of Riemannian geometry, and an ounce of category theory) which we will hopefully see how to use effectively in physics.

So we start with lectures on the basics of differential geometry following [Paulin1] (in French, see also [Lang99] and [Sharpe97] in English) and [Warner83] for the bit on Hodge theory.

  1. Manifolds
  2. From vector fields to tangent bundles
  3. From cotangent bundles to differential forms
  4. De Rham cohomology
  5. Hodge theory

Then we see how differential geometry can be used effectively in physics and then have first peeks at gauge theory, following [Sontz15, Chapters 12-13].

  1. Maxwell's equations
  2. Schrödinger's equation and Yang–Mills theory

In order to relate these elements of gauge theory to mathematical objects, we need to know more about principal bundles, their connections and related notions, which we will learn in [Husemoller94, Chapter 2-4], and [Paulin2] (in French) / [Sharpe97] and [GallotHulinLafontaine04].

  1. From fiber bundles to principal G-bundles (and back)
  2. Connections and their curvature

Finally, we will get to the the climax following [Sontz15, Chapter 14] and [Husemoller94, Chapter 7].

  1. Mathematical gauge theory

Requirement of differential calculus, see [Cartan67] in French or English: you need to know your grad, your div and your curl so that I can convince you that you don't have to compute some of their compositions.

2 Bibliography

2.1 Maths

  1. [Cartan67] Cartan, Henri Calcul différentiel. (French) Hermann, Paris 1967
  2. Cartan, Henri Differential calculus. Exercises by C. Buttin, F. Rideau and J. L. Verley. Translated from the French. Hermann, Paris; Houghton Mifflin Co., Boston, Mass., 1971.
  3. [GallotHulinLafontaine04] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry. Universitext. Springer-Verlag, Berlin, third edition, 2004.
  4. [Husemoller94] Husemöller, Dale Fibre Bundles. Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994.
  5. [Lang99] Lang, Serge Fundamentals of differential geometry. Graduate Texts in Mathematics, 191. Springer-Verlag, New York, 1999.
  6. [Paulin1] Paulin, Frédéric Variétés différentielles et formes différentielles
  7. [Paulin2] Paulin, Frédéric Groupes et géométries
  8. [Sharpe97] Sharpe, Richard W. Differential geometry. Cartan's generalization of Klein's Erlangen program. Graduate Texts in Mathematics, 166. Springer-Verlag, New York, 1997.
  9. [Warner83] 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.

2.2 Relation between gauge theories in maths and physics

  1. [BaezMuniain94] Baez John; Muniain, Javier P. Gauge fields, knots and gravity. Series on Knots and Everything – Vol. 4, World Scientific, Singapore, 1994.
  2. [EguchiGilkeyHanson80] Eguchi, Tohru; Gilkey, Peter B.; Hanson, Andrew J. Gravitation, gauge theories and differential geometry. Phys. Rep. 66 (1980), no. 6, 213–393.
  3. [RudolphSchmidt17] Rudolph, Gerd; Schmidt, Matthias Differential geometry and mathematical physics. Part II. Fibre bundles, topology and gauge fields. Theoretical and Mathematical Physics. Springer, Dordrecht, 2017.
  4. [Sontz15] Sontz Stephen Bruce Principal bundles. The classical case. Universitext, Springer, Cham, 2015.

2.3 Excellent books, related to the content of this course (not explicitly used though)

  1. Hirsch, Morris W. Differential topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1994.
  2. MacLane, Saunders Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York-Berlin, 1971.
  3. Milnor, John W. Topology from the differentiable viewpoint. Based on notes by David W. Weaver. Revised reprint of the 1965 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997.

Author: Rémi Leclercq

Created: 2024-01-16 mar. 11:33

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