Prof. Dr. Mark Hamilton

Lecture: Mathematical Gauge Theory I

Gauge theories play an important role in modern physics and mathematics. This course is an introduction to the mathematical foundations underlying such theories. The main topics include: Lie groups and Lie algebras, group actions, principal fibre bundles, vector bundles, connections and curvature, spinors and Dirac operators, and the Yang-Mills functional.

Depending on time and the interests of the audience we will also cover some applications in theoretical physics, like the mathematical foundations of the Standard Model of elementary particles, spontaneous symmetry breaking and the Higgs mechanism of mass generation.

• LSF entry
• Registration: Please register for the course in Moodle (password: bundle)
• Online format: This course is partly a reading course, based on my book Mathematical gauge theory (MGT) (see link below; please let me know if you do not have access to the pdf). The lectures will be asynchronous and consist of recorded videos that will be uploaded to Moodle every week. In the lectures I cover some topics from MGT in more detail, calculate examples, etc. There will also be a live tutorial, starting in the second week of the lecture period.
• Prior knowledge: Linear algebra, calculus, basic knowledge of topology and manifolds (can be improved parallel to the course)

Plan of lectures

The lectures are based on the following sections of the book Mathematical gauge theory:

• 1. week (April 12 - April 16): 1.1 - 1.3
• 2. week (April 19 - April 23): 1.4, 1.5
• 3. week (April 26 - April 30): 1.7, 2.1
• 4. week (May 3 - May 7): 2.1
• 5. week (May 10 - May 14): 2.2 - 2.5
• 6. week (May 17 - May 21): 3.2, 3.3
• 7. week (May 24 - May 28): 3.4, 3.5
• 8. week (May 31 - June 4): 3.7, 3.8, 4.1, 4.2
• 9. week (June 7 - June 11): 4.2, 4.3, 4.5
• 10. week (June 14 - June 18): 4.7, 5.1, 5.2
• 11. week (June 21 - June 25): 5.2, 5.4 - 5.6
• 12. week (June 28 - July 2): 5.8, 5.9
• 13. week (July 5 - July 9): 5.12, 5.13, 7.2, 7.5
• 14. week (July 12 - July 16): 7.2, 7.3, 7.6

Literature

Some references are (further references will be provided during the course):

• H. Baum, Eichfeldtheorie. Eine Einführung in die Differentialgeometrie auf Faserbündeln, Springer Spektrum (2014). Link to full text in library
• D. Bleecker, Gauge theory and variational principles, Addison-Wesley Publishing Company (1981).
• M. Hamilton, Mathematical gauge theory. With applications to the Standard Model of particle physics, Springer International Publishing (2017). Link to full text in library
• S. Kobayashi, K. Nomizu, Foundations of Differential Geometry I, II, Interscience Publishers, (1963-1969).
• W. Ziller, Lie Groups. Representation Theory and Symmetric Spaces, Lecture Notes, University of Pennsylvania, Fall 2010.