Description:

A theory of gauge fields forms the foundation of contemporary particle physics, namely of the Standard Model. The main goal of this course to to acquaint students with the mathematical apparatus required for its geometric description. We will focus on theory of principal fiber bundles and the interpretation of gauge fields as connection forms on principal fiber bundles. All theoretical concepts are demonstrated on particular examples, e.g. frame bundle, Hopf fibration and Yang-Mills field.

Contents:

1. Maxwell equations in the language of differential forms, gauge invariant action, local gauge invariance, minimal interaction with the complex scalar field
2. Lie group actions and their properties, fiber bundles
3. Principal fiber bundles, fundamental vector fields and the vertical subspace
4. Forms valued in vector spaces, forms of affine connections
5. Cartan structure equations, connection forms on principal fiber bundles
6. Smooth distributions and their integrability, horizontal distributions, horizontal lift and parallel transport
7. Exterior covariant derivative, curvature form, integrability of parallel transport, holonomy
8. Local connection and curvature forms, gauge transformation
9. Gauge invariant action, equations of motion of gauge theory, Yang-Mills field as an example
10. Reduction of vector bundles, associated fibration, mass fields in gauge theories.

Seminar contents:

Examples of mathematical structures defined during lectures, applications in theoretical physics.

Recommended literature:

Key references:
[1] J. Lee: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2012.
[2] S. B. Sontz: Principal Bundles: The Classical Case, Springer, 2015.

Recommended references:
[3] M. Fecko: Differential Geometry and Lie Groups for Physicists, Cambridge University Press, 2006.
[4] M. Nakahara: Geometry, Topology and Physics, CRC Press, 2003.

Keywords:

gauge field, Lie group actions, principal fiber bundle, connection and curvature, covariant derivative, parallel transport, Yang-Mills field

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar