Geometric Methods in Physics 2
Code  Completion  Credits  Range 

02GMF2  Z,ZK  5  2+2 
 Lecturer:
 Jan Vysoký, Libor Šnobl (guarantor)
 Tutor:
 Jan Vysoký, Libor Šnobl (guarantor)
 Supervisor:
 Department of Physics
 Synopsis:

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 apparautus 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 YangMills field.
 Requirements:

knowledge equivalent to Geometric Methods in Physics 1
 Syllabus of lectures:

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

Examples of mathematical structures defined during lectures, appliactions in theoretical physics.
 Study Objective:

Knowledge:
Geometry of classical gauge field theories.
Skills:
Application of contemporary geometrical methods in theoretical physics
 Study materials:

Key references:
[1] M. Fecko: Differential geometry and Lie groups for physicists, Cambridge University Press, 2006.
[2] S. B. Sontz: Principal Bundles: The Classical Case, Springer, 2015.
Recommended references:
[3] J. Lee: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2012.
[4] M. Nakahara: Geometry, topology and physics, CRC Press, 2003.
 Note:
 Timetable for winter semester 2020/2021:
 Timetable is not available yet
 Timetable for summer semester 2020/2021:
 Timetable is not available yet
 The course is a part of the following study plans:

 Matematická fyzika (compulsory course of the specialization)