Geometric Methods in Theoretical Physics
- a.y. 2015/16 and 2018/19
- Galilean School (LM students)
This course is no longer running; the page is kept as an archive.
General information
- Target skills and knowledge
- This course reviews mathematical concepts with interesting non-trivial applications in modern physics. The main objective is to show when and how mathematical rigour becomes effective in giving a better description of physical phenomena.
- Learning activities
- Lectures. Assignments.
- Exercises
- First part, second part (PDF).
Class schedule
| Day | Time | Room |
|---|---|---|
| Monday | 17:00–19:30 | Aula B |
Syllabus
1. Sigma models
Symmetry breaking: explicit and spontaneous breaking. Sigma models.
2. Homogeneous manifolds
Group manifolds. Cosets, coset representatives, coset manifolds. Pions.
3. Homotopy
Skyrmions. Fundamental group. Homotopy groups.
4. Fibre bundles
Fibre bundles. Connection, local connection, curvature, local curvature.
5. Simple topological solutions: solitons
The kink. Solitons with more scalars: skyrmions. Solitons in gauge theories. The vortex. Strings. Non-topological solitons.
6. Monopoles
SU(2) monopoles. Fibre-bundle interpretation. Other models. Dyons.
7. Tunnelling
Decay of metastable vacua in QM. Instantons. Determinant zero modes. Decay of the false vacuum in scalar field theory. Thin-wall approximation.
8. Instantons and sphalerons in gauge theories
Euclidean Yang–Mills theories. Instantons in Yang–Mills theories. Classical vacua and θ-vacua. Sphalerons.
9. Complex geometry
Almost complex manifolds. Complex manifolds. Symplectic manifolds. Kähler manifolds. Group structures and differential properties.