Target skills and knowledge: This course will review some mathematical concepts that have 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.

Planned learning activities and teaching methods: Lectures. Assignments.

Exercises

First part, Second part.

Lectures: Lectures will be delivered at the Physics and Astronomy department:

1. Sigma Models.

Symmetry breaking: explicit and spontaneous breakings. 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 theta-vacua. Sphalerons.

9. Complex geometry.

Almost complex manifolds. Complex Manifolds. Symplectic manifolds. Kaehler manifolds. Group structures and differential properties.