Topology in Physics

Course in spring 2018

Lecturers

Lectures: Marcel Vonk and Hessel Posthuma
Exerciseclasses: Niek de Kleijn and Beatrix Muhlmann

Place and time

Lectures: Tuesdays, 13.00-15.00, for the location please check datanose or the mastermath page.
Exercise classes: Tuesday 15.00-17.00 SP G0.23-G0.25

We will have a break in week 13 (March 27)

Aim of the course

This course is aimed at both physics and mathematics students. The aim of the course is to demonstrate how many current mathematical methods, that can be very broadly classified as "topological", play an important role in quantum field theory and other areas of modern physics, and conversely how ideas from physics are applied in modern mathematics. The course will focus on the following topics:

Characteristic classes (Chern-Weil)
Anomalies
Fermions and Dirac operators
Index theorems and their "physics proof"
If time permits, several further topics on the border line of mathematics and physics could be covered, such as topological quantum field theories, Chern-Simons theories and knot invariants.

Prerequisites

Some background knowledge in both physics and mathematics is assumed. On the mathematics side, we will assume some basic knowledge of:

Manifolds
Lie groups (following the simultaneous course in Lie Groups at the UvA math department is sufficient)
Topology and cohomology (cohomology will be briefly reviewed during the course)

On the physics side, we will assume some basic knowledge of:

Classical mechanics
Quantum mechanics
Quantum field theory

If you would like to know whether your background knowledge suffices to follow this course (or what you should do as a preparation if this is not the case), feel free to contact one of the lecturers.

Exam and homework

A written exam at the end of the course. Each week there will be homework consisting typically of one of the exercises of the exercise class. Homework counts as 30% (best 11 out of 14).

Lecture Notes / Literature

We will provide lecture notes.

Week by week schedule

Lecture 1: Calculus on manifolds (Lecture Notes) Exercises: Exercise sheet 1 Animations for these exercises: double pendulum (Exercises 1 & 6) hinge mechanism (Exercise 1)
Lecture 2: Path integrals, Maxwel theory (Lecture Notes) Exercises: Exercise sheet 2
Lecture 3: de Rham cohomology as a topological invariant. (Lecture Notes) Exercises: Exercise sheet 3
Lecture 4: Gauge theories (Lecture Notes) Exercises: Exercise sheet 4
Lecture 5: Bundles and Connections (Lecture Notes) Exercises: Exercise sheet 5
Lecture 6: Principal bundles, instantons & the Berry phase (Lecture Notes) Exercises: Exercise sheet 6
Lecture 7: Characteristic classes of vector bundles, Chern numbers(Lecture Notes) Exercises: Exercise sheet 7
Lecture 8: Chern-Simons theory, fermions & path integrals(Lecture Notes) Exercises: Exercise sheet 8
Lecture 9: Fermions and the Dirac operator(Lecture Notes) Exercises: Exercise sheet 9
Lecture 10: Elliptic operators and the index theorem(Lecture Notes) Exercises: Exercise sheet 10
Lecture 11: Dirac operators and supersymmetry(Lecture Notes) Exercises: Exercise sheet 11
Lecture 12: A path integral for the index(Lecture Notes) Exercises: Exercise sheet 12
Lecture 13: Spinors (Lecture Notes) Exercises: Exercise sheet 13
Lecture 14: The physics proof of the Atiyah-Singer index theorem(Lecture Notes) Exercises: Exercise sheet 14
Lecture 15: The geometry of gauge fields and a bit of knot theory (Lecture Notes) Exercises: Exercise sheet 15

Exam
The exam is on Thursday, May 31, 11.00-14.00 in SP C1.110. Here you can find a (mock exam).
Update: This was the (exam), and here is the (correctionmodel)