Topology in Physics

Course in spring 2019

Lecturers

Lectures: Marcel Vonk and Hessel Posthuma
Exercise classes: Bjarne Kosmeijer and Beatrix Muhlmann

Place and time

Lectures: Tuesdays, 14.00-16.00, SP A1.04. (This is in the big building at Science Park)
Exercise classes: Tuesday 16.00-17.00 in the same room

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 (see also the short recap here)
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). (Here you can find the exam of last year. (Here the solutions)

Lecture Notes / Literature

We will provide lecture notes.

Week by week schedule

Lecture 1: Cohomology of Manifolds Exercises: Exercise sheet 1
Lecture 2: Maxwell theory and differential forms Exercises: Exercise sheet 2
Lecture 3: Principal and vector bundles Exercises: Exercise sheet 3
Lecture 4: Gauge fields and Yang--Mills theory Exercises: Exercise sheet 4
Lecture 5: Connections on principal bundles Exercises: Exercise sheet 5 (New version March 13! Action in 1d) changed. Notice the new deadline!)
Lecture 6: Bundles and connections in physics, Instantons and Berry's phase. Exercises: Exercise sheet 6
Lecture 7: Characteristic classes. Exercises: Exercise sheet 7
March 26: No lecture:
Lecture 8: Chern-Simons theory and Fermions. Exercises: Exercise sheet 8
Lecture 9: The Dirac operator. Exercises: Exercise sheet 9
Lecture 10: The Atiyah-Singer index theorem. Exercises: Exercise sheet 10
Lecture 11: Dirac operators and supersymmetry. Exercises: Exercise sheet 11
Lecture 12: The index as a path integral Exercises: Exercise sheet 12 (For hand-in exercise 4, please see the notes on the exercise sheet)
Lecture 13: Spinors Exercises: Exercise sheet 13
Lecture 14: The path integral computation of the index Exercises: Exercise sheet 14
Lecture 15: Before the break: Anomalies. After the break: Chern-Simons & Knot invariants