Topology in Physics

6 EC

Semester 2, period 4, 5

5354TOIP6Y

Owner Master Physics and Astronomy (joint degree)
Coordinator dr. Marcel Vonk
Part of Master Mathematics, year 1Master Physics and Astronomy, track Theoretical Physics, year 1Master Mathematical Physics, year 1

Course manual 2017/2018

Course content

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)
  • 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, knot invariants and anomalies.

Study materials

Literature

  • Geometry, Topology and Physics - M. Nakahara

Syllabus

Objectives

Students will learn several mathematical methods from topology and how to apply those methods to physics problems. Vice versa, students will learn how to apply intuition from physics as a useful mathematical tool.

Teaching methods

  • Lecture
  • Seminar
  • Self-study

Learning activities

Activity

Number of hours

Hoorcollege

34

Werkcollege

34

Zelfstudie

100

Attendance

Requirements concerning attendance (OER-B).

  • In addition to, or instead of, classes in the form of lectures, the elements of the master’s examination programme often include a practical component as defined in article 1.2 of part A. The course catalogue contains information on the types of classes in each part of the programme. Attendance during practical components is mandatory.

    • Additional requirements for this course:

    No additional requirements.

    Assessment

    Item and weight Details

    Final grade

    30%

    Homework

    70%

    Tentamen

    The homework is optional but strongly recommended. If no homework is handed in (or in the exceptional situation where the homework lowers the grade) the exam counts for 100% of the grade. The student only passes if the exam grade is 4.5 or higher and the overall grade is 5.5 or higher.

    Inspection of assessed work

    Contact the course coordinator to make an appointment for inspection.

    Fraud and plagiarism

    The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

    Course structure

    Weeknummer Onderwerpen Studiestof
    1 Calculus on manifolds Lecture notes 1
    2 Path integrals, Maxwel theory Lecture notes 2
    3 de Rham cohomology as a topological invariant Lecture notes 3
    4 Gauge theories Lecture notes 4
    5 Bundles and Connections Lecture notes 5
    6 Principal bundles, instantons & the Berry phase Lecture notes 6
    7 Characteristic classes of vector bundles, Chern numbers Lecture notes 7
    8 Chern-Simons theory, fermions & path integrals Lecture notes 8
    9 No lecture  
    10 Fermions and the Dirac operator Lecture notes 9
    11 Elliptic operators and the index theorem Lecture notes 10
    12 Dirac operators and supersymmetry Lecture notes 11
    13 A path integral for the index Lecture notes 12
    14 Spinors Lecture notes 13
    15 The physics proof of the Atiyah-Singer index theorem Lecture notes 14
    16 The geometry of gauge fields and a bit of knot theory Lecture notes 15

    Timetable

    The schedule for this course is published on DataNose.

    Honours information

    N/A

    Additional information

    This course is also part of the national MasterMath program.

    Contact information

    Coordinator

    • dr. Marcel Vonk