# DIFFERENTIAL GEOMETRIC METHODS IN PHYSICS: AN INTRODUCTION TO YANG-MILLS
THEORIES

## Course Code: SI2320

## Credits: 7.5hp

The first lecture takes place on Friday, August 29, 10-12, in Theoretical
Physics seminar room A4:1069, AlbaNova. Preliminarily, the lectures will be on Thursdays and Fridays 10-12 every second week,
in periods 1-2. NEXT MEETINGS: Thursday 25/9 and Friday 26/9, 10-12.
Last meetings in period 1: Thursdays and Fridays 10-12: October 2,3,16, and 17.
Period 2: Same times on October 30,31; November 13,14, 27,28, and December 4,5.
Written examination in January 2009

The homework problems of the previous course can be dowloaded here
1.pdf
2.pdf
3.pdf
4.pdf
5.pdf
6.pdf
7.pdf
8.pdf
9.pdf
10.pdf

## Examiner:

Jouko Mickelsson (e-mail jouko@kth.se)
## Prerequisites:

Course SI2370 Relativity Theory is strongly recommended, a good
knowledge of multivariable differential and integral calculus is required.
## Background:

The basic theories in microphysics are based on the assumption
that the fundamental interactions between elementary particles are
described in terms of the so-called gauge fields (Yang-Mills fields);
these are in a certain sense genralizations of the familiar Maxwell field
in electrodynamics. Different models are specified by their characteristic
symmetry groups. In the case of Maxwell the symmetry is the commutative
group U(1) (rotations in the complex plane), in the case of the strong
nuclear forces it is the group SU(3) (complex unimodular 3x3 matrices),
and for electroweak forces (unified electromagnetic and weak interactions)
it is SU(2) x U(1). These groups appear in a localized form, i.e., at
each space-time point one can have an independent 'gauge transformation'.

The research in gauge field theories has been an important unifying link
between physics and mathematics. The problems in physics have given new
directions in topology and differential geometry and on the other hand
new results in mathematics have been quickly employed by physicists.
A googd example of this is the recent activity on Langlands program. The program
originates from number theory, involving deep ideas of Robert Langlands on field extensions and
representation theory, but it was later transformed to a geometric setting.
In the geometric disguise it was realized a couple of years ago that there are very
interesting links to conformal field theory and supersymmetric Yang-Mills theory
in physics and since then there has been intensive cross disciplinary research
on this circle of problems, see e.g. E. Frenkel: Lectures on the Langlands program and conformal
field theory, arXiv-hepth/0512172 and A. Kapustin and E. Witten: Electric-magnetic duality and the
geometric Langland's program, ArXiv-hepth/0604151.

## Contents:

1. Manifolds, vector fields, differential forms. 2. Riemannian
geometry, parallel transport and curvature. 3. Lie groups and Lie algebras.
4. Principal bundles and connections. 5. Yang-Mills functional, instantons
and monopoles. 6. Interaction between matter and gauge fields, Dirac's
equation. 7. Quantum effects; the determinant of a Dirac operator.
8. Quantum mechanical symmetry breaking and cohomology of gauge groups.
## Examination:

Written examination.
## Literature:

**Textbook:**
Theodore Frankel:
The geometry of physics. An introduction.
Cambridge University Press, Cambridge. Second Edition 2004.
xxvi+694 pp. ISBN: 0-521-83330-2; 0-521-53927-7
Also useful:
M. Nakahara: Geometry, Topology, and Physics, Graduate
Student Series in Physics, Institute of Physics Publ. Second Edition
2003. xxii+573 pp. ISBN: 0-7503-0606-8 58-02
Lecture notes, Chapter 1

Lecture notes, Chapter 2

Lecture notes, Chapter 3

Lecture notes, Chapter 4

Lecture notes, Chapter 5

Lecture notes, Chapter 6

### Other recommended reading:

- M. Goeckeler and T. Schuecker: Differential Geometry, Gauge Theories,
and Gravity, Cambridge University Press 1987.
- Jouko Mickelsson, Current Algebras and Groups, Plenum Press 1989.
- R. Bishop and S. Goldberg: Tensor Analysis on Manifolds, Dover Publ. 1980.

jouko@kth.se