# Pääsäiekimpuista ja Yang-Mills -teoriasta, kevät 2011

### (Principal bundles and Yang-Mills theory; can be given in English)

### Luennoitsija

### Laajuus

10 op.

### Tyyppi

Syventävä opinto

### Esitietovaatimukset

Matematiikan perus- ja aineopinnot; erityisesti useamman muuttujan funktioiden differentiaalilaskenta;

perusasioita ryhmäteoriasta (esim. Algebra 1).

### Luentoajat

Viikot 3-9 ja 11-18 ma 10-12, ti 10-12 B322, lisäksi laskuharjoituksia 2 viikkotuntia.

Pääsiäisloma 21.-27.4.

### Kokeet: NEXT EXAMINATION: Thursday, June 9, 10 - 14

### Kirjallisuus

Luentomuistiinpanot sekä T. Frankel: The Geometry of Physics, An Introduction

(Cambridge University Press, Second Edition 2004), ja

M. Nakahara: Geometry, Topology, and Physics (Graduate Student Series in Physics, Institute of Physics Publ. Second Edition 2003)

### Taustaa ja sisällöstä (in English)

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.

### Luentomuistiinpanot (Lecture notes)

PART 1

PART 2

PART 3

PART 4

PART 5

PART 6

### Ilmoittaudu

Unohditko ilmoittautua? Mitä tehdä.

### Laskuharjoitukset

Ryhmä |
Päivä |
Aika |
Paikka |
Pitäjä |
---|---|---|---|---|

1. |
ke |
10-12 |
B321 |
Henri Lipponen |

The set of solved problems should be given to Henri before the

start of the exercise session. You get a compensation from the succesfully solved problems,

corresponding maximally to the credits from one examination problem. First session 10-12,

January 26.

### Homework problems

Homework set 1

Homework set 2

Homework set 3

Homework set 4

(For Mayer-Vietoris, see http://en.wikipedia.org/wiki/Mayer-Vietoris_sequence

or the book by Bott and Tu)

Homework set 5

Homework set 6 (March 16)

Homework set 7 (March 23)

Homework set 8 (April 6)

Homework set 9 (April 13)

Homework set 10 (April 20)

Homework set 11 (May 4)