Written examination in January 2009
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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.
Lecture notes, Chapter 1
Lecture notes, Chapter 2
Lecture notes, Chapter 3
Lecture notes, Chapter 4
Lecture notes, Chapter 5
Lecture notes, Chapter 6