Representation of a gauge group as motions of a Hilbert space

Abstract

This is a survey article based on the author’s Master thesis on affine representations of a gauge group. Most of the results presented here are well-known to differential geometers and physicists familiar with gauge theory. However, we hope this short systematic presentation offers a useful self-contained introduction to the subject. In the first part we present the construction of the group of motions of a Hilbert space and we explain the way in which it can be considered as a Lie group. The second part is about the definition of the gauge group, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔊</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> , associated to a principal bundle, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> . In the third part we present the construction of the Hilbert space where the representation takes place. Finally, in the fourth part, we show the construction of the representation and the way in which this representation goes to the set of connections associated to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> .