The Noncommutative Yang–Mills SU(N)-Gauge Theory

Eberhard Zeidler
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Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 15 in Quantum Field Theory III: Gauge Theory, 2011, pp 843-870 from Springer

Abstract: Abstract Fix N=1,2,… Let $\mathcal{G}$ be a closed subgroup of the Lie matrix group GL(N,ℂ) of invertible complex (N×N)-matrices. Then, $\mathcal{G}$ is a Lie group. As a prototype, the reader should have the special case in mind where N=2 and $$\mathcal{G} = SU(2).$$ This gauge group was used by Yang and Mills in 1954. Recall that the Lie group SU(2) consists of all the unitary (2×2)-matrices U with det (U)=1. The corresponding Lie algebra su(2) consists of all the complex (2×2)-matrices A with A †=−A and tr (A)=0. Further examples for the Lie group $\mathcal{G}$ are the Lie groups U(N),SU(N),GL(N,ℂ),SL(N,ℂ), and SO(3) with N=3. We will show how the U(1)-gauge theory from Chap. 13 has to be modified in the case of a noncommutative gauge group $\mathcal{G}$ (e.g., $\mathcal{G}= SU(N)$ with N=2,3,…).

Keywords: Gauge Theory; Vector Bundle; Gauge Transformation; Parallel Transport; Principal Bundle (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-22421-8_16

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