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Yang-Mills equations of motion for the Higgs sector of \(\operatorname{SU}(3)\)-equivariant quiver gauge theories. (English) Zbl 1311.81180
Summary: We consider \(\operatorname{SU}(3)\)-equivariant dimensional reduction of Yang-Mills theory on spaces of the form \(\mathbb{R} \times \operatorname{SU}(3)/H\), with \(H\) equals either \(\operatorname{SU}(2) \times \operatorname{U}(1)\) or \(\operatorname{U}(1) \times \operatorname{U}(1)\). For the corresponding quiver gauge theory, we derive the equations of motion and construct some specific solutions for the Higgs fields using different gauge groups. Specifically, we choose the gauge groups \(\operatorname{U}(6)\) and \(\operatorname{U}(8)\) for the space \(\mathbb{R} \times \mathbb{C}P^2\), as well as the gauge group \(\operatorname{U}(3)\) for the space \(\mathbb{R} \times \operatorname{SU}(3)/\operatorname{U}(1) \times \operatorname{U}(1)\), and derive Yang-Mills equations for the latter one using a spin connection endowed with a nonvanishing torsion. We find that a specific value for the torsion is necessary in order to obtain nontrivial solutions of Yang-Mills equations. Finally, we take the space \(\mathbb{R} \times \mathbb{C}P^1 \times \mathbb{C}P^2\) and derive the equations of motion for the Higgs sector for the \(\operatorname{U}(3m+3)\) gauge theory.{
©2010 American Institute of Physics}

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
81V22 Unified quantum theories
16G20 Representations of quivers and partially ordered sets
58J52 Determinants and determinant bundles, analytic torsion
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