Gauge symmetry breaking with S2 extra dimensions

We consider symmetry breaking of arbitrary gauge groups on a six-dimensional space-time which consists of a four-dimensional Minkowski space-time <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>M</a:mi><a:mn>4</a:mn></a:msup></a:math> and a two-dimensional sphere <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>S</c:mi><c:mn>2</c:mn></c:msup></c:math>. We expand the gauge fields in the presence of a nontrivial background unique to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>S</e:mi><e:mn>2</e:mn></e:msup></e:math>. We analyze Kaluza-Klein (KK) modes of the gauge fields and derive the mass spectrum of the KK modes. We found that the gauge fields (not) commuting with the background fields (do not) remain symmetry operators in four dimensions. We also discuss the mass spectrum of the extra-dimensional components of the gauge fields and identify a physical scalar <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ϕ</g:mi></g:math> and a Nambu-Goldstone mode <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>χ</i:mi></i:math>. As a result, we obtain a method to break gauge symmetry due to the nontrivial solution for gauge fields which is a unique feature of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>S</k:mi><k:mn>2</k:mn></k:msup></k:math>.