Attempt at constructing a model of grand gauge-Higgs unification with family unification

We discuss a possibility whether a model of grand gauge-Higgs unification incorporating family unification in higher dimensions can be constructed. We first extend a five-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>6</a:mn><a:mo stretchy="false">)</a:mo></a:math> grand gauge-Higgs unification model to a five-dimensional <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>S</e:mi><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>7</e:mn><e:mo stretchy="false">)</e:mo></e:math> grand gauge-Higgs unification model compactified on an orbifold <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:msup><i:mrow><i:mi>S</i:mi></i:mrow><i:mrow><i:mn>1</i:mn></i:mrow></i:msup><i:mo>/</i:mo><i:msub><i:mrow><i:mi>Z</i:mi></i:mrow><i:mrow><i:mn>2</i:mn></i:mrow></i:msub></i:mrow></i:math> to obtain three generations of quarks and leptons after symmetry breaking of the larger family unified gauge group. A prescription of constructing a six-dimensional <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>S</k:mi><k:mi>U</k:mi><k:mo stretchy="false">(</k:mo><k:mi>N</k:mi><k:mo stretchy="false">)</k:mo></k:math> grand gauge-Higgs unification model including a five-dimensional <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>S</o:mi><o:mi>U</o:mi><o:mo stretchy="false">(</o:mo><o:mn>7</o:mn><o:mo stretchy="false">)</o:mo></o:math> grand gauge-Higgs unification after compactifying the sixth dimension on an orbifold <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mi>S</s:mi><s:mn>1</s:mn></s:msup><s:mo>/</s:mo><s:msub><s:mi>Z</s:mi><s:mn>2</s:mn></s:msub></s:math> is given. We find a six-dimensional <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>S</u:mi><u:mi>U</u:mi><u:mo stretchy="false">(</u:mo><u:mn>14</u:mn><u:mo stretchy="false">)</u:mo></u:math> grand gauge-Higgs unification model with a set of representations containing three generations of quarks and leptons. Published by the American Physical Society 2024