Gauge-Higgs unification models in six dimensions with $S^2/Z_2$ extra space and GUT gauge symmetry

In this article, we review gauge-Higgs unification models based on gauge theories defined on six-dimensional spacetime with $S^2/Z_2$ topology in the extra spatial dimensions. On the extra $S^2/Z_2$ space, non-trivial boundary conditions are imposed. This review considers two scenarios for constructing a four-dimensional theory from the six-dimensional model. One scheme utilizes the SO(12) gauge symmetry with a special symmetry condition imposed on the gauge field, whereas the other employs the E$_6$ gauge symmetry without requiring the additional symmetry condition. Both models lead to a Standard Model-like gauge theory with the SU(3) $\times$ SU(2)$_L$ $\times$ U(1)$_Y$($\times$ U(1)$^2$) symmetry and SM fermions in four dimensions. The Higgs sector of the model is also analyzed. The electroweak symmetry breaking can be realized, and the weak gauge boson and Higgs boson masses are obtained.