Towards Z₂-protected gauge--Higgs unification

In theories with flux compactification in eight or higher dimensions, the extra-dimensional components of the gauge field may be regarded as the Higgs field candidates. We suggest a way to protect these components from getting large tree-level masses by imposing a $Z_2$-symmetry acting on compact manifolds and background fields on them. In our scheme the infinite series of heavy KK modes naturally decouples from the light Higgs candidates, whose number is generically larger than one. We also present toy models with three families of leptons, illustrating that the Yukawa sector in our scheme is fairly strongly constrained. In one of these models, one fermion gets a tree-level mass after electroweak symmetry breaking, while two others remain naturally massless at the tree level.