Topology of electroweak vacua

In the Standard Model, the electroweak symmetry is broken by a complex, SU(2)-doublet Higgs field and the vacuum manifold SU(2)×U(1)/U(1) has the topology of a 3-sphere. We remark that there exist theoretical alternatives that are locally isomorphic, but in which the vacuum manifold is homeomorphic to an arbitrary non-trivial principal U(1)-bundle over a 2-sphere. These alternatives have non-trivial fundamental group and thus feature topologically-stable electroweak strings. An alternative based on the manifold RP3 (with fundamental group Z/2) allows custodial protection of gauge boson masses and their couplings to fermions and has an explicit realisation in the Minimal Composite Higgs Model, in the case of maximal electroweak symmetry breaking. We show that, in common with all alternatives to S3, such models have a problem with fermion masses.