Catalog of electroweak scalar manifolds

The local structure of the Higgs sector around the vacuum does not uniquely determine its global properties. Most of the current experimental data provides only local information, which allows for a rich variety of global features, including several distinct topologies of the scalar manifold, and the existence of zero, one, or two fixed points of the symmetry transformations. Here, I provide, under general conditions, a complete classification of realizations of the electroweak symmetry with minimal field content -- the three would-be Goldstone bosons and the Higgs -- and outline some of their physical consequences.