Superconnections for electroweak and extensions, and the mass of the Higgs

This review describes the theory of internal supersymmetry as first introduced by Ne’eman and Fairlie independently in 1979 and its various developments since then including applications and predictions of the theory to elementary particle physics. In particular we give an explanation of the electroweak isospin and hypercharge spectrum in several generations and a prediction of the Weinberg angle and the Higgs mass derived from the embedding of su ( 2 ) × u ( 1 ) as the even part of the superalgebra su ( 2 / 1 ) . The formulas leading to these predictions are not su ( 2 / 1 ) invariant and we do not present a theory of su ( 2 / 1 ) symmetry breaking. Rather, we use the embedding of su ( 2 ) × u ( 1 ) in su ( 2 / 1 ) to give a geometric interpretation of the Higgs field as part of a superconnection and interpret the quartic polynomial entering into the Higgs mechanism as deriving from a super-Yang–Mills functional. Despite the fact that our formulas are not su ( 2 / 1 ) invariant, we find these geometrical structures interesting and suggestive of the presence of Lie superalgebras in nature.