Superconnections and the Higgs Field

Abstract

Within the mathematical framework of Quillen, one interprets the Higgs field as part of the superconnection on a superbundle. We propose to take as superbundle the exterior algebra obtained from a Hermitian bundle with structure group U(n). Spontaneous symmetry breaking appears as a consequence of a non-vanishing scalar curvature. The U(1) Higgs model reformulates the Ginzburg-Landau theory, while the U(2) model relates to the electroweak theory with the relation $g^2=3g4^2$ for the gauge coupling constants, the formula $\sin^2\theta=1/4$ for the Weinberg angle, and the ratio $ m_W^2 : m_Z^2 : m_H^2 = 3 : 4 : 12 $ for the masses (squared) of the W, Z, and Higgs boson (at tree level).