Noncommutative Geometry, Superconnections and Riemannian Gravity as a Low-Energy Theory

A superconnection is a supermatrix whose evenpart contains the gaugepotential one-forms of a localgauge group, while the odd parts contain the (zero-form)Higgs fields breaking the local symmetry spontaneously. The combined grading is thus odd everywhere andthe superconnection can be directly derived from aformulation of Noncommutative Geometry, as theappropriate one-form in the relevant form calculus. The simple supergroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar P$$ \end{document} (4, ℝ) (rank = 3) in Kac' classification (evensubgroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {SL}$$ \end{document}(4,ℝ)) provides themost economical spontaneous breaking of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {SL}$$ \end{document}(4,ℝ) as gauge group leaving just local\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overline {SO}$$ \end{document}(1,3) unbroken. Post-Riemannian SKY gravity thereby yields Einstein's theory asa low-energy (longer range) effective theory. The theoryis renormalizable and may be unitary.