Noncommutative geometry, topology, and the standard model vacuum

As a ramification of a motivational discussion for previous joint work, in which equations of motion for the finite spectral action of the standard model were derived, we provide a new analysis of the results of the calculations therein, switching from the perspective of spectral triple to that of Fredholm module and thus from the analogy with Riemannian geometry to the premetrical structure of the noncommutative geometry. Using a suggested noncommutative version of Morse theory together with algebraic K theory to analyze the vacuum solutions, the first two summands of the algebra for the finite triple of the standard model arise up to Morita equivalence. We also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix. (c) 2006 American Institute of Physics.