Minimal extensions of the SM arise either by adding new fields, or by enlarging the local

gauge group (adding a right handed neutrino field constitute its simples extension, something that ameliorate, but not solve the two problems mentioned above).

The action for a topological theory on a manifold M with local

gauge group G is given by:

The sectors contributing to the

gauge group are the 0 sector and the 10 antiholomorphic sets:

For an integer k, let [P.sub.k] [right arrow] [summation]Y be the principal G-bundle induced by kf, and let [G.sub.k] be its

gauge group. If the order of the commutator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is m, then the number of distinct p-local homotopy types for the

gauge groups {[G.sub.k]} is at most [v.sub.p](m) + 1.

For historical reasons the local symmetry group is known as a

gauge group (strictly of the so-called second kind - the corresponding global symmetry is a gauge symmetry of the first kind) and the hope is widely shared that all interactions, including electromagnetic, strong, weak, and even gravitational, can be derived by imposing the appropriate local gauge symmetry.

Since we are dealing with an extended

gauge group, there will be additional gauge bosons, one of them being singly charged, W', due to the chosen charge operator parameter.

The apparent lack of a generic tunable parameter that allows to solve the theory perturbatively (like the electric coupling constant in electrodynamics, or the rank of the

gauge group in large-N Yang-Mills theory) is arguably the single most important obstacle for generic efficient approaches to the physics of strong gravity and black holes.

The very successful Standard Model (SM) local

gauge group SU[(2).sub.L] x U[(1).sub.[gamma]] x SU[(3).sub.C] defines an electroweak (EW) interaction part and a color interaction part.

Among the topics are deformations and cohomology, the

gauge group, strongly homotopy Lie algebras, and operads.

Other subjects explored are topological field theories with non-semisimple

gauge group of symmetry, continuity equations in Riemannian spaces, the non-Hamiltonian nature of nucleon dynamics in an effective field theory, and singular value decomposition and Lanczos potential.

The vector and axial-vector simplified models of (1), (2) are not in general invariant under the full SM

gauge group SU[(3).sub.c] x SU[(2).sub.L] x U[(1).sub.Y] but only under the unbroken subgroup SU[(3).sub.c] x U[(1).sub.e.m.].

Herein I determine the true source of the PMNS matrix elements by using the linear superposition of the generators for 3 discrete binary rotational subgroups of the Standard Model (SM) electroweak

gauge group SU[(2).sub.L], x U[(1).sub.Y].