Topological Dynamics and Grand Unified Theory

A formalism is presented to construct a non-perturbative Grand Unified Theory when gravitational Planck-scale phenomena are included. The fundamental object on the Planck scale is the threetorus T 3 from which the known properties of superstrings, such as the geometric action and duality, follow directly. The low energy theory is 11-dimensional and compactification to a Lorentzian fourmanifold is an automatic feature of the unified model. In particular, the simply-connected K3 Calabi-Yau manifold follows naturally from the model and provides a direct link with M-theory. The high energy theory is formulated on a T 3 lattice with handles which exhibits the necessary symmetry groups for the standard model, and yields a consistent amplitude for the cosmic microwave background fluctuations. The equation of motion for the supersymmetric, unified theory is derived and leads to the Higgs field. This formulation predicts a remnant, scalar topological defect of mass ≤ 9mPlanck/46 from the Planck epoch, which is a candidate for dark matter. Finally, it is shown that if the universe is quantum-mechanical, its spatial dimension is equal to three and the laws of nature are Lorentz invariant when gravity can be neglected.