Unified Gauge Field Theory and Topological Transitions

TL;DR

This paper explores conditions for a unified gauge field theory incorporating gravity, using higher-dimensional compactifications, topological indices, and analyzing topological transitions in gauge and spacetime structures.

Contribution

It introduces a framework for unifying interactions including gravity using compactification, topological indices, and topological transitions in higher-dimensional gauge theories.

Findings

Conditions for a unified gauge theory with gravity are formulated.

Topological Euler number transitions in compactified spaces are analyzed.

Partition functions for configurations are constructed and discussed.

Abstract

The search for a Unified description of all interactions has created many developments of mathematics and physics. The role of geometric effects in the Quantum Theory of particles and fields and spacetime has been an active topic of research. This paper attempts to obtain the conditions for a Unified Gauge Field Theory, including gravity. In the Yang Mills type of theories with compactifications from a 10 or 11 dimensional space to a spacetime of 4 dimensions, the Kaluza Klein and the Holonomy approach has been used. In the compactifications of Calabi Yau spaces and sub manifolds, the Euler number Topological Index is used to label the allowed states and the transitions. With a SU(2) or SL(2,C) connection for gravity and the U(1)*SU(2)*SU(3) or SU(5) gauge connection for the other interactions, a Unified gauge field theory is expressed in the 10 or 11 dimension space. Partition functions for the sum over all possible configurations of sub spaces labeled by the Euler number index and the Action for gauge and matter fields are constructed. Topological Euler number changing transitions that can occur in the gauge fields and the compactified spaces, and their significance are discussed. The possible limits and effects of the physical validity of such a theory are discussed.