Nonlinear Realization of the Local Conform-Affine Symmetry Group for Gravity in the Composite Fiber Bundle Formalism

TL;DR

This paper develops a gauge theory of gravity using a nonlinear realization of the Conform-Affine group within the composite fiber bundle formalism, deriving geometric structures and dynamical equations.

Contribution

It introduces a novel nonlinear realization approach for the Conform-Affine symmetry in gravity, incorporating auxiliary fields and boundary invariants within the fiber bundle framework.

Findings

Derived coframe fields and gauge connections for the theory

Constructed boundary topological invariants as gravitational Lagrangians

Obtained Bianchi identities, field equations, and gauge currents

Abstract

A gauge theory of gravity based on a nonlinear realization (NLR) of the local Conform-Affine (CA) group of symmetry transformations is presented. The coframe fields and gauge connections of the theory are obtained. The tetrads and Lorentz group metric are used to induce a spacetime metric. The inhomogenously transforming (under the Lorentz group) connection coefficients serve as gravitational gauge potentials used to define covariant derivatives accommodating minimal coupling of matter and gauge fields. On the other hand, the tensor valued connection forms serve as auxillary dynamical fields associated with the dilation, special conformal and deformational (shear) degrees of freedom inherent in the bundle manifold. The bundle curvature of the theory is determined. Boundary topological invariants are constructed. They serve as a prototype (source free) gravitational Lagrangian. The Bianchi identities, covariant field equations and gauge currents are obtained.