Overconnections and the energy-tensors of gauge and gravitational fields

TL;DR

This paper introduces a geometric method to extend connections to bundle of connections, leading to a natural, auxiliary-structure-free energy-tensor for gauge and gravitational fields, clarifying their symmetries and interactions.

Contribution

It presents a novel geometric construction for prolonging connections and defining a natural energy-tensor for interacting gauge and gravitational fields without auxiliary structures.

Findings

A geometric prolongation of connections is developed.

A natural energy-tensor for gauge and gravitational fields is defined.

The symmetry related to the Komar superpotential is characterized.

Abstract

A geometric construction for obtaining a prolongation of a connection to a connection of a bundle of connections is presented. This determines a natural extension of the notion of canonical energy-tensor which suits gauge and gravitational fields, and shares the main properties of the energy-tensor of a matter field in the jet space formulation of Lagrangian field theory, in particular with regards to symmetries of the Poincar\'e-Cartan form. Accordingly, the joint energy-tensor for interacting matter and gauge fields turns out to be a natural geometric object, whose definition needs no auxuliary structures. Various topics related to energy-tensors, symmetries and the Einstein equations in a theory with interacting matter, gauge and gravitational fields can be viewed under a clarifying light. Finally, the symmetry determined by the "Komar superpotential" is expressed as a symmetry of the gravitational Poincar\'e-Cartan form.