Gauge and spacetime connections in the Plebanski formulation of complex general relativity

TL;DR

This paper explores the relationship between gauge and spacetime connections within the Plebanski formulation of complex general relativity, clarifying how these structures are interconnected in a gauge-invariant, diffeomorphism-invariant framework.

Contribution

It provides a detailed analysis of how the Levi-Civita connection emerges from the gauge-theoretic variables in the Plebanski formulation, establishing their relationship.

Findings

Clarifies the link between gauge and spacetime connections

Shows how the Levi-Civita connection arises in this formulation

Provides a framework for relating internal gauge structures to spacetime geometry

Abstract

The Plebanski formulation of complex general relativity is given in terms of variables valued in the complexification of the $so(3)$ Lie algebra. Therefore, it is genuinely a gauge theory that is also diffeomorphism-invariant. For this reason, the way that the Levi-Civita connection emerges from this formulation is not direct because both the internal (gauge) and the spacetime connections are geometrical structures a priori not related, there is not a natural link between them. Any possible relationship must be put in by hand or must come from extra hypotheses. In this paper, we analyze the correct relationship between these connections and show how they are related.