Real Formulations of Complex Gravity and a Complex Formulation of Real Gravity

TL;DR

This paper explores gauge-invariant theories related to complex and real gravity, reformulating them to reveal their complex structures and connections to traditional real gravity within the Ashtekar framework.

Contribution

It introduces new gauge theories on the Yang-Mills phase space based on specific Lie algebras, demonstrating their equivalence to complex gravity and multiple copies thereof.

Findings

Theories are manifestly real but describe complex gravity.

Reformulations show the connection to real gravity.

Reality conditions are represented by constraint-like terms.

Abstract

Two gauge and diffeomorphism invariant theories on the Yang-Mills phase space are studied. They are based on the Lie-algebras $so(1,3)$ and $\widetilde{so(3)}$ -- the loop-algebra of $so(3)$. Although the theories are manifestly real, they can both be reformulated to show that they describe complex gravity and an infinite number of copies of complex gravity, respectively. The connection to real gravity is given. For these theories, the reality conditions in the conventional Ashtekar formulation are represented by normal constraint-like terms.