Metric Versus Ashtekar Variables in Two Killing Field Reduced Gravity

TL;DR

This paper explores the relationship between metric and Ashtekar formulations of two Killing field reduced Einstein gravity, revealing their equivalence and differences in the associated chiral models and Poisson brackets.

Contribution

It demonstrates the equivalence of SL(2,R)/SO(2) and SL(2,C) chiral models in reduced gravity and analyzes the impact on non-ultralocal terms in Poisson brackets.

Findings

Both chiral models are equivalent.

Transition affects non-ultralocal terms in Poisson brackets.

Disappearance of non-ultralocal terms in the SL(2,C) model.

Abstract

The relation between the SL(2,R)/SO(2)- and the SL(2,C)-chiral model that naturally arise within the metric respectively the Ashtekar formulation of two Killing field reduced Einstein gravity is revealed. Both chiral models turn out to be completely equivalent even though the transition from the coset- to the SL(2,C)-model is accompanied by a disappearance of the non-ultralocal terms in the Poisson brackets.