The Geroch group in the Ashtekar formulation

TL;DR

This paper explores the Geroch group within the Ashtekar formulation of general relativity, revealing connections between symmetries, conserved charges, and the algebraic structure in reduced models.

Contribution

It demonstrates the realization of Ehlers and Matzner-Misner symmetries using Ashtekar variables and clarifies the algebraic structure of the Geroch group in this framework.

Findings

The third column of the Ashtekar connection relates to the Ernst potential.

Ehlers' and Matzner-Misner's SL(2,R) symmetries are explicitly constructed.

The loop algebra previously associated with the Geroch group is not its Lie algebra.

Abstract

We study the Geroch group in the framework of the Ashtekar formulation. In the case of the one-Killing-vector reduction, it turns out that the third column of the Ashtekar connection is essentially the gradient of the Ernst potential, which implies that the both quantities are based on the ``same'' complexification. In the two-Killing-vector reduction, we demonstrate Ehlers' and Matzner-Misner's SL(2,R) symmetries, respectively, by constructing two sets of canonical variables that realize either of the symmetries canonically, in terms of the Ashtekar variables. The conserved charges associated with these symmetries are explicitly obtained. We show that the gl(2,R) loop algebra constructed previously in the loop representation is not the Lie algebra of the Geroch group itself. We also point out that the recent argument on the equivalence to a chiral model is based on a gauge-choice which cannot be achieved generically.