6 - Chiral Pure Connection Formulation

Summary

This Chapter describes the chiral pure connection formulation of 4D GR, which is singled out from all other reformulations because of the economy of the description that arises. We first obtain the chiral pure connection Lagrangian, and explain how the metric arises from a connection. We also discuss the reality conditions. We then introduce notions of definite and semi-definite connections, and discuss the question of whether the pure connection action can be defined non-perturbatively. The question is that of selecting an appropriate branch of the square root of a matrix that appears in the action. Many examples are looked at to get a better feeling for how this connection formalism works. Thus, we describe the Page metric, Bianchi I as well as Bianchi IX setups, and the spherically symmetric problem. All these are treated by the chiral pure connection formalism, to illustrate its power. We also give here the connection description of the gravitational instantons, and in particular describe the Fubini-Study metric. We also show how to use the connection formalism to describe some Ricci-flat metrics, and illustrate this on the examples of Schwarzschild and Eguchi-Hanson metrics. We finish with the description of the chiral pure connection perturbative description of GR.