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.

This is a short Chapter introducing an index-free notation for 2+1 gravity, which encodes all objects into 2x2 matrix-valued one-forms. We then describe the Chern-Simons formulation of 2+1 gravity, as well as the pure spin connection formulation.

Historical introduction into the topic of formulations of General Relativity with their associated mathematical formalisms. We review the history of the idea that gravity is geometry, as well as the history of development of main geometical ideas of modern differential geometry. We start with metric geometry of Riemann, Levi and Civita, and describe the main contributions of Cartan. We touch upon the fact that spinors and differential forms are closely related, first observed by Chevalley. We give arguments for why formalisms based on differential forms may be superrior to the metric one.

This is the longest Chapter of the book introducing the "chiral" formulations of 4D GR. The most important concept here is that of self-duality. We describe the associated decomposition of the Riemann tensor, and then the chiral version of Einstein-Cartan theory, together with its Yang-Mills analog. The geometry that is necessary to understand the fact that the knowledge of the Hodge star is equivalent to the knowledge of the conformal metric is explained in some detail. We also describe the different signature pseudo-orthogonal groups in 4 dimensions, and in particular explain that it is natural to put coordinates of a 4D space into a 2x2 matrix, the fact that is going to play central role in the later description of twistors. The notions of the chiral part of the spin connection, as well as the chiral soldering form are introduced. We give an example of a computation of Riemann curvature using the chiral formalism, to illustrate its power. We then describe the Plebanski formulation, as well as its linearisation. This allows to derive the linearisation of the chiral pure connection action, to be studied in full in the following Chapter. We describe coupling to matter in Plebanski formalism, and then some alternative descriptions related to Plebanski formalism.

This Chapter describes, in concise manner, aspects of differential geometry that are necessary to follow the developments of this book. We give several definitions of the concept of the manifold, illustrated by a number of examples. We then define differential forms, which are viewed as the most primitive objects one can put on a manifold. We define their wedge product and the operation of exterior differentiation. We then define the notions necessary to define the integration of differential forms. After this we define vector fields, their Lie bracket, interior product, then tensors. We then describe the Lie derivative. We briefly talk about distributions and their integrability conditions. Define metrics and isometries. Then define Lie groups, discuss their action on manifolds, then define Lie algebras. Describe main Cartan's isomoprhisms. Define fibre bundles and the Ehresmann connections. Define principal bundles and connections in them. Describe the Hopf fibration. Define vector bundles and give some canonical examples of the latter. Describe covariant differentiation. Briefly reivew Riemannian geometry and the affine connection. We end this Chapter with a description of spinors and their relation to differential forms.