SU(2) structures in four dimensions and Plebanski formalism for GR

An SU(2) structure in four dimensions can be described as a triple of 2-forms Sigma^i in Lambda^2(M), i=1,2,3 satisfying Sigma^i wedge Sigma^j ~ deltaⁱʲ. Such a triple defines a Riemannian signature metric on M. An SU(2) structure is said to be integrable if the holonomy of this Riemannian metric is contained in SU(2). It is well-known that this is the case if and only if the 2-forms are closed dSigma^i=0. The main purpose of the paper is to analyse the second order in derivatives diffeomorphism invariant action functionals that can be constructed for an SU(2) structure. The main result is that there is a unique such action functional if one imposes an additional requirement that the action is also SU(2) invariant, with SU(2) acting on the triple Sigma^i as in its vector representation. This action functional has a very simple expression in terms of the intrinsic torsion of the SU(2) structure. We show that its critical points are SU(2) structures whose associated metric is Einstein. The action we describe has also a first order in derivatives version, and we show how this is related to what in the physics literature is known as Plebanski formalism for GR.