  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 ~ delta^{ij}.
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.
