  In this paper we consider the Hilbert-Einstein-Dirac functional, whose
critical points are pairs, metrics-spinors, that satisfy a system coupling the
Riemannian and the spinorial part. Under some assumptions, on the sign of the
scalar curvature and the diameter, we prove a compactness result for this class
of pairs, in dimension three and four. This can be seen as the equivalent of
the study of compactness of sequences of Einstein manifolds as in
\cite{And0,N}. Indeed, we study the compactness of sequences of critical points
of the Hilbert-Einstein-Dirac functional which is an extension of the
Hilbert-Einstein functional having Einstein manifolds as critical points.
Moreover we will study the second variation of the energy, characterizing the
horizontal deformations for which the second variation vanishes. Finally we
will exhibit some explicit examples.

Differential Geometry;Analysis of PDEs