  We generalize previous works on the Dirac eigenvalues as dynamical variables
of the Euclidean gravity and N=1 D=4 supergravity to on-shell N=2 D=4 Euclidean
supergravity. The covariant phase space of the theory is defined as as the
space of the solutions of the equations of motion modulo the on-shell gauge
transformations. In this space we define the Poisson brackets and compute their
value for the Dirac eigenvalues.
