On local constraints of D = 4 Euclidean supergravity in terms of Dirac eigenvalues

It has recently been shown that in order to have Dirac eigenvalues as obser vables of Euclidean supergravity, certain constraints should be imposed on the covariant phase space as well as on Dirac eigenspinors. We investigate the relationships among the constraints in the first set and argue that the se relationships are not linear. We also derive a set of equations that sho uld be satisfied by some arbitrary functions that enter as coefficients in the equation expressing the linear dependency of the constraints in order t hat the second set of constraints be linearly independent.