1 Observables of D = 4 Euclidean Supergravity and Dirac

The observables of supergravity are defined to be the phase space gauge invariant objects of the theory. Dirac eigenvalues are obsevables of Euclidean gravity on a compact manifold [1,2,3]. We resume the basic results for super-gravity [4,5], [6,7]. The general setting which we consider is of minimal supergravity in four dimensions on a compact spin manifold without boundary endowed with an Euclidean metric g µν (x) = e a µ (x)e νa (x) where the indices of the tetrad e a µ (x) are spacetime indices µ = 1, · · · , 4 and internal Euclidean indices a = 1, · · · , 4, respectively. They are raised and lowered by the Euclidean metric δ ab. The gravitino is represented by a Euclidean spin-vector field ψ µ (x) and it should be defined by a modified Majorana condition, since the group SO(4) does not admit Majorana spinors. A standard condition is ¯ ψ = ψ T C [8] (see also [9,10]). By definition, the phase space of the theory is the space of the solutions of the equations of motion, modulo the gauge transformations which are: diffeomorphisms in four dimensions, local SO(4) rotations and the local N = 1 supersymmetry. The phase space is covariantly defined and its elements are all pairs (e, ψ) that are solutions of the equations of motion modulo gauge transformations. Therefore, it is sufficient to consider only on-shell supersymmetry and the supersymmetric algebra closes over graviton and gravitino. The observables of the theory are the functions on the phase space. The self-adjoint Dirac operator and the spin connection are defined as follows