Observables of Euclidean Supergravity

The set of constraints under which the eigenvalues of the Dirac operator can play the role of the dynamical variables for Euclidean supergravity is derived. These constraints arise when the gauge invariance of the eigenvalues of the Dirac operator is imposed. They impose conditions which restrict the eigenspinors of the Dirac operator. PACS 04.60.-m, 04.65.+e During the last years, the Dirac operator has become a very powerful tool for studying the geometrical properties of the manifolds as well as the fundamental physics that takes place on them. Not long ago, Connes showed, in the context of noncommutative geometry, that the Dirac operator contains full information about the geometry of space-time [1]. It turned out that this property makes the Dirac operator suitable for describing the dynamics of the general relativity and that, at least in principle, the Dirac operator can be used instead of the metric to describe the gravitational field [2]-[9]. However, there are still some major problems that must be solved before this point of view be totally accepted. One of the most important ones is raised by the fact that the spectrum of the Laplace type operators (like the squared Dirac operator) cannot uniquely determine the topology and the geometry of a four-dimensional Riemannian manifold (a detailed analysis on this topic can be found in [3]). Another problem arises when Riemannian manifolds without boundary are considered. They provide only an idealization of the manifolds encountered in physics since boundary terms play a crucial role in many important physical phenomena, as for example in determining the black-hole entropy from a perturbative evaluation of the path-integral for the partition function [4]. In the spectral geometry approach to Euclidean gravity, a major role is played by the eigenvalues λ’s of the Dirac operator D which are diffeomorphism-invariant functions of the geometry and thus can be considered as the observables of general relativity. In a very recent paper, Landi and Rovelli expressed the Poisson bracket of λ’s in terms of the components of the energy-momentum tensor of the corresponding eigenspinor, and derived the Einstein equations from a spectral action with no cosmological term [5]. This could be a new way to think of quantum gravity, but this method works at the moment being only for the Euclidean theory. There are, however, several attempts to implement