We generalize previous works on the Dirac eigenvalues as dynamical variables of 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 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.
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 A. Connes, Noncommutative Geometry (Academic Press, New York, 1994)
 P. Gilkey, Invariant Theory, The Heat Equation and the Atiyah-Singer Index Theorem (Chemical Rubber Company, Boca Raton, FL, 1995)
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