Local Supersymmetry in Perturbative Quantum Cosmology

Abstract

In Euclidean supergravity, the spin-3/2 potential has the pair of spatial components $\left( {\psi _i^A , \tilde \psi _i^{A'} } \right)$. We perform a one-loop calculation for gravitinos subject to the following local boundary conditions on S³: $S^3 :\sqrt 2 {}_en_A^{A'} \psi _i^A = \pm \tilde \psi _i^{A'}$. As in chapter five, the background is flat Euclidean space, and the physical degrees of freedom (PDF) are picked out imposing the supersymmetry constraints and choosing the gauge condition: $e_{AA'}^i \psi _i^A = 0,e_{AA'}^i \tilde \psi _i^{A'} = 0$. The boundary conditions are then found to imply the following eigenvalue condition: [J_n+2(E)]² − [J_n+3(E)]² = 0, ∨ _n ≥ 0, with degeneracy (n + 4)(n + 1). Thus we can apply again the technique of chapter eight. The ζ(0) value is given by the one for the massless Majorana spin-1/2 field plus two other terms, leading to the PDF result: ζ(0) = −289/360. Thus for the gravitino field the PDF method and local boundary conditions lead to a result for ζ(0) which is equal to the PDF value one obtains setting equal to zero on S³ all untwiddled coefficients of $\psi _i^A$ and $\tilde \psi _i^{A'}$.

Finally, we show that also extended supergravity theories do not lead to a vanishing result for the total PDF ζ(0). The boundary conditions used in this calculation are Dirichlet for scalar fields, magnetic for spin 1, Dirichlet for the perturbed three-metric for pure gravity, plus the other local boundary conditions for fermionic fields used in chapter eight and in this chapter. However, the question of the one-loop finiteness of extended supergravity, in the presence of boundaries, has not yet been settled. In fact the direct calculation of the full ζ(0), including gauge-averaging and ghost terms for gauge fields, remains a formidable problem.