One-loop divergences in simple supergravity: Boundary effects

Abstract

This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with a boundary, within the framework of -function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin 3/2, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-3/2 potentials, jointly with boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms. The axial gauge-averaging functional is used, which is then sufficient to ensure selfadjointness. One, thus, finds that the contributions of ghost and gauge modes vanish separately. Hence the contributions to the one-loop wave function of the universe reduce to those (0) values resulting from physical modes only. Another set of mixed boundary conditions, motivated instead by local supersymmetry and first proposed by Luckock, Moss, and Poletti, is also analyzed. In this case the contributions of gauge and ghost modes do not cancel each other. Both sets of boundary conditions lead to a nonvanishing (0) value, and spectral boundary conditions are also studied when two concentric three-sphere boundaries occur. These results seem to point out that simple supergravity is not even one-loop finite in the presence of boundaries.

Bicyclic Semigroup (Monoid)

“…By contrast, when the mixed boundary conditions (19)-(23) motivated by local supersymmetry are imposed, the contributions of gauge and ghost modes do not cancel each other. Both sets of boundary conditions lead to a non-vanishing ζ(0) value, and spectral boundary conditions have also been studied when two concentric three-sphere boundaries occur [2,3]. These results seem to point out that simple supergravity is not even one-loop finite in the presence of boundaries, apart from the case of those particular backgrounds bounded by two surfaces where pure gravity alone is one-loop finite.…”

“…The work of the authors [2,3] has found that, in the case of non-local boundary conditions (10) and (11) in the axial gauge, the contributions of ghost and gauge modes vanish separately, and hence contributions to the one-loop wave function of the universe reduce to those ζ(0) values resulting from transversetraceless perturbations only. By contrast, when the mixed boundary conditions (19)-(23) motivated by local supersymmetry are imposed, the contributions of gauge and ghost modes do not cancel each other.…”

“…This has been proved by finding a non-vanishing two-loop counterterm. It therefore seems that the issue of finiteness of quantum supergravity is far from being settled, both in the presence [1][2][3] and in the absence [5] of boundaries. Hopefully, the methods of modern spectral geometry [4] will make it possible, in the years to come, to perform a more careful assessment of all possible counterterms at various orders in perturbation theory.…”

“…Since recent calculations [2] have led to the correct values of the one-loop divergences for spin-1 2 fields, Euclidean Maxwell theory and Euclidean quantum gravity for various choices of boundary conditions, including all ghost and gauge modes whenever appropriate, the last open problem in this respect is the evaluation of one-loop divergences in simple supergravity in the presence of boundaries [2,3].…”

“…Thus, the specification of the whole gravitino potential at the boundary would lead to an over-determined problem. One thus has a choice of spectral or local boundary conditions for Rarita-Schwinger potentials, and they are both studied here in the presence of three-sphere boundaries [3].…”