One-Loop Divergences in Simple Supergravity: Boundary Effects

TL;DR

This paper investigates one-loop divergences in simple supergravity with boundaries, revealing that boundary conditions significantly affect finiteness, and suggesting supergravity may not be one-loop finite when boundaries are present.

Contribution

It analyzes how different boundary conditions impact one-loop divergences in simple supergravity, highlighting the role of spectral and local boundary conditions in finiteness.

Findings

Ghost and gauge mode contributions vanish under spectral boundary conditions.

Nonvanishing $z(0)$ values indicate potential divergence issues.

Boundary conditions influence the finiteness of supergravity at one loop.

Abstract

This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with boundary, within the framework of $\zeta$-function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin ${3\over 2}$, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-${3\over 2}$ 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 self-adjointness. 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 $\zeta(0)$ 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 $\zeta(0)$ 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.