---
author:
- Finn Larsen and
- Pedro Lisbão
bibliography:
- 'ads4\_bibliography.bib'
title: Divergences and Boundary Modes in N=8 Supergravity
---

Introduction and Summary {#sec:introduction}
========================

Divergences in quantum gravity are famously severe and suggestive that long distance physics depends sensitively on the shortest lengths. Supersymmetry mitigates the divergences so effectively that for maximal ${\cal N}=8$ supergravity in four asymptotically flat dimensions it has not yet been established what divergences remain, if any [@Bern:2006kd; @Bern:2007xj; @Kallosh:2008mq; @BjerrumBohr:2009zz; @Kallosh:2009db; @Bjornsson:2010wm; @Dixon:2010gz]. On the other hand, it has long been known that in curved backgrounds, highly relevant for gravity, even the one-loop vacuum amplitude diverges [@Christensen:1978gi; @Christensen:1978md; @Christensen:1979iy; @Sezgin:1980tp; @Bastianelli:2005vk; @Bastianelli:2005uy]. The apparent incompatibility between these results created controversy already in the 1980’s [@Siegel:1980jj; @Duff:1980qv; @Siegel:1980ax; @Gibbons:1984dg; @Inami:1984vp; @Fradkin:1984ai; @Buchbinder:2008jf]. In this paper we revisit this tension from a modern perspective informed by the AdS/CFT correspondence [@Maldacena:1997re].

To exhibit the central issue in more detail it is convenient to focus on the anomalous contribution to the trace of the energy momentum tensor $$\label{eq:emtrace}
\langle T_\mu^\mu \rangle_{\rm an}= \frac{1}{(4\pi)^2} \left( c W^2 - a E_4\right)~,$$ where the square of the Weyl tensor $W^2={\rm Riem}^2 - 2{\rm Ric}^2 + \frac{1}{3}R^2$ and the Euler density 