Is d=4 Maximal Supergravity Special?

TL;DR

This paper investigates the UV divergence structure of maximal supergravities in various dimensions, highlighting the unique case of four dimensions where divergences are absent up to high loop orders, suggesting special properties of d=4 supergravity.

Contribution

It provides a detailed analysis of candidate counterterms and their symmetry properties across dimensions, revealing the special nature of four-dimensional maximal supergravity.

Findings

UV divergences occur below a critical loop order in dimensions >4

Counterterms preserve symmetries only at or above critical loop order

Four-dimensional case shows no divergences up to high loop orders

Abstract

We study candidate counterterms (CT)'s in maximal supergravities in diverse integer dimensions $d \geq 4 $. We find that UV divergences in these theories occur at the number of loops $L$ below certain critical value $L_{cr}$. At $L\geq L_{cr}$ the CT's have nonlinear local supersymmetry and local $H$ symmetry, but the ones below $L_{cr}$ break these symmetries. Therefore deforming the theories by adding the CT's to the action to cancel UV divergences would be inconsistent with BRST. Such divergences with $L < L_{cr}$ were found in maximal supergravities for all integer $d > 4$, but not for $N=5,6,8$ in $d = 4$ so far, which renders the case $d = 4$ special.