Is d=4 Maximal Supergravity Special?

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.