Non-renormalisation Conditions in Type II String Theory and Maximal Supergravity
Résumé
This paper considers general features of the derivative expansion of Feynman diagram contributions to the four-graviton scattering amplitude in eleven-dimensional supergravity compactified on a two-torus. These are translated into statements about interactions of the form D^{2k} R^4 in type II superstring theories, assuming the standard M-theory/string theory duality relationships, which provide powerful constraints on the effective interactions. In the ten-dimensional IIA limit we find that there can be no perturbative contributions beyond k string loops (for k>0). Furthermore, the genus h=k contributions are determined exactly by the one-loop eleven-dimensional supergravity amplitude for all values of k. To the extent that these results reflect exact properties of M-theory, they indicate that low energy maximally extended supergravity at h loops only has power-behaved ultraviolet divergences in dimensions d > 4 + 6/h, the same bound as for N=4 Yang--Mills. This implies that four-dimensional N=8 supergravity should contain, at worst, logarithmic divergences and might well be ultraviolet finite.