Non-renormalisation conditions in type II string theory and maximal supergravity

This paper considers general features of the derivative expansion of Feynmandiagram contributions to the four-graviton scattering amplitude ineleven-dimensional supergravity compactified on a two-torus. These aretranslated into statements about interactions of the form D^2k R^4 in type IIsuperstring theories, assuming the standard M-theory/string theory dualityrelationships, which provide powerful constraints on the effectiveinteractions. In the ten-dimensional IIA limit we find that there can be noperturbative contributions beyond k string loops (for k>0). Furthermore, thegenus h=k contributions are determined exactly by the one-loopeleven-dimensional supergravity amplitude for all values of k. A plausibleinterpretation of these observations is that the sum of h-loop Feynman diagramsof maximally extended supergravity is less divergent than might be expected andcould be ultraviolet finite in dimensions d < 4 + 6/h -- the same bound as forN=4 Yang--Mills.Comment: 30 pages, 4 figures. v3 Some rewording, typos corrected and some references added. v4: Abstract rephrased. More typos corrected. Version to be publishe