Non-renormalisation Conditions in Type II String Theory and Maximal Supergravity

Abstract

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. A plausible interpretation of these observations is that the sum of h-loop Feynman diagrams of maximally extended supergravity is less divergent than might be expected and could be ultraviolet finite in dimensions d < 4 + 6/h -- the same bound as for N=4 Yang--Mills.

Figures (4)

• Figure 1: The scalar field theory box diagram and the counter-term that subtracts the $\Lambda^3$ divergence.
• Figure 2: The two-loop four-graviton amplitude in eleven dimensions is given by the sum of scalar field theory double-box diagrams and counterterms that subtract the primitive divergence and subdivergences.
• Figure 3: Two diagrams with one-loop subdivergences that contain terms that contribute to the $s \ln (-{\alpha'} s)$ threshold in the ten-dimensional IIB limit.
• Figure 4: The string one-loop $s^4\ln(s)$ originates from the two-loop supergravity amplitude by resumming the Kaluza-Klein modes in one loop and using the finite $\zeta(3) \, D^4 { R}^4/R_{11}^3$ contribution from the one-loop amplitude, as in figure (a). The string two-loop contribution arises by picking the $\Lambda^3$ counterterm at one-loop, as in figure (b).