One-loop four-graviton amplitude in eleven-dimensional supergravity

We find explicit expression for the one-loop four-graviton amplitude in eleven-dimensional supergravity compactified on a circle. Represented in terms of the string coupling (proportional to the compactification radius) it takes the form of an infinite sum of perturbative string loop corrections. We also compute the amplitude in the case of compactification on a 2-torus, which is given by an SL(2, Z) invariant expansion in powers of the torus area, We discuss the structure of quantum corrections in eleven-dimensional theory and their relation to string theory. (C) 1997 Elsevier Science B.V.