Construction of SL(2, Z) invariant amplitudes in type IIB superstring theory

The construction of SL(2, Z) invariant amplitudes that generalize the Virasoro amplitude is investigated in detail. We describe a number of mathematical properties that characterize the simplest example, and present pieces of evidence that it represents the tree-level fourgraviton scattering amplitude in membrane theory on R × T 2 in the limit that the torus area goes to zero. In particular, we show that the poles of the S-dual amplitude are in precise correspondence with the states of membrane theory that survive in the type IIB limit. These are shown to be the states that span the Cartan subspaces of area preserving diffeomorphisms of the 2-torus; all other states become infinitely massive, and membrane world-volume theory acquires the structure of a free theory. February 1998 ∗ e-mail address: [email protected]