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

TL;DR

This paper constructs SL(2,Z) invariant amplitudes in type IIB superstring theory, linking them to membrane theory on a torus and analyzing their mathematical properties and physical implications.

Contribution

It provides a detailed construction of SL(2,Z) invariant amplitudes and connects them to membrane theory, offering evidence for their interpretation as four-graviton scattering amplitudes.

Findings

SL(2,Z) invariant amplitudes are constructed and characterized.

Poles of the amplitude correspond to membrane theory states.

In the zero-area limit, the amplitude matches membrane scattering results.

Abstract

The construction of $SL(2,Z)$ invariant amplitudes that generalizethe 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 four-graviton scattering amplitude in membrane theory on ${\bf R}^9\times T^2$ 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.