A COMMENT ON COMPACTIFICATION OF M-THEORY ON AN (ALMOST) LIGHT-LIKE CIRCLE

In perturbative quantum field theory the limit of compactification on an almost light-like circle has recently been shown to be plagued by d ivergences. We argue that the light-like limit for M-theory probably i s free of such divergences due to, among others, the existence of the wrapping modes of the membranes. To illustrate this, we consider super string theory compactified on an almost light-like circle. Specificall y, we compute a one-loop four-point amplitude in type II theory. As is well known, if the external states have vanishing momenta in the comp act dimension, the divergence in the light-like limit is even stronger than in field theory. However, in the case of present interest, where these external momenta are non-vanishing, there is a subtle compensat ion and the resulting amplitude has a well defined and finite light-li ke limit. The net effect of taking the light-like limit is to replace the integration over one of the moduli of the four-punctured torus by a sum over a discrete modulus taking values in a finite lattice on the torus. The same result can also be obtained from a suitably ''Wick ro tated'' amplitude computed directly with a compact light-like circle. (C) 1998 Elsevier Science B.V.