A Comment on Compactification of M-Theory on an (Almost) Light-Like Circle

TL;DR

This paper investigates the behavior of M-theory compactified on an almost light-like circle, demonstrating that the light-like limit can be finite and well-defined due to membrane wrapping modes, contrasting with divergences seen in field theory.

Contribution

It provides a detailed analysis of the light-like limit in M-theory, showing that membrane wrapping modes can resolve divergences and yield finite amplitudes.

Findings

The light-like limit can be finite with non-vanishing external momenta.

Membrane wrapping modes play a crucial role in divergence cancellation.

The amplitude's moduli space reduces to a finite lattice in the light-like limit.

Abstract

In perturbative quantum field theory the limit of compactification on an almost light-like circle has recently been shown to be plagued by divergences. We argue that the light-like limit for M-theory probably is free of such divergences due to, among others, the existence of the wrapping modes of the membranes. To illustrate this, we consider superstring theory compactified on an almost light-like circle. Specifically, 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 compact 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 compensation and the resulting amplitude has a well-defined and finite light-like limit. The net effect of taking the light-like limit is to replace the integration over one of the moduli of the 4-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 rotated" amplitude computed directly with a compact light-like circle.