$R^4$ couplings, the fundamental membrane and exceptional theta correspondences

TL;DR

This paper explores deriving exact BPS $R^4$ couplings in M-theory from membrane instantons, revealing hidden symmetries and proposing an exceptional theta correspondence to unify membranes and five-branes.

Contribution

It introduces a novel approach using exceptional theta correspondences to compute M-theory couplings, extending string theory methods to membranes.

Findings

Identifies $Sl(3, ext{Z})$ modular invariance in membrane theory.

Recovers classical spectrum and membrane instantons.

Proposes exceptional theta lifts to correct the membrane amplitude.

Abstract

This letter is an attempt to carry out a first-principle computation in M-theory using the point of view that the eleven-dimensional membrane gives the fundamental degrees of freedom of M-theory. Our aim is to derive the exact BPS $R^4$ couplings in M-theory compactified on a torus $T^{d+1}$ from the toroidal BPS membrane, by pursuing the analogy with the one-loop string theory computation. We exhibit an $Sl(3,\Zint)$ modular invariance hidden in the light-cone gauge (but obvious in the Polyakov approach), and recover the correct classical spectrum and membrane instantons; the summation measure however is incorrect. It is argued that the correct membrane amplitude should be given by an exceptional theta correspondence lifting $Sl(3,\Zint)$ modular forms to $\exc(\Zint)$ automorphic forms, generalizing the usual theta lift between $Sl(2,\Zint)$ and $SO(d,d,\Zint)$ in string theory. The exceptional correspondence $Sl(3)\times E_{6(6)}\subset E_{8(8)}$ offers the interesting prospect of solving the membrane small volume divergence and unifying membranes with five-branes.