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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 toroidal compactifications of M-theory 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 off. We argue 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, and outline the construction of such objects. The exceptional correspondence $Sl(3)imes E_6subset E_8$ offers the interesting prospect of solving the membrane small volume divergence and unifying membranes with five-branes.