R-4 couplings, the fundamental membrane and exceptional theta correspondences

This letter is an attempt to carry out a first-principle computation in M-t heory using the point of view that the eleven-dimensional membrane gives th e fundamental degrees of freedom of M-theory. Our aim is to derive the exac t BPS R-4 couplings in M-theory compactified on a torus Td+1 from the toroi dal BPS membrane, by pursuing the analogy with the one-loop string theory c omputation. We exhibit an Sl(3, Z) modular invariance hidden in the light-c one gauge (but obvious in the Polyakov approach), and recover the correct c lassical spectrum and membrane instantons; the summation measure however is incorrect. It is argued that the correct membrane amplitude should be give n by an exceptional theta correspondence lifting Sl(3, Z) modular forms to Ed+1(d+1) (Z) automorphic forms, generalizing the usual theta lift between Sl(2, Z) and SO(d, d, Z) in string theory. The exceptional correspondence S l(3) x E-6(6) subset of E-8(8) offers the interesting prospect of solving t he membrane small volume divergence and unifying membranes with five-branes .