R 4 couplings , the fundamental membrane and exceptional theta correspondences

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 couplings in M-theory compactified on a torus T from the toroidal BPS membrane, by pursuing the analogy with the one-loop string theory computation. We exhibit an Sl(3,Z) 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,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 Sl(3) × E6(6) ⊂ E8(8) offers the interesting prospect of solving the membrane small volume divergence and unifying membranes with five-branes.