R⁴ couplings, the fundamental membrane and exceptional theta correspondences

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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 R⁴ couplings in M-theory compactified on a torus Tᵈ⁺¹ from the toroidal BPS membrane, by pursuing the analogy with the one-loop string theory computation. We exhibit an Sl(3,) 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,) modular forms to () automorphic forms, generalizing the usual theta lift between Sl(2,) and SO(d,d,) 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.