R4 couplings, the fundamental membrane and exceptional theta correspondences

This letter is an attempt to carry out a first-principle computation inM-theory using the point of view that the eleven-dimensional membrane gives thefundamental 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 toroidalBPS membrane, by pursuing the analogy with the one-loop string theorycomputation. We exhibit an $Sl(3,\Zint)$ modular invariance hidden in thelight-cone gauge (but obvious in the Polyakov approach), and recover thecorrect classical spectrum and membrane instantons; the summation measurehowever is incorrect. It is argued that the correct membrane amplitude shouldbe given by an exceptional theta correspondence lifting $Sl(3,\Zint)$ modularforms to $\exc(\Zint)$ automorphic forms, generalizing the usual theta liftbetween $Sl(2,\Zint)$ and $SO(d,d,\Zint)$ in string theory. The exceptionalcorrespondence $Sl(3)\times E_{6(6)}\subset E_{8(8)}$ offers the interestingprospect of solving the membrane small volume divergence and unifying membraneswith five-branes.Comment: Latex2e, 17 pages, JHEP.cls; v3: final version for JHEP, (iii) p.14 improved, plus cosmetic