The Automorphic Membrane

We present a 1-loop toroidal membrane winding sum reproducing the conjectured M-theory, four-graviton, eight derivative, R amplitude. The U -duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for M-theory compactified on a 3-torus, where the target-space Sl(3,Z)× Sl(2,Z) U -duality and Sl(3,Z) world-volume modular groups are embedded in E6(6)(Z). Unlike previous semi-classical expansions, U -duality is built in manifestly and realized at the quantum level thanks to Fourier invariance of cubic characters. In addition to winding modes, a pair of new discrete, flux-like, quantum numbers are necessary to ensure invariance under the larger group. The action for these modes is of Born-Infeld type, interpolating between standard Polyakov and Nambu-Goto membrane actions. After integration over the membrane moduli, we recover the known R amplitude, including membrane instantons. Divergences are disposed of by trading the non-compact volume integration for a compact integral over the two variables conjugate to the fluxes – a constant term computation in mathematical parlance. As byproducts, we suggest that, in line with membrane/fivebrane duality, the E6 theta series also describes five-branes wrapped on T 6 in a manifestly U-duality invariant way. In addition we uncover a new action of E6 on ten dimensional pure spinors, which may have implications for ten dimensional super Yang–Mills theory. An extensive review of Sl(3) automorphic forms is included in an Appendix.