M2-branes on M-folds

We argue that the moduli space for the Bagger-Lambert A4 theory at level k is (R8 x R8)/D2k, where D2k is the dihedral group of order 4k. We conjecture that the theory describes two M2-branes on a Z2k ''M-fold'', in which a geometrical action of Z2k is combined with an action on the branes. For k = 1, this arises as the strong coupling limit of two D2-branes on an O2- orientifold, whose worldvolume theory is the maximally supersymmetric SO(4) gauge theory. Finally, in an appropriate large-k limit we show that one recovers compactified M-theory and the M2-branes reduce to D2-branes.