A toy model of open membrane field theory in constant 3-form flux

Based on an explicit computation of the scattering amplitude of four open membranes in a constant 3-form background, we construct a toy model of the field theory for open membranes in the large C field limit. It is a generalization of the noncommutative field theories which describe open strings in a constant 2-form flux. The noncommutativity due to the B-field background is now replaced by a nonassociative triplet product. The triplet product satisfies the consistency conditions of lattice 3d gravity, which is inherent in the world-volume theory of open membranes. We show the UV/IR mixing of the toy model by computing some Feynman diagrams. Inclusion of the internal degree of freedom is also possible through the idea of the cubic matrix.