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

TL;DR

This paper develops a toy model for open membrane field theory in a constant 3-form flux, extending noncommutative string theories to a nonassociative framework with potential implications for 3d gravity.

Contribution

It introduces a nonassociative triplet product as a generalization of noncommutative geometry for open membranes in a 3-form background, connecting to lattice 3d gravity.

Findings

Demonstrates the nonassociative triplet product satisfies gravity-like consistency conditions.

Shows UV/IR mixing in the toy model through Feynman diagram analysis.

Proposes inclusion of internal degrees of freedom via cubic matrix approach.

Abstract

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.