Linear Flux Origins of κ-Geometry: From Lie Non-Commutativity to 3-Lie Nambu Space

Abstract

Linear background fluxes offer a unified origin for non-commutative and non-associative geometries in string and M-theory. Revisiting the open string with a linearly varying Kalb-Ramond field, we recover kappa-Minkowski space as a Jordanian deformation of Poincar & eacute; symmetry. Lifting the construction to an open M2-brane in a linear three-form background yields a kappa-Nambu triple bracket that realises the three-Lie algebra of the Bagger-Lambert-Gustavsson model and exposes an underlying quasi-Hopf two-algebra. A Bopp-shift quantisation reveals that isotropic potentials are unaltered at leading order, whereas anisotropic ones receive Planck-suppressed corrections, and gerbe holonomy discretises the deformation scale. Inspired by the membrane bracket, we build a cubic-matrix extension of the BFSS model whose higher-Lie structure supplies the long-missing five-brane central charge and reproduces the expected energy scaling of coincident M5-branes. These results establish linear fluxes as the minimal ingredients that interpolate smoothly from Moyal to Lie and Nambu deformations, providing concrete, anomaly-free models for non-associative field theories and five-brane dynamics.