Linear Flux Origins of κκ‐Geometry: From Lie Non‐Commutativity to 3‐Lie Nambu Space

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 Poincaré 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.