We consider D1-branes in a RR flux background and show that there is a low energy -large flux double scaling limit where the D1-branes action is dominated by a Chern-Simons-Myers coupling term. As a classical solution to the matrix model, we find a novel quantized geometry characterized by a quantum Nambu 3-bracket. Infinite dimensional representations of the quantum Nambu geometry are constructed which demonstrate that the quantum Nambu geometry is intrinsically different from the ordinary Lie algebra type noncommutative geometry. Matrix models for the IIB string, IIA string and M-theory in the corresponding backgrounds are constructed. A classical solution of a quantum Nambu geometry in the IIA Matrix string theory gives rise to an expansion of the fundamental strings into a system of multiple D4-branes and the fluctuation is found to describe an action for a non-abelian 3-form field strength which is a natural non-abelian generalization of the PST action for a single D4-brane. In view of the recent proposals [1, 2] of the M5-branes theory in terms of the D4-branes, we suggest a natural way to include all the KK modes and propose an action for the the multiple M5-branes in a constant C-field. The worldvolume of the M5-branes in a C-field is found to be described by a quantum Nambu geometry with self-dual parameters. It is intriguing that our action is naturally formulated in terms of a 1-form gauge field living on a six dimensional quantum Nambu geometry.

The supersymmetric actions of closed multiple M2 branes with flux for the BL and the ABJM theories have been constructed recently by Lambert and Richmond in [1]. In this paper we extend the construction to the case of open M2-branes with flux and derive the boundary conditions. This allows us to derive the modified Basu-Harvey equation in the presence of flux. As an example, we consider the Lorentzian BL model. A new feature of the fuzzy funnel solution describing a D2-D4 intersection is obtained as a result of the flux.