  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 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.
