Abstract
We consider D1-branes in an RR flux background and show that there is a low energylarge flux double-scaling limit where the D1-brane action is dominated by a ChernSimonsMyers 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 II B string, II A string and M-theory in the corresponding backgrounds are constructed. A classical solution of a quantum Nambu geometry in the II A 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 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. © 2012 IOP Publishing Ltd.