Towards the quantum geometry of the M5-brane in a constant C-field from multiple membranes

We show that the Nahm equation which describes a fuzzy D3-brane in the presence of a B-field can be derived as a boundary condition of the F1-strings ending on the D3-brane, and that the modifications of the original Nahm equation by a B-field can be understood in terms of the noncommutative geometry of the D3-brane. Naturally this is consistent with the alternative derivation by quantising the open strings in the B-field background. We then consider a configuration of multiple M2-branes ending on an M5-brane with a constant 3-form C-field. By analogy with the case of strings ending on a D3-brane with a constant B-field, one can expect that this system can be described in terms of the boundary of the M2-branes moving within a certain kind of quantum geometry on the M5-brane worldvolume. By repeating our analysis, we show that the analogue of the B-field modified Nahm equation, the C-field modified Basu-Harvey equation can also be understood as a boundary condition of the M2-branes. We then compare this to the M5-brane BIon description and show that the two descriptions match provided we postulate a new type of quantum geometry on the M5-brane worldvolume. Unlike the D-brane case, this is naturally expressed in terms of a relation between a 3-bracket of the M5-brane worldvolume coordinates and the C-field.