Non-commutative Weitzenboeck geometry, gerbe modules, and WZW branes

We study the non-commutative matrix model which arises as the low-energy effective action of open strings in WZW models. We re-derive this fuzzy effective gauge dynamics in two different ways, without recourse to conformal field theory. The first method starts from a linearised version of the WZW sigma model, which is classically equivalent to an action of Schild type, which in turn can be quantised in a natural way to yield the matrix model. The second method relies on purely geometric symmetry principles -- albeit within the non-commutative spectral geometry that is provided by the boundary CFT data: we show that imposing invariance under extended gauge transformations singles out the string-theoretic action up to the relevant order in the gauge field. The extension of ordinary gauge transformations by tangential shifts is motivated by the gerbe structure underlying the classical WZW model and standard within Weitzenboeck geometry -- which is a natural reformulation of geometry to use when describing strings in targets with torsion.