D-particles on T⁴/Z_n orbifolds and their resolutions

We formulate the effective field theory of a D-particle on orbifolds of $T^4$ by a cyclic group as a gauge theory in a $V$-bundle over the dual orbifold. We argue that this theory admits Fayet-Iliopoulos terms analogous to those present in the case of noncompact orbifolds. In the $n=2$ case, we present some evidence that turning on such terms resolves the orbifold singularities and may lead to a $K3$ surface realized as a blow up of the fixed points of the cyclic group action.