M2-brane Probe Dynamics and Toric Duality

We study the dynamics of a single M2-brane probing toric Calabi-Yau four-fold singularity in the context of the recently proposed M-theory crystal model of AdS4/CFT3 dual pairs. We obtain an effective abelian gauge theory in which the charges of the matter fields are given by the intersection numbers between loops and faces in the crystal. We argue that the probe theory captures certain aspects of the CFT3 even though the true M2-brane CFT is unlikely to be a usual gauge theory. In particular, the moduli space of vacua of the gauge theory coincides precisely with the Calabi-Yau singularity. Toric duality, partial resolution, and a possibility of new RG flows are also discussed.