Toric Fano varieties and Chern-Simons quivers

In favourable cases the low energy dynamics of a stack of M2-branes at a toric Calabi-Yau fourfold singularity can be described by an N = 2 supersymmetric Chern-Simons quiver theory, but there still does not exists an "inverse algorithm" going from the toric data of the CY4 to the CS quiver. We make progress in that direction by deriving CS quiver theories for M2-branes probing cones over a large class of geometries Y-p,Y- q (B-4), which are S-3/Z(p) bundles over toric Fano varieties B-4. We rely on the type IIA understanding of CS quivers, giving a firm string theory footing to our CS theories. In particular we give a derivation of some previously conjectured CS quivers in the case B-4 = CP1 x CP1, as field theories dual to M-theory backgrounds with nontrivial torsion G(4) fluxes.