Probing Non-Toric Geometry with Rotating Membranes

Recently Martelli and Sparks presented the first non-toric AdS_4/CFT_3 duality relation between M-theory on AdS_4 x V_{5,2}/Z_k and a class of three-dimensional N=2 quiver Chern-Simons-matter theories. V_{5,2} is a seven-dimensional homogeneneous Sasaki-Einstein manifold with isometry group SO(5)xU(1)_R, which is in general broken to SU(2)xU(1)xU(1)_R by the orbifold projection if k>1. The dual field theory is described by the A_1 quiver, U(N)_k x U(N)_{-k} gauge group, four bifundamentals, two adjoint chiral multiplets interacting via a cubic superpotential. We explore this proposal by studying various classical membrane solutions moving in V_{5,2}. Rotating membrane solutions of folded, wrapped, spike, and giant magnon types are presented with their dispersion relations. We also discuss their dual operators in the Chern-Simons-matter theory.