Membrane solitons in eight-dimensional hyper-Kähler backgrounds

We derive the BPS equations satisfied by lump solitons in(2 + 1)-dimensional sigma models with toric 8-dimensional hyper-Kähler(HK ₈) target spaces and check they preserve 1/2 of the supersymmetry. We show how these solitons are realised in M theory as M2-branes wrapping holomorphic 2-cycles in the double-struck E sign^1,2 × HK ₈ background. Using the k-symmetry of a probe M2-brane in this background we determine the supersymmetry they preserve, and note that there is a discrepancy in the fraction of supersymmetry preserved by these solitons as viewed from the low energy effective sigma model description of the M2-brane dynamics or the full M theory. Toric HK₈ manifolds are dual to a Hanany-Witten setup of D3-branes suspended between 5-branes. In this picture the lumps correspond to vortices of the three dimensional N = 3 or N = 4 theory.