Membrane solitons in eight-dimensional hyper-Kaehler backgrounds

We derive the BPS equations satisfied by lump solitons in $(2+1)$-dimensionalsigma models with toric 8-dimensional hyper-K\"ahler (${HK}_8$) target spacesand check they preserve 1/2 of the supersymmetry. We show how these solitonsare realised in M theory as M2-branes wrapping holomorphic 2-cycles in the$\bE^{1,2}\times {HK}_8$ background. Using the $\kappa$-symmetry of a probeM2-brane in this background we determine the supersymmetry they preserve, andnote that there is a discrepancy in the fraction of supersymmetry preserved bythese solitons as viewed from the low energy effective sigma model descriptionof the M2-brane dynamics or the full M theory. Toric ${HK}_8$ manifolds aredual to a Hanany-Witten setup of D3-branes suspended between 5-branes. In thispicture the lumps correspond to vortices of the three dimensional ${\mathcalN}=3$ or ${\mathcal N}=4$ theory.Comment: 12+1 pages. LaTex. v2: Typos corrected and references adde