Sigma-model soliton intersections from exceptional calibrations

A first-order 'BPS' equation is obtained for 1/8 supersymmetric intersections of soliton-membranes (lumps) of supersymmetric (4 + 1)-dimensional massless sigma models, and a special non-singular solution is found that preserves 1/4 supersymmetry. For 4-dimensional hyper-Kähler target spaces (HK₄) the BPS equation is shown to be the low-energy limit of the equation for a Cayley-calibrated 4-surface in script E sign⁴ × HK₄. Similar first-order equations are found for stationary intersections of Q-lump-membranes of the massive sigma model, but now generic solutions preserve either 1/8 supersymmetry or no supersymmetry, depending on the time orientation.