$delta$-sunspots, with highly complex magnetic structures, are very productive in energetic eruptive events, such as X-class flares and homologous eruptions. We here study the formation of such complex magnetic structures by numerical simulations of
magnetic flux emergence from the convection zone into the corona in an active-region-scale domain. In our simulation, two pairs of bipolar sunspots form on the surface, originating from two buoyant segments of a single subsurface twisted flux rope, following the approach of Toriumi et al. (2014). Expansion and rotation of the emerging fields in the two bipoles drive the two opposite polarities into each other with apparent rotating motion, producing a compact $delta$-sunspot with a sharp polarity inversion line. The formation of the $delta$-sunspot in such a realistic-scale domain produces emerging patterns similar to those formed in observations, e.g. the inverted polarity against Hales law, the curvilinear motion of the spot, strong transverse field with highly sheared magnetic and velocity fields at the PIL. Strong current builds up at the PIL, giving rise to reconnection, which produces a complex coronal magnetic connectivity with non-potential fields in the Delta-spot overlaid by more relaxed fields connecting the two polarities at the two ends.
