The collective dynamics of dislocations is the underlying mechanism of plastic deformation in metallic crystals. Dislocation motion in metals generally occurs on multiple slip systems. The simultaneous activation of different slip systems plays a crucial role in crystal plasticity models. In this contribution, we study the energetic interactions between dislocations on different slip systems by deriving the free energy in a multi-slip geometry. In this, we restrict ourselves to straight and parallel edge dislocations. The obtained free energy has a long-range mean-field contribution, a statistical contribution and a many-body contribution. The many-body contribution is a local function of the total dislocation density on each slip system, and can therefore not be written in terms of the net dislocation density only. Moreover, this function is a strongly non-linear and non-convex function of the density on different slip systems, and hence the coupling between slip systems is of great importance.