Classification of BPS instantons in N=4 D=4 supergravity

This talk is based on the recent work in collaboration with M. Azreg-A\"{\i}nou and G. Cl\'ement devoted to extremal instantons in the one-vector truncation of the Euclidean $\mathcal{N}=4,\, D=4$ theory. Extremal solutions satisfying the no-force condition can be associated with null geodesic curves in the homogeneous target space of the three-dimensional sigma model arising in toroidal reduction of the four-dimensional theory. Here we (preliminarily) discuss the case of two vector fields sufficient to find all relevant metrics in the full $\mathcal{N}=4,\, D=4$ theory. Classification of instanton solutions is given along the following lines. The first is their possible asymptotic structure: asymptotically locally flat (ALF), asymptotically locally Euclidean (ALE) and ALF or ALE with the dilaton growing at infinity. The second is the algebraic characterization of matrix generators according to their rank and the nature of the charge vectors in an associated Lorentzian space. Finally, solutions are distinguished by the number of independent harmonic functions with unequal charges (up to four).