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Classifying Supersymmetric Solutions in 3D Maximal Supergravity

String theory contains various extended objects. Among those, objects of codimension two (such as the D7-brane) are particularly interesting. Codimension two objects carry non-Abelian charges which are elements of a discrete U-duality group and they may not admit a simple space-time description, in which case they are known as exotic branes. A complete classification of consistent codimension two objects in string theory is missing, even if we demand that they preserve some supersymmetry. As a step toward such a classification, we study the supersymmetric solutions of 3D maximal supergravity, which can be regarded as approximate description of the geometry near codimension two objects. We present a complete classification of the types of supersymmetric solutions that exist in this theory. We found that this problem reduces to that of classifying nilpotent orbits associated with the U-duality group, for which various mathematical results are known. We show that the only allowed supersymmetric configurations are 1/2, 1/4, 1/8, and 1/16 BPS, and determine the nilpotent orbits that they correspond to. One example of 1/16 BPS configurations is a generalization of the MSW system, where momentum runs along the intersection of seven M5-branes. On the other hand, it turns out exceedingly difficult to translate this classification into a simple criterion for supersymmetry in terms of the non-Abelian (monodromy) charges of the objects. For example, it can happen that a supersymmetric solution exists locally but cannot be extended all the way to the location of the object. To illustrate the various issues that arise in constructing supersymmetric solutions, we present a number of explicit examples.