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We study the space of stability conditions on $K3$ surfaces from theperspective of mirror symmetry. It is done in the so called attractorbackgrounds (moduli) which can be far from the conventional large complexlimits and are selected by the attractor mechanism for certain black holes. Wefind certain highly non-generic behaviors of stability walls (a key notion inthe study of wall crossings) in the space of stability conditions. Theycorrespond via mirror symmetry to some non-generic behaviors of specialLagrangians in an attractor background. The main results can be understood as amirror correspondence in a synthesis of homological mirror conjecture and SYZmirror conjecture.

Stability Conditions and Mirror Symmetry of K3 Surfaces in Attractor Backgrounds