Given an involution on a complex variety, the Smith-Thom inequality says that the total \mathbb{F}_2-Betti number of the fixed locus is no greater than the total \mathbb{F}_2-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. On a Hyper-Kähler manifold X a holomorphic or a anti-holomorphic involution is referred to as a brane involution. While examples of non-compact hyper-Kähler manifolds admitting maximal branes are known, the compact case is more intriguing. In particular, although there exist some K3 surfaces admitting maximal brane involutions, the main result that I will show you is the non-existence of maximal branes on Hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of points on a K3 surface. This talk is based on a joint work with S. Billi, L. Fu and V. Kharlamov.