Three-Charge Black Holes and Quarter BPS States in Little String Theory

We show that the system of $k$ NS5-branes wrapping ${\mathbb{T}}^4\times S^1$ has non-trivial vacuum structure. Different vacua have different spectra of 1/4 BPS states that carry momentum and winding around the $S^1$. In one vacuum, such states are described by black holes; in another, they can be thought of as perturbative BPS states in Double Scaled Little String Theory. In general, both kinds of states are present. We compute the degeneracy of perturbative BPS states exactly, and show that it differs from that of the corresponding black holes. We comment on the implication of our results to the black hole microstate program, UV/IR mixing in Little String Theory, string thermodynamics, the string/black hole transition, and other issues.