  Instantons and W-bosons in 5d maximally supersymmetric Yang-Mills theory
arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein
modes and winding self-dual strings, respectively. We study an index which
counts BPS instantons with electric charges in Coulomb and symmetric phases. We
first prove the existence of unique threshold bound state of (noncommutative)
U(1) instantons for any instanton number, and also show that charged instantons
in the Coulomb phase correctly give the degeneracy of SU(2) self-dual strings.
By studying SU(N) self-dual strings in the Coulomb phase, we find novel
momentum-carrying degrees on the worldsheet. The total number of these degrees
equals the anomaly coefficient of SU(N) (2,0) theory. We finally show that our
index can be used to study the symmetric phase of this theory, and provide an
interpretation as the superconformal index of the sigma model on instanton
moduli space.
