Symmetries of the refined D1/D5 BPS spectrum

We examine the large N 1/4-BPS spectrum of the symmetric orbifold CFT Sym(N) (M) deformed to the supergravity point in moduli space for M = K3 and T-4. We consider refinement under both left- and right-moving SU(2)(R) symmetries of the superconformal algebra, and decompose the spectrum into characters of the algebra. We find that at large N the character decomposition satisfies an unusual property, in which the degeneracy only depends on a certain linear combination of left- and right-moving quantum numbers, suggesting deeper symmetry structure. Furthermore, we consider the action of discrete symmetry groups on these degeneracies, where certain subgroups of the Conway group are known to play a role. We also comment on the potential for larger discrete symmetry groups to appear in the large N limit.