S-duality Action on Discrete T-duality Invariants

In heterotic string theory compactified on T , the T-duality orbits of dyons of charge (Q,P ) are characterized by O(6, 22; R ) invariants Q, P 2 and Q · P together with a set of invariants of the discrete T-duality group O(6, 22; ZZ). We study the action of S-duality group on the discrete T-duality invariants and study its consequence for the dyon degeneracy formula. In particular we find that for dyons with torsion r, the degeneracy formula, expressed as a function of Q, P 2 and Q · P , is required to be manifestly invariant under only a subgroup of the Sduality group. This subgroup is isomorphic to Γ(r). Our analysis also shows that for a given torsion r, all other discrete T-duality invariants are characterized by the elements of the coset SL(2, ZZ)/Γ(r).