(F,D5) bound state, SL(2,Z)invariance and the descendant states in type IIB and type IIA string theory

Recently the space-time configurations of a set of nonthreshold bound states, called the $(F,\mathrm{D}p)$ bound states, have been constructed explicitly for every p with $2<~p<~7$ in both type IIA (for p even) and type IIB (for p odd) string theories by the present authors. By making use of the $\mathrm{SL}(2,Z)$ symmetry of type IIB theory we construct a more general $\mathrm{SL}(2,Z)$ invariant bound state of the type $\left(\right(F,\mathrm{D}1\mathrm{}),$ (NS5, D5)$)$ in this theory from the $(F,\mathrm{D}5\mathrm{})$ bound state. There are actually an infinite number of $\mathrm{}(m,n)\mathrm{}$ strings forming bound states with ${(m}^{\prime }{,n}^{\prime })$ 5-branes, where strings lie along one of the spatial directions of the 5-branes. By applying T duality along one of the transverse directions we also construct the bound state $\left(\right(F,\mathrm{D}2\mathrm{}),\hspace{0.5em}(\mathrm{K}\mathrm{K},\mathrm{}\mathrm{D}6\mathrm{}\left)\right)$ in type IIA string theory. Then we give a list of possible bound states which can be obtained from these newly constructed bound states by applying T dualities along the longitudinal directions as well as S dualities to those in type IIB theory.