Noncommutative and Ordinary Super Yang-Mills on (D$(p-2)$, D$p$) Bound States

We study properties of (D$(p-2)$, D$p$) nonthreshold bound states ($2 \le p \le 6 $) in the dual gravity description. These bound states can be viewed as D$p$-branes with a nonzero NS $B$ field of rank two. We find that in the decoupling limit, the thermodynamics of the $N_p$ coincident D$p$-branes with $B$ field is the same not only as that of $N_p$ coincident D$p$-branes without $B$ field, but also as that of the $N_{p-2}$ coincident D$(p-2)$-branes with two smeared coordinates and no $B$ field, for $N_{p-2}/N_p= \tilde{V}_2/[(2\pi)^2 \tilde{b}]$with$\tilde{V}_2$ being the area of the two smeared directions and $\tilde{b}$ a noncommutativity parameter. We also obtain the same relation from the thermodynamics and dynamics by probe methods. This suggests that the noncommutative super Yang-Mills with gauge group $U(N_p)$ in ($p+1$) dimensions is equivalent to an ordinary one with gauge group $U(\mathrm{\infty })$ in ($p-1$) dimensions in the limit ${\tilde{V}}_2\to \mathrm{\infty }$. We also find that the free energy of a D$p$-brane probe with $B$ field in the background of D$p$-branes with $B$ field coincides with that of a D$p$-brane probe in the background of D$p$-branes without $B$ field.