  It is well-known that ($N$, $M$) 5-branes of type IIB supergravity form a
non-threshold bound state with ($N'$, $M'$) strings called the (F, D1, NS5, D5)
bound state where the strings lie along one of the spatial directions of the
5-branes (hep-th/9905056). By taking low energy limits in appropriate ways on
this supergravity configuration, we obtain the supergravity duals of various
decoupled theories in (5+1) dimensions corresponding to noncommutative open
string (NCOS) theory, open D-string (OD1) theory and open D5-brane (OD5)
theory. We then study the $SL(2, Z)$ transformation properties of these
theories. We show that when the asymptotic value of the axion ($\chi_0$) is
rational (for which $\chi_0$ can be put to zero), NCOS theory is always related
to OD1 theory by strong-weak or S-duality symmetry. We also discuss the
self-duality conjecture (hep-th/0006062) of both NCOS and OD1 theories. On the
other hand, when $\chi_0$ is irrational, we find that $SL(2, Z)$ duality on
NCOS theory gives another NCOS theory with different values of the parameters,
but for OD1 theory $SL(2,Z)$ duality always gives an NCOS theory. $SL(2, Z)$
transformation on OD5 theory reveals that it gives rise to Little String Theory
(LST) when $\chi_0$ = rational, but it gives another OD5 theory with different
values of the parameters when $\chi_0$ is irrational.

High Energy Physics - Theory