  The partition functions of refined topological strings(A-models) are
computed, which give rise to the circle-compactified five-dimensional
supersymmetric linear quiver gauge theories in generic (not necessarily
self-dual) Omega backgrounds. Based on the slicing independence conjecture of
refined topological string partition functions, it is demonstrated explicitly
that the duality exists between
  $SU(N)^{M-1}$ and $SU(M)^{N-1}$ supersymmetric linear quiver gauge theories,
even in generic Omega backgrounds. It is found that the relations between
string moduli and gauge moduli are deformed from the self-dual case. However if
the duality map which preserves the ratio of the Omega background parameters q
and t, is considered, duality maps of the gauge moduli are not changed from the
self-dual case.
