  In this article, we consider the infinite dimensional vector-valued resonant
nonlinear Schr\"odinger system, which arises from the study of the asymptotic
behavior of the defocusing nonlinear Schr\"{o}dinger equation on "wave guide"
manifolds like $\mathbb{R}^2\times \mathbb{T}$ in [7]. We show global
well-posedness and scattering for this system by long time Strichartz estimates
and frequency localized interaction Morawetz estimates. As a by-product, our
results make the arguments of scattering theory in [7] closed as crucial
ingredients for compactness of the critical elements.
