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

Global well-posedness and scattering for mass-critical, defocusing, infinite dimensional vector-valued resonant nonlinear Schr\"odinger system