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In this paper, we will obtain a rigorous derivation of the defocusing cubicnonlinear Schr\"{o}dinger equation on the three-dimensional torus$\mathbb{T}^3$ from the many-body limit of interacting bosonic systems. Thistype of result was previously obtained on $\mathbb{R}^3$ in the work ofErd\H{o}s, Schlein, and Yau \cite{ESY2,ESY3,ESY4,ESY5}, and on $\mathbb{T}^2$and $\mathbb{R}^2$ in the work of Kirkpatrick, Schlein, and Staffilani\cite{KSS}. Our proof relies on an unconditional uniqueness result for theGross-Pitaevskii hierarchy at the level of regularity $\alpha=1$, which isproved by using a modification of the techniques from the work of T. Chen,Hainzl, Pavlovi\'{c} and Seiringer \cite{ChHaPavSei} to the periodic setting.These techniques are based on the Quantum de Finetti theorem in the formulationof Ammari and Nier \cite{AmmariNier1,AmmariNier2} and Lewin, Nam, and Rougerie\cite{LewinNamRougerie}. In order to apply this approach in the periodicsetting, we need to recall multilinear estimates obtained by Herr, Tataru, andTzvetkov \cite{HTT}.  Having proved the unconditional uniqueness result at the level of regularity$\alpha=1$, we will apply it in order to finish the derivation of thedefocusing cubic nonlinear Schr\"{o}dinger equation on $\mathbb{T}^3$, whichwas started in the work of Elgart, Erd\H{o}s, Schlein, and Yau \cite{EESY}. Inthe latter work, the authors obtain all the steps of Spohn's strategy for thederivation of the NLS \cite{Spohn}, except for the final step of uniqueness.Additional arguments are necessary to show that the objects constructed in\cite{EESY} satisfy the assumptions of the unconditional uniqueness theorem.Once we achieve this, we are able to prove the derivation result. Inparticular, we show \emph{Propagation of Chaos} for the defocusingGross-Pitaevskii hierarchy on $\mathbb{T}^3$ for suitably chosen initial data.

A rigorous derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation on $\mathbb{T}^3$ from the dynamics of many-body quantum systems