Cancellations of Resonances and Long Time Dynamics of Cubic Schr\"odinger Equation on $\mathbb{T}$

We prove a vanishing property of the normal form transformation of the 1D cubic nonlinear Schr"odinger (NLS) equation with periodic boundary conditions on $[0,L]$. We apply this property to quintic resonance interactions and obtain a description of dynamics for time up to $T=\frac{L^2}{{ϵ}^4}$, if $L$ is sufficiently large and size of initial data $ϵ$ is small enough. Since $T$ is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of 1D cubic NLS. Our approach can be adapted to other integrable systems without too many difficulties. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively.