Unconditional uniqueness for the cubic nonlinear Schrödinger equation in $H^{1/2}(\Bbb T^3)$

In this paper, we establish unconditional uniqueness for the cubic nonlinear Schrödinger equation (NLS) in the scaling-critical class $H^{1/2}(\mathbb{T}^3)$. Unlike approaches based on the GP-hierarchy or normal form reductions, which typically involve higher-order expansions, our proof works directly with the original cubic NLS, where we also make use of several new a priori Euclidean-like Strichartz estimates. More precisely, we show that every distributional solution in $C_tH_x^{1/2}$ belongs a priori to the class $L^2_{t,\mathrm{loc}}L_x^6$. The key ingredients are a suitable high-low modulation decomposition and the compactness of solution trajectories on bounded time intervals. We also exploit the preservation of the canonical symplectic form under the truncated NLS flow, in order to obtain necessary stability estimates in the rough Sobolev space $H^{-1/2}(\mathbb{T}^3)$.