Authors’ summary: We prove global well-posedness for low regularity data for the

${L}^{2}$-critical defocusing nonlinear Schrödinger equation (NLS) in 2D. More precisely, we show that a global solution exists for initial data in the Sobolev space

${H}^{s}\left({\mathbb{R}}^{2}\right)$ and for any

$s>\frac{2}{5}$. This improves the previous result of

*Y. F. Fang* and

*M. G. Grillakis* [ J. Hyperbolic Differ. Equ. 4, No. 2, 233–257 (2007;

Zbl 1122.35132)] where global well-posedness was established for any

$s\ge \frac{1}{2}$. We use the

$I$-method to take advantage of the conservation laws of the equation. The new ingredient is an interaction Morawetz estimate similar to one that has been used to obtain global well-posedness and scattering for the cubic NLS in 3D. The derivation of the estimate in our case is technical since the smoothed out version of the solution

$Iu$ introduces error terms in the interaction Morawetz inequality. A by-product of the method is that the

${H}^{s}$ norm of the solution obeys polynomial-in-time bounds.