  We exhibit new functions of the eigenvectors of the Dyson Brownian motion
which follow an equation similar to the Bourgade-Yau eigenvector moment flow.
These observables can be seen as a Fermionic counterpart to the original
(Bosonic) ones. By analyzing both Fermionic and Bosonic observables, we obtain
new correlations between eigenvectors. The fluctuations $\sum_{\alpha\in
I}u_k(\alpha)^2-{\vert I\vert}/{N}$ decorrelate for distinct eigenvectors as
the dimension $N$ grows and an optimal estimate on the partial inner product
$\sum_{\alpha\in I}u_k(\alpha)u_\ell(\alpha)$ between two eigenvectors is
given. These static results obtained by integrable dynamics are stated for
generalized Wigner matrices and should apply to wide classes of mean field
models.
