  The semiclassical motion of electrons in phase space, x=(R, k), is influenced
by Berry phases described by a 6-component vector potential, A=(A^R, A^k). In
chiral magnets Dzyaloshinskii-Moriya (DM) interactions induce slowly varying
magnetic textures (helices and skyrmion lattices) for which all components of A
are important inducing effectively a curvature in mixed position and momentum
space. We show that for smooth textures and weak spin-orbit coupling phase
space Berry curvatures determine the DM interactions and give important
contributions to the charge. Using ab initio methods we calculate the strength
of DM interactions in MnSi in good agreement with experiment and estimate the
charge of skyrmions.
