  We reconsider the derivation of Ward identities for spin stiffnesses, which
determine the non-linear sigma model of magnetic degrees of freedom of
interacting electrons in the presence of antiferromagnetic or incommensurate
correlations. We emphasize that in the approaches, which do not break
explicitly spin symmetry of the action, the spatial components of gauge kernel,
which is used to obtain spin stiffnesses, remain gauge invariant even in case
of spontaneous spin symmetry breaking. We derive the corrected Ward identities,
which account for this gauge invariance. We emphasize that the frequency
dependence of temporal spin stiffnesses is not fixed by the obtained
identities, and show that the infinitesimally small external staggered field is
crucially important to obtain finite static uniform transverse susceptibility.
On the other hand, we find that the spatial spin stiffnesses are determined by
the gauge kernel of the Legendre transformed theory, which is in general {\it
different} from the gauge kernel of the original theory and obtain an explicit
expressions for spatial spin stiffnesses through susceptibilities and current
correlation functions. We verify numerically the obtained results within
dynamic mean field theory, and obtain doping dependencies of the resulting spin
stiffnesses for antiferromagnetic and incommensurate phase.
