Geometric derivation of nonlinear sigma model for the 1D antiferromagnet

It is argued that a purely geometric derivation of the long-range action for the 1D antiferromagnet is available in terms of a Kähler potential. The derivation allows for a natural extension to the t-J model. In particular, it follows that a relevant long-wavelength action of the t-J model exhibits at least at the SUSY (J=2t) point the su(2∣1) invariance rather than the so(5) one.