The geometry of antiferromagnetic spin chains

It has been a conjecture of F.D.M. Haldane that long-wavelength excitations around antiferromagnetic vacua of certain spin chains are related to relativistic two-dimensional $\sigma$-models. We construct a class of such spin chains and associated $\sigma$-models, using symplectic geometry as a main tool. The target space of the $\sigma$-model can be an arbitrary complex flag manifold, and we find universal expressions for the metric and $\theta$-term. The derivation relies on the fact that the flag manifold is a Lagrangian submanifold in a direct product of Grassmannians associated (via geometric quantization) to the representations of spins at consecutive sites of the spin chain.