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In the context of higher dimensional WZW models the following 1-dimensional sigma-models are seen to be examples
the free non-relativistic particle;
the massive Green-Schwarz superparticle.
See in (AzcarragaIzqierdo) section 8.3 and 8.7.
Write
for the coset obtained as the quotient of the Galilei group in some dimension $d$ by the grup of rotations. This $H$ has a canonical global coordinate chart $(t,x, \dot x)$. We may regard it as the first order jet bundle to the bunde $\mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$ whose sections are trajectories in Cartesian space $\mathbb{R}^d$.
Among the $H$-left invariant 2-forms on $H$ is
for some $m \in \mathbb{R}$ (where a contraction of vectors is understood).
This is a representative of degree-2 Lie algebra cohomology of $Lie(H)$. Taking it to be the curvature of a WZW 1-bundle with connection 1-form
Hence the value of the action functional of the corresponding 1d pure (topological) WZW model on a field configuration is
where $L(x, \dot x) d t = \frac{1}{2}m \dot x^2 d t$ is the Lagrangian of the the free non-relativistic particle of mass $m$.
Applied to jet-prolongations of sections of the field bundle for which $d x = \dot x d t$ the first term vabishes and so the WZW-type action is that of the free non-relativistic particle.
See (Azcarraga-Izqierdo, section 8.3) for a useful account.
Section 8.3 and 8.7 of