physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
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