1d WZW model
Quantum field theory
In the context of higher dimensional WZW models the following 1-dimensional sigma-models are seen to be examples:
(Azcarraga-Izqierdo 95, section 8.3 and 8.7) .
Free massive non-relativistic particle
for the coset obtained as the quotient of the Galilei group in some dimension by the group of rotations. This has a canonical global coordinate chart . We may regard it as the first order jet bundle to the bundle whose sections are trajectories in Cartesian space (the field bundle for the 1d sigma-model with target space ).
Among the -left invariant differential 2-forms on is
for some (where a contraction of vectors is understood).
This is a representative of a degree-2 cocycle in the Lie algebra cohomology of . We may regard this as the curvature of a connection 1-form
Hence the value of the action functional of the corresponding 1d pure (topological) WZW model on a field configuration is
where is the Lagrangian of the the free non-relativistic particle of mass .
Evaluated on jet prolongations of sections of the field bundle, for which the relation holds, then the first term of this expression vanishes and so the resulting WZW-type action functional is that of the free non-relativistic particle.
See (Azcarraga-Izqierdo, section 8.3) for a useful account.
Revised on January 10, 2017 16:33:57
by Urs Schreiber