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
See in (AzcarragaIzqierdo) 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 grup of rotations?. This has a canonical global coordinate chart . We may regard it as the first order jet bundle to the bunde whose sections are trajectories in Cartesian space .
Among the -left invariant 2-forms on is
for some (where a contraction of vectors is understood).
This is a representative of degree-2 Lie algebra cohomology of . 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 is the Lagrangian of the the free non-relativistic particle of mass .
Applied to jet-prolongations of sections of the field bundle for which 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
- J.A. Azcárraga, J. Izqierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge monographs of mathematical physics, (1995)
Revised on June 14, 2012 17:21:47
by Urs Schreiber