# Contents

## Idea

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.

## Examples

### Free massive non-relativistic particle

Write

$H≔G/R$H \coloneqq G/R

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 $\left(t,x,\stackrel{˙}{x}\right)$. We may regard it as the first order jet bundle to the bunde ${ℝ}^{d}×ℝ\to ℝ$ whose sections are trajectories in Cartesian space ${ℝ}^{d}$.

Among the $H$-left invariant 2-forms on $H$ is

${\omega }_{m}≔m\left({d}_{\mathrm{dR}}x-\stackrel{˙}{x}{d}_{\mathrm{dR}}t\right)\wedge {d}_{\mathrm{dR}}\stackrel{˙}{x}$\omega_m \coloneqq m (d_{dR} x - \dot x d_{dR} t) \wedge d_{dR} \dot x

for some $m\in ℝ$ (where a contraction of vectors is understood).

This is a representative of degree-2 Lie algebra cohomology of $\mathrm{Lie}\left(H\right)$. Taking it to be the curvature of a WZW 1-bundle with connection 1-form

$A≔m\stackrel{˙}{x}\left({d}_{\mathrm{dR}}x-\frac{1}{2}\stackrel{˙}{x}{d}_{\mathrm{dR}}t\right)\phantom{\rule{thinmathspace}{0ex}}.$A \coloneqq m \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t) \,.

Hence the value of the action functional of the corresponding 1d pure (topological) WZW model on a field configuration is

$m{\int }_{{\Sigma }_{1}}\stackrel{˙}{x}\left({d}_{\mathrm{dR}}x-\frac{1}{2}\stackrel{˙}{x}{d}_{\mathrm{dR}}t\right)=m{\int }_{{\Sigma }_{1}}\frac{\partial L}{\partial \stackrel{˙}{x}}\left(dx-\stackrel{˙}{x}\right)+Ldt\phantom{\rule{thinmathspace}{0ex}},$m \int_{\Sigma_1} \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t) = m \int_{\Sigma_1} \frac{\partial L}{ \partial \dot x}(d x - \dot x)+ L d t \,,

where $L\left(x,\stackrel{˙}{x}\right)dt=\frac{1}{2}m{\stackrel{˙}{x}}^{2}dt$ 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 $dx=\stackrel{˙}{x}dt$ 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.

## References

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 (94.136.12.233)