# nLab 1d WZW model

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# Contents

## Idea

In the context of higher dimensional WZW models the following 1-dimensional sigma-models are seen to be examples:

## Examples

### Free massive non-relativistic particle

Write

$H \coloneqq G/R$

for the coset obtained as the quotient of the Galilei group in some dimension $d$ by the group 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 bundle $\mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$ whose sections are trajectories in Cartesian space $\mathbb{R}^d$ (the field bundle for the 1d sigma-model with target space $\mathbb{R}^d$).

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

$\omega_m \coloneqq m (d_{dR} x - \dot x d_{dR} t) \wedge d_{dR} \dot x$

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

This is a representative of a degree-2 cocycle in the Lie algebra cohomology of $Lie(H)$. We may regard this as the curvature of a connection 1-form

$A \coloneqq m \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t) \,.$

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

$m \int_{\Sigma_1} \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t) = m \int_{\Sigma_1} \frac{\partial L}{ \partial \dot x}(d x - \dot x)+ L d t \,,$

where $L(x, \dot x) d t = \frac{1}{2}m \dot x^2 d t$ is the Lagrangian of the the free non-relativistic particle of mass $m$.

Evaluated on jet prolongations of sections of the field bundle, for which the relation $d x = \dot x d t$ 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.

## References

Last revised on January 10, 2017 at 16:33:57. See the history of this page for a list of all contributions to it.