nLab
1d WZW model

Idea

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.

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

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

$${\omega }_m≔m(d_{\mathrm{dR}}x-\dot{x}{d}_{\mathrm{dR}}t)\wedge d_{\mathrm{dR}}\dot{x}$$\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 $\mathrm{Lie}(H)$. Taking it to be the curvature of a WZW 1-bundle with connection 1-form

$$A≔m\dot{x}(d_{\mathrm{dR}}x-\frac{1}{2}\dot{x}{d}_{\mathrm{dR}}t).$$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

$$m{\int }_{{\Sigma }_1}\dot{x}(d_{\mathrm{dR}}x-\frac{1}{2}\dot{x}{d}_{\mathrm{dR}}t)=m{\int }_{{\Sigma }_1}\frac{\partial L}{\partial \dot{x}}(dx-\dot{x})+Ldt,$$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 $L(x,\dot{x})dt=\frac{1}{2}m{\dot{x}}^{2}dt$ 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 $dx=\dot{x}dt$ 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)