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 (the D0 super p-brane).

(Azcarraga-Izqierdo 95, section 8.3 and 8.7) .

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

• José de Azcárraga, J. Izqierdo, sections 8.3 and 8.7 of Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge monographs of mathematical physics, (1995)