1d WZW model


\infty-Wess-Zumino-Witten theory

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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


HG/R H \coloneqq G/R

for the coset obtained as the quotient of the Galilei group in some dimension dd by the grup of rotations. This HH has a canonical global coordinate chart (t,x,x˙)(t,x, \dot x). We may regard it as the first order jet bundle to the bunde d×\mathbb{R}^d \times \mathbb{R} \to \mathbb{R} whose sections are trajectories in Cartesian space d\mathbb{R}^d.

Among the HH-left invariant 2-forms on HH is

ω mm(d dRxx˙d dRt)d dRx˙ \omega_m \coloneqq m (d_{dR} x - \dot x d_{dR} t) \wedge d_{dR} \dot x

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

This is a representative of degree-2 Lie algebra cohomology of Lie(H)Lie(H). Taking it to be the curvature of a WZW 1-bundle with connection 1-form

Amx˙(d dRx12x˙d dRt). 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 Σ 1x˙(d dRx12x˙d dRt)=m Σ 1Lx˙(dxx˙)+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,x˙)dt=12mx˙ 2dtL(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 mm.

Applied to jet-prolongations of sections of the field bundle for which dx=x˙dtd x = \dot x d t 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 (