nLab
1d WZW model

Context

\infty-Wess-Zumino-Witten theory

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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

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.

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)