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1d WZW model

Context

\infty-Wess-Zumino-Witten theory

Models

  • 2d

Quantum field theory

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Physics

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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:

(Azcarraga-Izqierdo 95, 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 group 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 bundle d×\mathbb{R}^d \times \mathbb{R} \to \mathbb{R} whose sections are trajectories in Cartesian space d\mathbb{R}^d (the field bundle for the 1d sigma-model with target space d\mathbb{R}^d).

Among the HH-left invariant differential 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 a degree-2 cocycle in the Lie algebra cohomology of Lie(H)Lie(H). We may regard this as the curvature of a 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.

Evaluated on jet prolongations of sections of the field bundle, for which the relation dx=x˙dtd 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.