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:

(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

Revised on January 10, 2017 16:33:57 by Urs Schreiber (87.191.41.20)