# Abelian Higgs model and conserved vortex current

1. Dec 5, 2011

### crackjack

Refer page-24 in http://arxiv.org/abs/1010.0443v3

Eq-6.19 is supposed to be the Lagrangian of Abelian-Higgs models. Eg-6.19 is clearly a conserved current of U(1) symmetry. But I cant show how eq-6.21 is a conserved vortex current.

Note: The last term in the Lagrangian is not the usual $F_{\mu \nu} F^{\mu \nu}$.

2. Dec 6, 2011

### crackjack

I see it now. I missed the fact that it is a 2+1 dimensional model

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If it helps anyone...

In 2+1 dimensions, $\epsilon^{\mu \nu \lambda} \partial_\nu A_\lambda$ (cf. 6.21) is proportional to the dual field strength tensor:
$$\hat{J}^\mu \propto \tilde{F}^\mu = \star F_{\nu \lambda}$$

So,  $\hat{J}^\mu$is conserved simply by the Bianchi identity (or the homogeneous Maxwell equation):
$$\partial_\mu \tilde{F}^\mu = 0$$

PS: In 2+1 dim, $( \epsilon^{\mu \nu \lambda} \partial_\nu A_\lambda )^2 \propto F_{\mu \nu} F^{\mu \nu}$ (last term in the lagrangian of eq-6.19)

Last edited: Dec 6, 2011