What is the `physical' meaning of **consistent anomalies** and **covariant anomalies**?

Perhaps a good Reference is: Consistent and covariant anomalies in gauge and gravitational theories - William A. Bardeen and Bruno Zumino

I kind of remember (and used to think) that: $$ \text{consistent anomaly} =(1/2) (\text{covariant anomaly}) $$

So the physical picture I have is, for example a 1+1D system. See a Reference doi.org/10.1103/PhysRevB.107.014311. Consider this 1+1D theory lives as the edge theory on the boundary of a 2+1D spatial cylinder. There is an (integer) quantum hall state with charge U(1) symmetry.

On the left edge, there is a left-moving current with a `consistent' anomaly $$ \partial_\mu J_L^\mu =(e/4\pi)\epsilon^{\mu\nu} F_{\mu\nu}(=\text{consistent anomaly}?) $$

On the right edge, there is a right-moving current with another `consistent' anomaly $$ \partial_\mu J_R^\mu =-(e/4\pi)\epsilon^{\mu\nu} F_{\mu\nu}(=-\text{consistent anomaly}?) $$

Consider putting these two edges more-or-less together as the same 1+1D (but without direct interactions), shows axial anomaly: $$ \partial_\mu J_A^\mu=\partial_\mu (J_L^\mu-J_R^\mu) =(e/2\pi)\epsilon^{\mu\nu} F_{\mu\nu}(=\text{covariant anomaly}?) $$

while vector current conserved: $$ \partial_\mu J_V^\mu=\partial_\mu (J_L^\mu+J_R^\mu) =0 $$

At least, this physical picture produces: $$ \text{consistent anomaly} =(1/2) (\text{covariant anomaly}) $$

Can someone inform whether this is a right picture or not for the **consistent anomalies** and **covariant anomalies**?

This post imported from StackExchange Physics at 2014-06-04 11:40 (UCT), posted by SE-user Idear