Zumino's consistent and covariant anomalies - applied to quantum hall?

Question

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 arXiv:1307.7480. 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?

Answer 1

In the Hall effect, the edge modes that possess an anomaly are connected to the bulk in such a way that the total system is gauge invariant and and has a conserved current. The Bardeen Zumino consistancy conditions arise from considering the current $J_{\mu {\rm consistent}}$ as the functional derivative with respect to $A_\mu$ of the edge effective action on the edge while ignoring the bulk effective action. When one computes the currents in the bulk by functionally differentiating the Chern-Simons bulk effective action with respect to $A_\mu$ you get the bulk Hall current whose inflow to the edge gives the anomaly, --- but you will have integrated by parts to get this expression and the integrated-out boundary term contributes to the edge currents. The bulk effective action contribution to the edge current are precisely the "Bardeen Polynomial" terms that when added to the currents in the "consistent anomaly" convert it to the current $J_{\mu {\rm covariant}}$" that appears in the covariant anomaly". On its own the edge theory is physically inconsistent, but is "consistent" in the sense of Bardeen and Zumino. The combined bulk plus edge theory is physically consistent, even though B and Z regard as "inconsistent". The combined theory gives the true physical edge current in which the covariant anomaly arises from the inflow from the bulk. For more details see section III of arXiv:1201.4095