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

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?

• You will find interesting informations page $3$ of the paper you first cited in free access here – Trimok Nov 13 '13 at 10:24

## 1 Answer

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

• Thanks for the reply, +1, I will have a thought on this. Is your definition the same as the bertlmann's book "Anomalies in QFT?" – wonderich Jun 19 '15 at 14:35
• Yes. I think so. It's a nice book, but does not explain the physical distinction between consistent and covariant anomalies. I think that was first figured out in: S.G. Nachulich, "Axionic Strings: Covariant anomalies and bosonization of chiral zero modes" Nucl Phys.B296 837-867 (1988). – mike stone Jun 26 '15 at 16:11
• Sorry for asking, but what is the notion of the covariant anomly in the abelian case? The consistent anomaly is indeeed gauge covariant, so I don't understand how people describe the anomalous Hall effect by adding the mystical Bardeen-Zumino polynomial, which we don't need for the abelian case (there are few articles, which operate the notions of consistent and covariant anomalies and say that by using the definition of the covariant anomaly we explain the AHE). – Name YYY Nov 18 '16 at 9:13
• I think that in the Abelian case the boundary term that usually arises when we functionally diffrentiate the bulk Chern Simons term is ${\rm tr}(A^2)$. This is zero as the 1-forms $A$ anticommute. Thus there is no Bardeen Polynomial and no distinction between consistent an covariant. Which AHE papers are you thinking of that make a distinction? – mike stone Dec 4 '16 at 16:49