# Why is the Wess-Zumino condition on anomalies called a "consistency condition"?

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I mean, consistency of what with what ? Why is it important that a given anomaly is (or is not) consistent in this sense?

recategorized Apr 7, 2016

Please give some context. Who called it that where?

@ArnoldNeumaier It's not uncommon to see it being called a consistency condition, e.g. hep-th:0704-2472

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The beginning of the paper

Wess, Julius, and Bruno Zumino. "Consequences of anomalous Ward identities." Physics Letters B 37.1 (1971): 95-97.

It is known [1] that the naive or normal Ward identities are not always satisfied in renormalized perturbation theory. In other words, the Green's functions do not always satisfy the identities which correspond to invariance or partial invariance of the basic Lagrangian. In the case of the partial conservation equation for the axial vector current, anomalies occur if there are fundamental spinor fields in the theory. They are due to the singularities of the spinar triangle graph and, in the non-abelian case of SU(3)$\otimes$SU(3), of higher polygonal graphs. It is likely that the anomalies due to these single loop graphs are not modified by higher order many loop corrections.

In this note we observe that the anomalies must satisfy consistency or integrability relations which follow from the structure of the gauge group and which are non trivial in the case of nonabelian groups.

The anomalies (nonvanishing divergence terms) must match the integrability conditions listed on the top right of p.96. An arbitrary potential assignment of values to all these divergence terms is consistent (i.e., can possibly correspond to an anomaly) only if the assignments are chosen to satisfy these conditions. Thus one can consider these integrability conditions as well as consistency conditions. The abstract of the paper already introduces this terminology:

The anomalies of Ward identities are shown to satisfy consistency or integrability relations, which restrict their possible form.

The paper  hep-th:0704-2472 mentioned by dimension10 interprets these consistency conditions in terms of cohomology.

answered Apr 13, 2016 by (13,979 points)
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Sorry to offer answer to my own question, but i have found the most cogent answer, and i think i should share it.
In ticciatti's book, "Quntum field theory for mathematicians," p.427,
one finds,
"The fundamental distinction between the various currents involved
is whether or not they can be obtained by varying the action with respect to a field.
If they can, then they must satisfy an integrability condition, first formulated by
Wess and Zumino."

It seems that wess-zumino condition, is a necessary condition for the anomaly in question being derivable by varriation with respect to *local*
fields.

As opposed to other, globally derived, anomaliles.

Thiat it has
a cohomological interpretation does not seem to
have played a role, a priory, in calling it a "consistency condition".
answered Apr 21, 2016 by (90 points)

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