Anomalies in gauge symmetries[2] lead to an inconsistency, since a gauge symmetry is required in order to cancel degrees of freedom with a negative norm which are unphysical (such as a photon polarized in the time direction).
For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.
the operator corresponding to an infinitesimal gauge transformation by ε, then the Frobenius consistency condition requires that for any functional
, including the (semi)effective action S where [,] is the Lie bracket.
Let us make the further assumption (which turns out to be valid in all the cases of interest) that this functional is local (i.e. Ω(d)(x) only depends upon the values of the fields and their derivatives at x) and that it can be expressed as the exterior product of p-forms.
If we then arbitrarily extend the fields (including ε) as defined on Md to Md+1 with the only condition being they match on the boundaries and the expression Ω(d), being the exterior product of p-forms, can be extended and defined in the interior, then The Frobenius consistency condition now becomes As the previous equation is valid for any arbitrary extension of the fields into the interior, Because of the Frobenius consistency condition, this means that there exists a d+1-form Ω(d+1) (not depending upon ε) defined over Md+1 satisfying Ω(d+1) is often called a Chern–Simons form.
Once again, if we assume Ω(d+1) can be expressed as an exterior product and that it can be extended into a d+1 -form in a d+2 dimensional oriented manifold, we can define in d+2 dimensions.