 If there is a 't Hooft anomaly for formula_5, the effective action formula_7 is not invariant under the formula_5 gauge transformation on the background gauge field formula_6 and it cannot be restored by adding any four-dimensional local counter terms of formula_6.Wess–Zumino consistency condition shows that we can make it gauge invariant by adding the five-dimensional Chern–Simons action.With the extra dimension, we can now define the effective action formula_7 by using the low-energy effective theory that only contains the massless degrees of freedom by integrating out massive fields.