Naive Ward identities for the topological gluon current

The question of which symmetry can be associated with the topological current Kmu of the gluons is discussed. It is shown that in the case of a supersymmetric theory the matrix elements of ∂μKμ can be found by varying the effective action with respect to the bare coupling constant. The relation obtained this way is an analog of the naive conservation equation for the axial vector current of the fermions: the contribution of the triangle diagram does not satisfy the relation and represents an anomaly. For the case of pure gluodynamics it is shown that one can introduce conserved chiralities, but only at the one-loop level and for special choices of the background field. The vanishing of the corresponding matrix elements of ∂μKμ can naturally be considered as naive Ward identities for Kμ in this case. It follows from them, in particular, that the contribution of the triangle diagram vanishes