Suppose the theory of chiral Weyl fermion (say, left) $\psi_{L}$, which interacts with abelian gauge field. This theory contains the **gauge anomaly**, which I write in the form

$$

\tag 1 \frac{dQ_{L}}{dt} = \text{A},

$$

where $Q_{L}$ is the left charge and $A$ is anomaly function.

The same thing is true about right fermion $\psi_{R}$. If the gauge field is vector (not axial vector), then

$$

\tag 2 \frac{dQ_{R}}{dt} = -\text{A},

$$

The underlying reason for this is that the dynamics of theory generates anomalous commutator between canonical momentums of EM field (the electric field $\mathbf E$): pcecisely,

$$

\tag 3 [E_{i}(\mathbf x), E_{j}(\mathbf y)]_{L/R} = -i\Delta^{ij}_{L/R}(\mathbf A, \mathbf y)\delta (\mathbf x - \mathbf y),

$$

where $L,R$ denotes the subspaces of left and right fermions. This gives (see the question) the anomaly $\text{A}$:

$$

\tag 4 \frac{dQ_{L/R}}{dt} = \text{A} = \int d^{3}\mathbf r E_{i}(\mathbf r)\partial_{j}\Delta^{ij}_{L/R}(\mathbf A, \mathbf r)

$$

Suppose now we take the "direct sum" of left and right representations:

$$

\psi = \psi_{L} \oplus \psi_{R}

$$

In this case, by using $(1), (2)$, we see, that there is no gauge anomaly of vector charge $Q_{\text{vector}}$,

$$

\tag 5 \frac{dQ_{\text{vector}}}{dt} = \frac{dQ_{L}}{dt} + \frac{dQ_{R}}{dt} = \text{A} - \text{A} = 0,

$$

but there is the chiral anomaly of axial charge $Q_{\text{axial}}$,

$$

\tag 6 \frac{dQ_{\text{axial}}}{dt} =\frac{dQ_{L}}{dt} - \frac{dQ_{R}}{dt} =\text{A}+\text{A}= 2\text{A}

$$

Since the vector current is the gauge current, then the **total** contribution into anomalous commutator from the left and right particles must vanish:

$$

\tag 7 [E_{i}(\mathbf x), E_{j}(\mathbf y)]_{L\oplus R} = -i\Delta^{ij}_{L}(\mathbf A, \mathbf y)\delta (\mathbf x - \mathbf y)-i\Delta^{ij}_{R}(\mathbf A, \mathbf y)\delta (\mathbf x - \mathbf y) = 0

$$

Although we assume the left and right particles with the same charge and mass, this looks like anomaly cancellation. The left and right fermions remain anomalous separately.

**My question is following**. Although the anomalous commutator $(3)$ exists on subspaces $L$ and $R$, it vanishes for their direct sum, as is shown by $(7)$. But the chiral anomaly $(6)$ exists. In terms of broken canonical commutator $(3)$ I can understand this phenomena as the fact that this commutator violates chiral symmetry. This is the direct consequence of Eqs. $(4)$, $(6)$. But to me is very strange that the gauge anomaly of left and right fermions give the ungauged anomaly for the difference of their fermion number. Is my understanding correct?