quantum anomalies - Choice of basis for Fujikawa method to derive chiral anomaly

I am studying the Fujikawa method of determining the chiral anomalies in a $U(1)$ theory. As we know the basis vectors selected are the eigenstates of the Dirac operator. One of the reasons given is that the eigenstates diagonalize the action which is needed for determining an exact quantity such as Ward-Takahashi identities. Anyone care to explain? I am referring to Path Integrals and Quantum Anomalies by Kazuo Fujikawa and Hiroshi Suzuki.

Answer

Alright, I will try to answer why we need Dirac eigenstates in this procedure, but I am not sure if it is anything more that the tautology that the Fujikawa method is precisely defined by using the Dirac eigenstates.

Let me briefly recap what the idea is: (All of this is for a Euclidean theory.)

We consider the infinitesimal local transformaton

$$\psi(x) \mapsto (1 + \mathrm{i}\alpha(x)\gamma_5)\psi(x) \equiv \psi'(x)$$

which, as it is a gauge transformation, should not change the value of the path integral:

$$ Z = \int \mathcal{D}\psi\mathcal{D}\bar\psi \mathrm{e}^{\int\bar\psi \mathrm{i}\not{D}\psi\mathrm{d}^4x} \overset{!}{=} \int\mathcal{D}\psi'\mathcal{D}\bar\psi'\mathrm{e}^{\int \bar\psi\not{D}\psi\mathrm{d}^4x + \int\alpha\partial_\mu j_5^\mu\mathrm{d}^4x}$$

To derive the anomaly term, we must examine what the change of the measure $\mathcal{D}\psi \mapsto \mathcal{D}\psi'$ is. In general, we can say that it is, by the usual transformation formulae for Grassmannian integrations, $\det(M)^{-1}$ for the operator acting as $\psi' = M\psi$.

Now, we must recall that the functional measure $\mathcal{D}\psi$ is only defined in the limit of some (UV) regularized theory, it does not exist on its own.. So to actually calculate the correct $\det(M)$, we must obtain it as the limit $\lim_{\Lambda \rightarrow \infty}\det(M_\Lambda)$ of some $M_\Lambda$ and some UV cutoff scale (not necessarily hard) $\Lambda$. UV regulators suppress high momenta. And what are the eigenstates of the Dirac operator? ... Right, they are the momentum modes! So, the most natural regularisation of our theory is to exponentially suppress states with high Dirac eigenvalues as per

$$ \widetilde{\psi}_n = \mathrm{e}^{-\frac{\not{D}^2}{2\Lambda}}\psi_n$$

where the $\psi_n$ are the unregularised Dirac eigenstates. And by choosing the regulator that way, the only consistent way to define the measure is to see it as the integral over the coefficients of these modes, i.e.

$$\mathcal{D}\psi = \lim_{\Lambda \rightarrow \infty}\prod_n \mathrm{d}\widetilde{a}_n$$

where $\psi = \sum_n \widetilde{a}_n\widetilde{\psi}_n$. (The sum over the $n$ is actually an integral if we use no IR cutoff, but it doesn't matter here, you could as well use one, but it clutters notation a bit and contributes nothing insightful.)

Now, the rest of the Fujikawa anomaly method follows through as hopefully described in your book. I hope this is at least an approximate answer to your question.