Index theorem, UV and IR face of chiral anomaly

Question

The index theorem in theory with fermions and gauge fields implies the relation between the index $n_{+}-n_{-}$ of Dirac operator and the integral $\nu$ over EM field chern characteristic class: $$ \tag 1 n_{+} - n_{-} = \nu $$ Let's focus on 4D. The index theorem is obtained by computing anomalous jacobian $$ J[\alpha] = \text{exp}\left[-2i\alpha \sum_{n = 1}^{N = \infty}\int d^{4}x_{E}\psi^{\dagger}_{n}\gamma_{5}\psi_{n}\right] $$ Here $n$ denotes the number of eigenfunction of the Dirac operator
$$
D_{I}\gamma_{I}, \quad D_{I} \equiv i\partial_{I} - A_{I} 
$$

From the one side, this is bad defined quantity,
$$
J[\alpha] \simeq \text{exp}\left[i\alpha \lim_{x \to y}\text{Tr}(\gamma_{5})\delta (x - y)\right],
$$
so it requires the UV regularization. The explicit form of this regularization is fixed by the requirements of gauge and ''euclidean'' invariance, leading to introducing the function $f\left( \left(\frac{D_{I}\gamma_{I}}{M}\right)^{2}\right)$, with $M$ being the regularization parameter. From the other side, by using the regularization, it is not hard to show that the exponent is equal to the $-2i\alpha (n_{+}-n_{-})$. Since this number defines the difference of zero modes, it depends only on IR property of theory. Moreover, $\nu$ is also determined by the behavior of gauge fields on infinities, being IR defined number. 

Because of this ''puzzle'', I want to ask: does the index theorem provide the relation between IR (zero modes, large scale topology) nature and UV (regularization required) nature of chiral anomaly? Precisely, I know the "spectral flow" interpretation of chiral anomaly, according to which an anomaly is the collective motion of chiral charge from UV world to IR one. Does the index theorem provide this interpretation?

Comments

In short: yes.

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@40227 : I know the "spectral flow" interpretation of chiral anomaly, according to which an anomaly is the collective motion of chiral charge from UV world to IR one. Does the index theorem provide this interpretation?