Index theorem and 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?

Answer 1

The index theorem implies that in a given topological sector $\nu$ there are $n_L,n_R$ L/R zero modes such that $n_L-n_R=\nu$. These are solutions of the 4D euclidean Dirac equation $\gamma\cdot D\psi=0$. In particular, $\psi$ must be normalizable in 4D.

Now (for simplicity) go to temporal gauge and look at the associated Dirac equation $\partial_t\psi = i\alpha\cdot D\psi$. For smoothly varying fields the 4D solutions must correspond to adiabatic solution of the type $$ \psi(x,t) = \psi(x,-\infty) \exp(-\int^t_{-\infty}\epsilon(t') dt')\, . $$ Now the only way that $\psi$ is normalizable is that $\epsilon$ changes sign as $t$ goes from $-\infty$ to $+\infty$. This means that the spectral flow of the Dirac Hamiltonian $H$ is equal to the chiral imbalance of the 4D zero modes modes, which by the index theorem is determined by 4D topology.