The index theorem for chiral anomaly is written as $$ n_{+}-n_{-} = \nu, $$ where $n_{+} - n_{-}$ is the index of Dirac operator $$ D = \gamma_{j}(i\partial_{j} - A_{j}), $$ while $\nu$ is the winding number of the Chern character of gauge fields.
What is the importance of index theorem? Also, what is the physical sense of index theorem?
The fact which I know is that the index theorem relates the chiral anomaly to the topology of classical gauge fields: since the left hand-side is integer number, the right hand-side can't be changed continuously due to small perturbations under gauge field configurations.
However, it's not clear for me what is the opportunity of index theorem. Precisely, the chiral anomaly actually says current $J_{\mu}$ has anomalous conservation law with anomaly $\text{A}$, so that $$ \partial_{\mu}J^{\mu} = \text{A}(x), $$ which after integration over 4-space says that $$ Q(t = +\infty) - Q(t = -\infty) = \int d^{4}x\text{A}(x) $$ The right hand-side now has to be integer, without any index theorems, and thus is determined from the topology gauge fields. Therefore, the physical meaning of index theorem is still not clear for me.
