I'm posting this on MathOverflow because I'm not sure if it would get much of a response on Math.SE. Please feel free to remove it if it is not "research-level".

I was studying Donaldson's self-dual equations from the book *Instantons and Four-Manifolds* by Freed and Uhlenbeck and I came across the following elliptic complex, whose index is computed by the Atiyah-Singer Index Theorem:

$$0 \to \Omega^0(\text{ad} \eta)_\ell \xrightarrow{D} \Omega^1(\text{ad} \eta)_{\ell - 1} \xrightarrow{d_1P} \Omega^2_-(\text{ad} \eta)_{\ell - 2}.$$

The "$\ell$" subscripts denote Sobolev regularity, and the "minus" subscript on the rightmost space denotes the space of anti-self-dual forms. To unpack the rest, we have $\eta$ is a principal $SU(2)$-bundle and $D$ is some fixed covariant derivative. Finally, $d_1P$ is the differential in the first factor of the operator $$P: \Omega^1(\text{Ad} \eta)_\ell \oplus C \to \Omega^2_-(\text{ad} \eta)_{\ell - 2}$$ where $C$ is the space of perturbations of the metric.

Could anyone familiar with this computation give some hint/insight/resource explaining how it is done?

"Topology and Analysis: Atiyah-Singer index formula and gauge-theoretic physics"was made for this, especially since I don't know what background you already have. $\endgroup$ – Chris Gerig Jun 21 '17 at 6:11