In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a Hausdorff manifold, and if it is not the empty set, then the dimension is given by
$$p\_1(\text{Ad}(P))-\frac{1}{2}\dim G(\chi(M)-\tau(M))$$
Where $\chi(M)$ is the Euler characteristic and $\tau(M)$ the signature.


**EDIT:** It turns out the original paper contained an error/typo. It should in fact be
$$2p\_1(\text{Ad}(P))-\frac{1}{2}\dim G(\chi(M)-\tau(M))$$
**End of edit.**


Although I would love to be able to understand the full paper, I am not in a position to be able to do so yet, I am only trying to understand the computation of this dimension, because I am interested in some applications of the Atiyah-Singer index theorem.  

To compute this dimension, the following is utilised in the paper:
Let $D:\Gamma(V\_-\otimes E)\to\Gamma(V\_+\otimes E)$ be the Dirac operator for a spinor bundle with values in some auxiliary bundle $E$. By the index theorem,
$$\text{ind}(D)=\int\_M\text{ch}(E)\widehat{A}(M)$$
In dimension four, we have $\widehat{A}(M)=1-\frac{1}{24}p\_1(M)$ (but where is this used?). For the proof, we take $E=V\_-\otimes\text{Ad}(P)$. Then $\text{ch}(E)=\text{ch}(\text{Ad}(P))\text{ch}(V\_-)$. So far, so good. I lose track in the following computation:
$$\text{ind}(D)=\int\_M\text{ch}(\text{Ad}(P))\text{ch}(V\_-)\widehat{A}(M)\\
\color{red}{=p\_1(\text{Ad}(P))+\dim G(\text{ind}(D'))}=\\
p\_1(\text{Ad}(P))-\frac{1}{2}\dim G(\chi-\tau)$$
Where $D':\Gamma(V\_+\otimes V\_-)\to\Gamma(V\_-\otimes V\_-)$. I have been trying to find a result that explains the red coloured part of the equation, because this step seems completely non-trivial, and in spite of that, it is not elaborated upon within the paper at all, and I am not able to find any sources that explain this step. In [Index of Dirac operator and Chern character of symmetric product twisting bundle](https://mathoverflow.net/questions/351498/index-of-dirac-operator-and-chern-character-of-symmetric-product-twisting-bundle) the accepted answer seems to give an answer that goes some way towards explaining how this result is obtained, in a very particular case. However, I am not very experienced in this area and I don't know how to generalise the result to an arbitrary principal $G$-bundle. I am looking for an explanation of the above, whether someone is able to provide their own response or a reference. Either one would be greatly appreciated.



Atiyah's proof of the moduli space of SD irreducible YM connections
Hopefully I remember this well. My adviser explained this computation to me I don't even want to think how many years ago.


The deformation complex of the SD equation is $\DeclareMathOperator{\Ad}{Ad}$


$$L=d\_A^-\oplus d\_A^\*:\Omega^1\big(\, \Ad(P)\,\big)\to\Omega^2\_-\big(\; \Ad(P)\;\big)\oplus \Omega^0\big(\;\Ad(P)\;\big). $$


The dimension of the moduli space of self-dual connections is the index of this operator. $\DeclareMathOperator{\ind}{ind}$ $\DeclareMathOperator{\ch}{ch}$ $\DeclareMathOperator{\hA}{\widehat{A}}$This operator is obtained by twisting with $\Ad(P)$ the operator


$$ D=d^-+d^\*:\Omega^1(M)\to \Omega^2\_-(M)\oplus \Omega^0(M) $$


This is the operator $D: \Gamma(V\_+\oplus V\_-)\to \Gamma(V\_-\oplus V\_-)$ in the paper you mentioned.


The Atiyah-Singer index theory shows that $\ind L$ is


$$\ind L= \int\_M \big[\; \ch(\Ad(P)) \hA(X)\ch(V\_-)\;\big]\_4, $$


where $[--]\_4$ denotes the degree $4$ part of a non-homogeneous differential form.


We deduce


$$\ch(\Ad(P))=\dim G +\ch\_2(\Ad(P))+\cdots = \dim G+p\_1(\Ad(P))+\cdots, $$


$$\ind L= \int\_M \big(\; p\_1(\Ad(P))+(\dim G)\rho\_D\;\big) $$


where the degree $4$ from $\rho\_D= [\hA(X)\ch(V\_-)]\_4$ is the index density of $D$ appearing in the Atiyah-Singer index theorem
$$
\ind D=\int\_M \rho\_D.
$$


Thus


$$
\ind L=\int\_M p\_1(\Ad(P))+\dim G\ind D= \int\_M p\_1(\Ad(P))+\dim G(b\_1 -b\_2^--b\_0). $$


Now express $(b\_1-b\_2^--b\_0)$ in terms of the signature $\tau=b\_2^+-b\_2^-$ and the Euler characteristic $\chi=2b\_0-2b\_1+b\_2^++b\_2^-$.


