I have a very stupid and non-specific question, as follows. And let me know if I am asking in a wrong way.
We known that, for instance, if one is interested in computing dimension of moduli space of 4d instanton, then one can study the deformation complex, and find it elliptic, and then usual index theorems applies.
Now suppose I am interested in some $d$(odd)-dimensional instanton-like equation, and would like to know its moduli dimension. But it turns out that the deformation complex is far from elliptic, which looks like
$0 \to {\Omega ^0} \otimes \mathfrak{g}\xrightarrow{{{d_A}}}{\Omega ^1} \otimes \mathfrak{g}\xrightarrow{{\pi ^\circ {d_A}}}H \otimes \mathfrak{g} \to 0$
where $H$ is a sub-bundle of $\Omega^2$ with unfortunate $rank\left( H \right) \ne d - 1$. At some special points on the moduli space, I can "choose a good gauge" and replace the above bad complex with an elliptic once, but in general I am faced with this bad one.
So the question is: Does it mean that the dimension of moduli space cannot be described? Is there other ways (other index theorem) to compute the dimension?
