Atiyah-Hitchin-Singer Ref 1 states that the number of
**virtual dimensions** of the instanton moduli space
for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given by
$$
D=4N \mathcal{Q} - (N^2-1)\frac{\chi(X) + \sigma(X)}{2} \;,
$$
where $\chi$ and $\sigma$ are the Euler number and the signature of the 4-manifold $X$.

Questions:

If I understand correctly, the

**virtual dimensions**coincide with the actual**dimensions for irreducible solutions**when the gauge group is completely broken by the solutions. What is the mathematical definition of*virtual dimension*, precisely, of the instanton moduli space?If I understand correctly, $\mathcal{Q}$ shall be related to the Pontryagin class $p_1(E)$? (in p.428 or in p.431) $$ \mathcal{Q} \sim p_1(E) = -\frac{1}{ 4 \pi^2}\int \text{tr}(\Omega^2)= \frac{1}{ 4 \pi^2}\int |\Omega|^2 \omega, $$ where $\Omega$ is the curvature 2-form and $\omega$ is the volume form?

How to digest this dimension equality $ D=4N \mathcal{Q} - (N^2-1)\frac{\chi(X) + \sigma(X)}{2}$?

Since $D$ is a dimension, $D$ is an integer.

(1) Can $D=0$?

(2) $\chi(X) + \sigma(X)$ must be an even integer, for the $D$ to be an integer.

(3) Since $D \geq 0$, and the integer $D \in \mathbb{N}$, we have $$4N \mathcal{Q} > (N^2-1)\frac{\chi(X) + \sigma(X)}{2}$$ always. Why is this true intuitively? How do we see that increasing the $\frac{\chi(X) + \sigma(X)}{2}$, we obtain a smaller $D$?

"The geometry of four-manifolds"). $\endgroup$ – Chris Gerig Oct 17 '18 at 22:58