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:

1. 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 virtual dimension precisely of the instanton moduli space?

2. 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?

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

(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 $D$?

Ref 1: M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Selfduality in Four-Dimensional Riemannian Geometry, Proc. Roy. Soc. Lond. A362 (1978) 425–461.

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:

1. 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?

2. 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?

3. 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 $D$?

Ref 1: M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Selfduality in Four-Dimensional Riemannian Geometry, Proc. Roy. Soc. Lond. A362 (1978) 425–461.