# $U(1)$ v.s. $SU(N)$ v.s. $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of:

1. Chern class (1st, 2nd), and

2. Pontryagin class

3. And their normalization and mapping between the instanton numbers. For example, the unit instanton number 1 in $SO(3)$ gauge theory may map to the instanton number 1/4 in $SU(2)$ gauge theory on a non-spin manifold, while the unit instanton number 1 in $SO(3)$ gauge theory may map to the instanton number 1/2 in $SU(2)$ gauge theory on a non-spin manifold etc.

More precisely, can we compare the quantization of the charcteristic classes $$p_1(SO(N)),$$ $$c_2(SU(N)=\frac{1}{8 \pi^2}\int \text{tr}(F_{(SU(N))}\wedge F_{(SU(N))}),$$ $$c_1^2(U(1))=\int \frac{F_{U(1)}}{2 \pi} \wedge \frac{F_{U(1)}}{2 \pi} =\frac{1}{4 \pi^2} \int {F_{U(1)}} \wedge {F_{U(1)}} ,$$ on the spin manifolds and non-spin manifolds?

We can also embed the $U(1) \subset SU(N)$, $U(1) \subset SO(N)$ and $SO(N) \subset SU(N)$ to compare the instanton numbers of $G_a$ evaluated as the instanton numbers of $G_b$ whenever $G_a \subset G_b$. So how are the instanton number maps from that of $G_a$ to that of $G_b$?

Here $F=dA + A \wedge A$ is the 2-form curvature of the 1-form connection $A$.