Tag: numbers

It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n in mathbb{Z}$
$$
n = { 1 over 8pi^2} int_{mathcal{M}_{4}} text{tr} left(F wedge Fright) = {1 over 64 pi^2} int_{mathcal{M}_{4}} d^4 x , epsilon^{munurholambda} F_{munu}^alpha F_{rholambda}^alpha
in mathbb{Z}
$$

Here $F$ is the curvature 2-form $F=d A + A wedge A$ and $A$ is the gauge 1-connection of the gauge bundle of gauge group $G$. Also $F=F^alpha T^alpha$ where $alpha$ is the Lie algebra indices, with repeated indices summed over.

We know that:

• If the gauge group is $G=SU(2)$, the instanton number $n in mathbb{Z}.$ I think this can be understood as
  $$
  n = c_2(V_{SU(2)}) in mathbb{Z}?
  $$
• If the gauge group is $G=SO(3)$ on the spin 4-manifold $mathcal{M}_{4}$, then $n in frac{1}{2}mathbb{Z}$. (I use the notation to mean that $n in frac{1}{2}mathbb{Z}$ as $n$ takes the half integer values.)
  $$
  n = p_1(V_{SO(3)})/4 in frac{1}{2}mathbb{Z}?
  $$
• If the gauge group is $G=SO(3)$ on the non-spin 4-manifold $mathcal{M}_{4}$, then $n in mathbb{Z}/4$.
  $$
  n = p_1(V_{SO(3)})/4 in frac{1}{4}mathbb{Z}?
  $$

What are the general statements we can make for other general $G$ and other manifolds?

Questions:

1. If the gauge group is $G=U(1)$, the instanton number $n in 2 mathbb{Z}.$ True or False? We can express this $n$ as the first Chern class $c_1$ square of associated vector bundle $V_{U(1)}$ as
   $$
   n = c_1(V_{U(1)})^2 in 2 mathbb{Z}?
   $$
   Is this correct?

2. If the gauge group is $G=SU(N)$, the instanton number $n in mathbb{Z}.$ True or False? We can express this $n$ as the second Chern class $c_2$ of associated vector bundle $V_{SU(N)}$ as
   $$
   n = c_2(V_{SU(N)}) in mathbb{Z}?
   $$

3. If the gauge group is $G=PSU(N)$ on the spin 4-manifold $mathcal{M}_{4}$, the instanton number can be $1/N$ fractional of $mathbb{Z}$ values:
   $$n in frac{1}{N} mathbb{Z},$$
   True or False? What is the precise mathematical invariant to characterize this $n in frac{1}{N} mathbb{Z}$? Is that Pontryagin class $p_1$ when $G=PSU(N)$ is real $mathbb{R}$? Or some $c_2(V_{PSU(N)})$ when $PSU(N)$ is complex $mathbb{C}$?

4. If the gauge group is $G=PSU(N)$ on the non-spin 4-manifold $mathcal{M}_{4}$, the instanton number can be $1/N^2$ fractional of $mathbb{Z}$ values: $$n in frac{1}{N^2} mathbb{Z},$$
   True or False? What is the precise mathematical invariant to characterize this $n in frac{1}{N^2} mathbb{Z}$? Is that Pontryagin class $p_1$ when $G=PSU(N)$ is real $mathbb{R}$? Or some $c_2(V_{PSU(N)})$ when $PSU(N)$ is complex $mathbb{C}$?

5. If the gauge group is $G=U(N)$ on the spin or non-spin 4-manifold $mathcal{M}_{4}$, the instanton numbers carry both the $U(1)$ and $PSU(N)$ part with constraints. So there are two instanton numbers
   $n_{U(1)}$ and $n_{PSU(N)}$.
   What can be their constraints?
   $$n_{U(1)} in ?$$
   $$n_{PSU(N)} in ?$$