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

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? $$n_{U(1)} \in \text{what characteristic class or invariant} ?$$ $$n_{PSU(N)} \in \text{what characteristic class or invariant} ?$$

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  • 2
    $\begingroup$ For $G = O(N)$ or $SO(N)$, we have $n = -\frac{1}{2}\int_Mp_1(E)$ where $E$ is a real (respectively orientable) vector bundle of rank $N$. For $G = U(N)$ we have $n = -\frac{1}{2}\int_Mp_1(E) = -\frac{1}{2}\int_Mc_1(E)^2 - 2c_2(E)$ where $E$ is a complex vector bundle of rank $N$, and for $G = SU(N)$ we have $n = -\frac{1}{2}\int_Mp_1(E) = \int_Mc_2(E)$ where $E$ is a complex vector bundle of rank $N$ with $c_1(E) = 0$. $\endgroup$ Jul 4, 2020 at 23:43
  • 2
    $\begingroup$ For $G = PSU(N)$, one has natural projective bundles, but I don't know if you can get vector bundles in any natural way. You need a real vector bundle to talk about $p_1$ or a complex vector bundle to talk about $c_1$ or $c_2$. In particular, if you have a group homomorphism $PSU(N) \to O(M)$ or $U(M)$, then we can construct the associated vector bundle and talk about its Pontryagin or Chern classes. $\endgroup$ Jul 4, 2020 at 23:44
  • $\begingroup$ many thanks @Michael Albanese! a systematic answer is still welcome $\endgroup$
    – wonderich
    Jul 5, 2020 at 0:06
  • 2
    $\begingroup$ What would you consider a systematic answer to contain? Do you have references for the quantization in the $PSU(N)$ case? $\endgroup$ Jul 5, 2020 at 0:28
  • $\begingroup$ No, Refs are welcome. I knew the Book by Freed and Uhlenbeck $\endgroup$
    – wonderich
    Jul 5, 2020 at 0:50

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