# Why are gauge anomalies characterised by the non-triviality of $\pi_5(\mathcal G)$?

The folklore in 4-dimensional gauge theories is that the existence of potential gauge anomalies from the triangle diagrams that need to be cancelled are characterised by the non-triviality of the fifth homotopy group of the gauge group: $$\pi_5(\mathcal G)\ne0$$ (with the exception of $$\mathrm U(1)$$ - so $$\mathrm{SU}(n>2)$$, $$\mathrm{SO}(4n+2)$$ and $$E_6$$). Why is this the case?

I know that a necessary condition for the appearance of the $$\mathcal G^3$$ anomaly is that the gauge group needs to have complex representations for the fermions to live in. This is because we cannot add a gauge-invariant mass term with which we can use Pauli-Villars regularisation to kill the anomaly. More rigorously, it can be seen as via a particular symmetrised trace over the generators: $$\mathcal D_{abc}=\mathrm{tr}_\mathcal R t_{(a}t_bt_{c)}=0$$ for real and pseudo-real representations, automatically guaranteeing safety from anomalies. Are these two conditions related? I can't think of any naïve mathematical equivalence between the two, so I assume it has some physical motivation rooted in anomalies and/or obstructions.

I don't think this is true, $$\pi_5(G)$$ has little to do with anomalies, at least not in any direct way.

The general statement is: triangle anomalies for a given symmetry $$G$$, in $$d$$ dimensions, are classified by the free part of $$\Omega_{d+2}(G)$$. Here $$\Omega_n(G)$$ denotes the cobordism group of $$G$$, namely the collection of $$n$$ dimensional manifolds $$M_n$$ with a prescribed $$G$$-bundle, modulo the identification $$M_n\sim M'_n$$, where $$\sim$$ denotes the existence of an $$(n+1)$$-dimensional manifold $$M_{n+1}$$ such that $$\partial M_{n+1}=M_n\sqcup \bar M_n'$$, and such that the $$G$$-bundle extends smoothly.

Both $$\Omega_\bullet(G)$$ and $$\pi_\bullet(G)$$ measure topological obstructions of manifolds equipped with $$G$$-structures. So they carry some overlapping information. But the correct object to look at is $$\Omega$$, not $$\pi$$.

[More precisely, if you want $$G$$-anomalies you probably want to look at the reduced cobordism group $$\Omega_{d+2}(G)=\tilde \Omega_{d+2}(G)\oplus \Omega_{d+2}(pt)$$. I believe $$\Omega_n(pt)$$ may have a free part only if $$n$$ is a multiple of $$4$$, but the details (which are above my paygrade) might depend on which structure you want your manifolds to carry. The $$G$$-independent part $$\Omega_n(pt)$$ measures purely gravitational anomalies. This only plays a role in $$d=2\mod4$$ dimensions so irrelevant in $$d=4$$; but it is essential in e.g. $$d=2$$ where $$\Omega_4(pt)=\mathbb Z$$ is the famous central charge of a CFT, which constrains the number of spacetime dimensions in String Theory to $$D=10$$ or $$D=26$$.]

In four dimensions, for example for the ABJ anomaly, one has $$\Omega_6(U(1))=\mathbb Z^2$$, where one factor is the cubic anomaly $$J^3$$ and the other one is the mixed $$U(1)$$-grav part, $$JT^2$$, where $$J$$ is the current for $$U(1)$$ and $$T$$ is the energy-momentum tensor. But the group $$U(1)=S^1$$ has trivial $$\pi_i$$ for $$i>1$$ so it is clear that $$\pi$$ will miss anomalies. You really want to look at $$\Omega$$, not $$\pi$$.

To make things even more confusing, one can also look at homology groups $$H_\bullet(G)$$, which are some sort of linear approximation to $$\pi_\bullet(G)$$. These homology groups also measure topological obstructions, so they also carry some information in $$\Omega_\bullet$$. But, again, the correct object to look at is $$\Omega$$, not $$\pi$$ nor $$H$$.

That being said, it is true that $$H^n(G,\mathbb Z)$$ is a decent first order approximation to $$\Omega_n(G)$$, so it captures some anomalies. It misses all the mixed $$G$$-grav anomalies, but it does see many pure $$G$$-anomalies, at least for sufficiently well-behaved $$G$$. In this sense, triangle anomalies are approximately characterised by $$H^{d+2}(G,\mathbb Z)$$. This group turns out to be isomorphic to $$H^{d+1}(G,U(1))$$, which famously classifies Chern-Simons terms in $$d+1$$ dimensions. And these terms do indeed characterise some triangle anomalies in $$d$$ dimensions, via the Wess-Zumino descent procedure. So, to this approximation, it is true that triangle anomalies in $$d=4$$ are classified by $$H^5(G,U(1))$$ (note, again, $$H^5$$ and not $$\pi_5$$!)

• Brilliant answer, thanks! A supplementary point I found was in arXiv:0509097, pages 29-30, although the argument is sketched – Nihar Karve Apr 11 at 7:06