When is a Wess-Zumino term well-defined?

Question

According to wikipedia, a Wess-Zumino term is well-defined when the Lie group (target space) $G$ is compact and simply connected, because that implies that $\pi_2(G)$ is trivial. But there are Lie groups with trivial $\pi_2(G)$ that are not simply connected (e.g., $S^1$). Is $S_\mathrm{WZ}(G)$ well-defined in such a case? Is a trivial $\pi_2(G)$ necessary and sufficient, or only necessary? If it is not sufficient, then what is the sufficient condition on $G$ for it to admit a WZW model?

On a second thought, it appears that the Wikipedia page contains a few misleading statements. For example, it says that $\pi_2(G)$ is trivial because $G$ is simply-connected; but, as mentioned by user gj255 in the comment section, $\pi_2(G)$ is trivial for any $G$, simply-connected or not. Moreover, in the sub-section Topological obstructions, Wikipedia claims that $kS_\mathrm{WZ}(G)$ is well-defined if $k\in\pi_3(G)$, so it seems that the relevant homotopy group is the third one instead of the second one. If this is correct, then I would guess $kS_\mathrm{WZ}(S^1)$ is only well-defined at level zero (because $\pi_3(S^1)=\{0\}$), and therefore there is no Wess-Zumino term for such a target space. Is this correct? More generally, is $kS_\mathrm{WZ}(G)$ well-defined if and only if $k\in\pi_3(G)\neq\{0\}$?

Answer 1

It's wrong to study the homotopy groups to classify WZW terms because we're not just interested in spherical spacetimes.

Let $X$ be the target space and $M$ an arbitrary closed spacetime $n$-manifold equipped with a map $\sigma:M \to X$. Let $\omega$ be a closed $(n+1)$-form on $X$ with integer periods.

If $\sigma$ is null-homotopic, we can extend it to the cone $\hat\sigma:CM \to X$, which is defined by forming the prism $M \times [0,1]$ and collapsing one end to a point. Then we may define $$WZW(M,\sigma) = \exp 2\pi i \int_{CM} \hat \sigma^*\omega.$$ And because $\omega$ is closed and has integer periods, this can be shown to be independent of the extension $\hat \sigma$.

It's very important for this definition that we can extend $\sigma$ to the cone, or at least to some $n+1$-chain whose boundary is $M$ (we can use a homology theory whose chains are singular manifolds with corners so it still makes sense to form the pullback $\hat \sigma^*\omega$ everywhere except a measure zero set). The condition for this more general situation is a homological one: we need $\sigma_* [M] = 0 \in H_n(X,\mathbb{Z})$, where $[M]$ is the fundamental class of $M$. If $H_n(X,\mathbb{Z}) = 0$, then we're always in business.

Note that there was recently some study of sigma model topological terms coming from cobordism that was pretty interesting: https://arxiv.org/abs/1707.05448 . It explains what happens to the theta angle one expects in a 2+1D theory with $X = S^2$ from $\pi_3 S^2 = \mathbb{Z}$.