WZW terms without anomalies: generalised symmetries in chiral Lagrangians

Abstract: We consider a 4d non-linear sigma model on the coset $(\mathrm{SU}(N)L \times \mathrm{SU}(N)_R \times \mathrm{SU}(2))/(\mathrm{SU}(N){L+R}\times \mathrm{U}(1))\cong \mathrm{SU}(N) \times S^2$, that features a topological Wess-Zumino-Witten (WZW) term whose curvature is $\frac{n}{24\pi^2}\mathrm{Tr}(g^{-1}dg)^3 \wedge \mathrm{Vol}{S^2}$ where $g$ is the $\mathrm{SU}(N)$ pion field. This WZW term, unlike its familiar cousin in QCD, does not match any chiral anomaly, so its microscopic origin is not obviously QCD-like. We find that generalised symmetries provide a key to unlocking a UV completion. The $S^2$ winding number bestows the theory with a 1-form symmetry, and the WZW term intertwines this with the $\mathrm{SU}(N)^2$ flavour symmetry into a 2-group global symmetry. Like a 't Hooft anomaly, the 2-group symmetry should match between UV and IR, precluding QCD-like completions that otherwise give the right pion manifold. We instead construct a weakly-coupled UV completion that matches the 2-group symmetry, in which an abelian gauge field connects the QCD baryon number current to the winding number current of a $\mathbb{C}P^1$ model, and explicitly show how the mixed WZW term arises upon flowing to the IR. The coefficient is fixed to be the number of QCD colours and, strikingly, this matching must be 'tree-level exact' to satisfy a quantization condition. We discuss generalisations, and elucidate the more intricate generalised symmetry structure that arises upon gauging an anomaly-free subgroup of $\mathrm{SU}(N){L+R}$. This WZW term may even play a phenomenological role as a portal to a dark sector, that determines the relic abundance of dark matter.