Global Anomalies in Sigma Models with Majorana--Weyl Fermions

We investigate a global sigma model anomaly in two-dimensional sigma models with Majorana--Weyl fermions coupled to a sigma model field with target space~$G$. The anomaly originates from the nontrivial topology of the space of maps and manifests as a phase in the fermion path integral. Using the global anomaly formula expressed in terms of the reduced~$\eta$-invariant, we demonstrate that this anomaly modifies the standard quantization condition of the Wess--Zumino term on~$G$, in close analogy with the three-dimensional parity anomaly. However, our situation is more refined and highlights a qualitatively new phenomenon in two dimensions: whereas in three dimensions the anomalous quantization restricts the level to lie in a half-integer lattice, here it can force the level to take \emph{arbitrary real values}. Furthermore, our results support the consistency of the low-energy description proposed by Gaiotto, Johnson-Freyd, and Witten for three-dimensional $\mathcal{N}=1$ supersymmetric Yang--Mills theory on an interval, by highlighting a subtle and qualitatively distinct nature of the sigma-model anomalies.