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Generalized 't Hooft anomalies on non-spin manifolds

We study the mixed anomaly between the discrete chiral symmetry and general baryon-color-flavor (BCF) backgrounds in $SU(N_c)$ gauge theories with $N_f$ flavors of Dirac fermions in representations ${\cal R}_c$ of $N$-ality $n_c$, formulated on non-spin manifolds. We show how to study these theories on $\mathbb{CP}^2$ by turning on general BCF fluxes consistent with the fermion transition functions. We consider several examples in detail and argue that matching the anomaly on non-spin manifolds places stronger constraints on the infrared physics, compared to the ones on spin manifolds (e.g.~$\mathbb{T}^4$). We also show how to consistently formulate various chiral gauge theories on non-spin manifolds.