Class $\mathcal{S}$ Anomalies from M-theory Inflow

We present a first principles derivation of the anomaly polynomials of 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories of type $A_{N-1}$ with arbitrary regular punctures, using anomaly inflow in the corresponding M-theory setup with $N$ M5-branes wrapping a punctured Riemann surface. The labeling of punctures in our approach follows entirely from the analysis of the 11d geometry and $G_4$ flux. We highlight the applications of the inflow method to the AdS/CFT correspondence.