Perturbative Anomaly Inflow on Orbifolds

We study perturbative anomaly inflow for effective theories supported on orbifold fixed loci of the internal geometry. Donnelly’s equivariant APS theorem gives a fixed-point density, whose degree-$(d+2)$ component is identified as the anomaly polynomial of the $d$-dimensional effective theory. We apply the construction to six-dimensional A-type orbi-instanton theories engineered by $N$ M5-branes probing a transverse ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_k$ singularity at an end-of-the-world M9-brane. For a flat $E_8$ connection specified by $\rho :{\mathbb{Z}}_k\to E_8$, we compute the non-identity ALE fixed-point class on the M9 wall. Its degree-eight component agrees with the complete Kac-label-dependent remainder conjectured in \cite{MOTZ}, including the $SU(2{)}_R$ and tangent-bundle curvatures. Together with the known M5/Hořava–Witten and ALE/Hořava–Witten terms, this reproduces the unflavored tensor-branch anomaly polynomial. This local equality further suggests a complete M-theory corner-inflow interpretation. The same prescription also be extended to include flavor symmetries as the centralizer of $E_8$.