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

## TL;DR

This paper derives the anomaly polynomials for 4d $ ext{N}=2$ class $ ext{S}$ theories of type $A_{N-1}$ from M-theory inflow, providing a geometric approach that connects 11d geometry, fluxes, and puncture data.

## Contribution

It offers a first principles derivation of anomaly polynomials for class $ ext{S}$ theories using M-theory inflow, clarifying the geometric origin of puncture labeling.

## Key findings

- Derived anomaly polynomials for class $ ext{S}$ theories from M-theory.
- Connected puncture data to 11d geometry and flux analysis.
- Highlighted applications to the AdS/CFT correspondence.

## Abstract

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.