Anomaly of strings of 6d $\mathcal{N}{=}(1,0)$ theories

TL;DR

We provide a general inflow-based formula for the 2d anomaly polynomial of strings in 6d $\mathcal{N}=(1,0)$ theories, requiring only bulk anomaly data. The paper explicitly derives the formula, then validates it by matching with known 2d gauge theory descriptions for E-string, M-string, the $n=4$ minimal theory, and 6d string chains, establishing consistency across setups. It further explores implications, including an ADE-type structure for 6d $\mathcal{N}=(2,0)$ theories, the automatic emergence of $E_8$ flavor symmetry for the smallest $\mathcal{N}=(1,0)$ theory, and the worldsheet dynamics of minimal-theory strings, notably the instanton-moduli/Higgs-branch interpretation with the relation $h^\vee_G = 3(n-2)$. These results unify bulk-boundary anomaly data, illuminate string dynamics in 6d theories, and connect to F-theory numerology and ADE classifications.

Abstract

We obtain the anomaly polynomial of strings of general 6d $\mathcal{N}{=}(1,0)$ theories in terms of anomaly inflow. Our computation sheds some light on the reason why the simplest 6d $\mathcal{N}{=}(1,0)$ theory has $E_8$ flavor symmetry, and also partially explains a curious numerology in F-theory.