Anomaly polynomial of E-string theories

TL;DR

This work determines the anomaly polynomial of the rank-$Q$ E-string theories realized by $Q$ M5-branes in the $E_8$ end-of-the-world brane, including the decoupled center-of-mass hypermultiplet. By merging Horava–Witten and Freed–Harvey–Minasian–Moore inflow analyses, the authors derive $A_{E_8+\text{free}}(Q)= Q^3 \frac{p_2(N)}{6} + Q^2 \frac{\chi_4(N) I_4}{2} + Q \left(\frac{I_4^2}{2}-I_8\right)$ with $I_4=\frac{1}{4}(p_1(N)+p_1(T)+\mathrm{Tr}F^2)$ and $I_8=\frac{1}{48}(p_2(N)+p_2(T)-\frac{1}{4}(p_1(N)-p_1(T))^2)$. The interacting E-string anomaly is obtained by subtracting the free hypermultiplet, yielding a compact formula in terms of Pontrjagin classes and characteristic forms $p_2(N)$, $\chi_4(N)$, $I_4$, and $I_8$, and the authors perform nontrivial consistency checks via heterotic/K3 reductions, known $(\mathrm{Tr}F^2)^2$ coefficients, and the special case $Q=1$. The results enable further explorations of 6d $\mathcal{N}=(1,0)$ dynamics, holographic comparisons, and compactifications to 4d $\mathcal{N}=1$ theories, where the anomaly data constrain central charges and RG flows.

Abstract

We determine the anomaly polynomial of the E-string theory and its higher-rank generalizations, that is, the 6d $\mathcal{N} =(1, 0)$ superconformal theories on the worldvolume of one or multiple M5-branes embedded within the end-of-the-world brane with $E_8$ symmetry.