M-theory boundaries beyond supersymmetry

The chiral worldvolume theory of an M-theory boundary (the so-called M9 brane) is uniquely determined by supersymmetry and anomaly inflow. In this brief note we investigate whether alternative chiral boundary field contents may be allowed by anomaly cancellation once supersymmetry is dropped. Even then, anomaly inflow places stringent constraints on the gauge group $G$ and matter content of the boundary worldvolume theory, which we determine explicitly. We find the most general solution to these constraints in the case where all matter fields are of the same chirality, for all simple Lie algebras except $\mathfrak{sp}_2$, $\mathfrak{su}_{n\leq5}$, and $\mathfrak{so}_{n}$ with $7\leq n\leq 12$, and find no solutions other than the supersymmetric $E_8$ boundary of Ho\v{r}ava and Witten. However, when we extend our search to allow for any chirality in the matter fields, we find one minimal solution with gauge group $G_2$, charged matter in the $\mathsf{14}$, $\mathsf{27}$ and $\mathsf{77}$ representations, which satisfies all constraints in a non-trivial way. Therefore, it could in principle describe the low-energy theory of a novel nonsupersymmetric M-theory boundary condition, different from the Ho\v{r}ava-Witten proposal. We briefly discuss some consequences if this was indeed the case, such as the existence of a non-supersymmetric, exotic "$G_2$-string" CFT in 6d, and a novel, non-perturbative, heterotic-like 10d string with gauge group $G_2\times G_2$.