QCD String Axions and $M$-theory

In $SU(N)$ Yang-Mills theory without matter, there exist stable long electric fluxtube strings which carry a 1-form symmetry charge. Over the past decade or so, there has been increasing evidence from lattice calculations that the worldsheet theories of such QCD strings contain a massive pseudoscalar (axion), at least when $N\geq 3$. This has so far been puzzling from the perspective of holographic realizations of strings in confining gauge theories. In this note, we will show how such axions appear naturally in the realization of 4D $\mathcal{N}=1$ Super-Yang-Mills (SYM) from $M$-theory spacetimes in which the extra dimensions are modeled by certain complete metrics of $G_2$-holonomy. This picture predicts that QCD string axions exist only if the gauge group is $SU(N\geq 3)$, $SO(4N+2)$, or $E_6$ (or a quotient/double-cover thereof), and is absent from the spectrum of stable QCD strings if the gauge group is $SU(2)$, $SO(2N+1)$, $SO(4N)$, $Sp(N)$, or $E_7$. We argue why we expect this pattern to persist for non-supersymmetric Yang-Mills strings, at least for large $N$, something which could be tested in future lattice studies of QCD strings for gauge algebras of B, C, D, and E-type.