On TCS $G_2$ manifolds and 4D Emergent Strings

In this note, we study the Swampland Distance Conjecture in TCS $G_2$ manifold compactifications of M-theory. In particular, we are interested in testing a refined version -- the Emergent String Conjecture, in settings with 4d $N=1$ supersymmetry. We find that a weakly coupled, tensionless fundamental heterotic string does emerge at the infinite distance limit characterized by shrinking the $K3$-fibre in a TCS $G_2$ manifold. Such a fundamental tensionless string leads to the parametrically leading infinite tower of asymptotically massless states, which is in line with the Emergent String Conjecture. The tensionless string, however, receives quantum corrections. We check that these quantum corrections do modify the volume of the shrinking K3-fibre via string duality and hence make the string regain a non-vanishing tension at the quantum level, leading to a decompactification. Geometrically, the quantum corrections modify the metric of the classical moduli space and are expected to obstruct the infinite distance limit. We also comment on another possible type of infinite distance limit in TCS $G_2$ compactifications, which might lead to a weakly coupled fundamental type II string theory.