F-theory and Unpaired Tensors in 6D SCFTs and LSTs

We investigate global symmetries for 6D SCFTs and LSTs having a single "unpaired" tensor, that is, a tensor with no associated gauge symmetry. We verify that for every such theory built from F-theory whose tensor has Dirac self-pairing equal to -1, the global symmetry algebra is a subalgebra of $\mathfrak{e}_8$. This result is new if the F-theory presentation of the theory involves a one-parameter family of nodal or cuspidal rational curves (i.e., Kodaira types $I_1$ or $II$) rather than elliptic curves (Kodaira type $I_0$). For such theories, this condition on the global symmetry algebra appears to fully capture the constraints on coupling these theories to others in the context of multi-tensor theories. We also study the analogous problem for theories whose tensor has Dirac self-pairing equal to -2 and find that the global symmetry algebra is a subalgebra of $\mathfrak{su}(2)$. However, in this case there are additional constraints on F-theory constructions for coupling these theories to others.