Global anomalies in $6D$ gauged supergravities

There exists a rare class of R-symmetry gauged $N=(1,0)$ supergravities in six dimensions with gauge group $G\times U(1)_R$, where $G$ is semisimple with rank greater than one, and the number of tensor multiplets $n_T=1$, which are free from all local anomalies. We find new members of this family, in which $G$ contains up to four factors. We study the global anomalies of these models in a framework in which the Dirac quantization of the anomaly coefficients and the well-definedness of the Green-Schwarz anomaly counterterm in generic backgrounds play key roles, and we apply the anomaly freedom criteria that takes this into account as formulated by Monnier and Moore in \cite{Monnier:2018nfs}. To this end we use the result derived by Yuji Tachikawa in the appendix which states that the spin cobordism group $\Omega_7^{\rm Spin}(BG)$ vanishes for $G=G_1\times \cdots \times G_n$ where $G_i$ is $U(1)$ or any simple simply-connected non-Abelian compact Lie group. We also require correct sign for the vector field kinetic terms, and positive Gauss-Bonnet term. We find that among the models considered here only one of them fails to satisfy all the stated criteria. We also find that, in general, the requirement that the anomaly coefficients defined by the factorized anomaly polynomial are elements of a unimodular charge lattice imposes a constraint on the number of vector multiplets given by $n_{V}=8\ {\rm mod}\ 12$ for $U(1)_{R}$, and $n_{V} = 60$ for $Sp(1)_{R}$ gauged models.