Anomaly-free six-dimensional $Sp(1)_R$-gauged supergravities

Abstract

We study anomaly constraints in six-dimensional $N=(1,0)$ supergravity with gauged $Sp(1)_R$ symmetry, one tensor multiplet, $n_V$ vector multiplets and $n_H$ hypermultiplets. Imposing cancellation of local gravitational, gauge and mixed anomalies, together with the embedding of the anomaly coefficients in a unimodular rank-two string charge lattice, the number of vector multiplets is sharply constrained: the only allowed positive values are $n_V=12$ and $n_V=96$. For a neutral abelian factor, or drone, the anomaly equations force the corresponding row and column of the abelian gauge-kinetic matrix to vanish. Such a factor therefore cannot be retained as an ordinary propagating vector multiplet. For $n_V=96$, within the class of semisimple non-$R$ gauge algebras containing no $A_1$ or $A_2$ factor, the exhaustive spectra-assembly calculation yields no drone-free spectrum. For $n_V=12$, the semisimple non-$R$ branch is necessarily $A_1^3$. In the doublet/adjoint truncation described in the text, we find $4{,}337{,}331$ consistent models, classified by $102$ anomaly-vector classes. We also consider the reductive branch $A_2\times U(1)\times Sp(1)_R$. For $A_2$ representations in $\{\mathbf 1,\mathbf 3,\overline{\mathbf 3}, \mathbf 6,\overline{\mathbf 6},\mathbf 8\}$ and primitive $U(1)$ charge assignments with $|q|\le 2$, we find $178{,}538$ solutions, classified by $21$ sets of anomaly vectors.