Towards a complete classification of 6D supergravities

The constraints arising from anomaly cancellation are particular strong for chiral theories in six dimensions. We make progress towards a complete classification of 6D supergravities with minimal supersymmetry and non-abelian gauge group. First, we generalize a previously known infinite class of anomaly-free theories which has $T\gg 9$ to essentially any semi-simple gauge group and infinitely many choices for hypermultiplets. The construction relies on having many decoupled sectors all selected from a list of four simple theories which we identify. Second, we use ideas from graph theory to rephrase the task of finding anomaly-free theories as constructing cliques in a certain multigraph. A branch-and-bound type algorithm is described which can be used to explicitly construct, in a $T$-independent way, anomaly-free theories with an arbitrary number of simple factors in the gauge group. We implement these ideas to generate an ensemble of $\mathcal{O}(10^7)$ irreducible cliques from which anomaly-free theories may be easily built, and as a special case obtain a complete list of $19,\!847$ consistent theories for $T=0$, for which the maximal gauge group rank is $24$. Modulo $\mathrm{U}(1)$, $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ simple factors and the new infinite families, we give a complete characterization of anomaly-free theories and show that the bound $T\leq 273$ is sharp.