6d Supergravity Blocks

We propose a systematic framework for constructing six-dimensional supergravity theories with eight supercharges that respect all known consistency constraints, including anomaly cancellation and the non-perturbative ${\it H}$-string constraints recently discovered by Kim, Vafa, and Xu. The basic objects in this framework are ${\it supergravity\,blocks}$, which are minimal collections of tensor multiplets consisting of a single little string theory sharing the ${\it H}$-string charge together with additional tensors whose string charges intersect it positively. A characteristic feature of each supergravity block is that its Gram matrix has exactly one positive eigenvalue, and therefore it necessarily contains gravitational BPS strings that cannot become tensionless anywhere in tensor moduli space. Any consistent 6d $(1,0)$ supergravity theory can then be obtained by gluing compatible blocks and subsequently enhancing the gauge algebras and matter content. As a first step toward establishing this framework concretely, we provide a complete classification of the ${\it non\text{-}Higgsable\,supergravity\,blocks}$, (or ${\it non\text{-}Higgsable\,gravity\,blocks}$ for short) namely those built from tensor multiplets that support only non-Higgsable gauge algebras.