Swampland Constraints on the SymTFT of Supergravity

We consider string/M-theory reductions on a compact space $X=X^\text{loc} \cup X^\circ$, where $X^\text{loc}$ contains the singular locus, and $X^\circ$ its complement. For the resulting supergravity theories, we construct a suitable Symmetry Topological Field Theory (SymTFT) associated with the boundary $\partial X^\text{loc} \coprod \partial X^\circ$. We propose that boundary conditions for different BPS branes wrapping the same boundary cycles must be correlated for the SymTFT to yield an absolute theory consistent with quantum gravity. Using heterotic/M-theory duality, this constraint can be translated into a field theoretic statement, which restricts the global structure of $d\geq 7$, $\mathcal{N}=1$ supergravity theories to reproduce precisely the landscape of untwisted toroidal heterotic compactifications. Furthermore, for 6d $(2,0)$ theories, we utilize a subtle interplay between gauged 0-, 2-, and 4-form symmetries to provide a bottom-up explanation of the correlated boundary conditions in K3 compactifications of type IIB.