Swampland constraints on the symmetry topological field theory of supergravity

We consider string/M-theory reductions on a compact space 𝑋=𝑋loc∪𝑋∘, where 𝑋loc contains the singular locus and 𝑋∘ is its complement. For the resulting supergravity theories, we construct a suitable symmetry topological field theory (SymTFT) associated with the boundary ∂𝑋loc⁢∐∂𝑋∘. We propose that boundary conditions for different Bogomol’nyi-Prasad-Sommerfield 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≥7, 𝒩=1 supergravity theories to reproduce precisely the landscape of untwisted toroidal heterotic compactifications. Furthermore, for six-dimensional (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.