Swampland constraints on the symmetry topological field theory of supergravity

We consider string/M-theory reductions on a compact space <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>X</a:mi><a:mo>=</a:mo><a:msup><a:mi>X</a:mi><a:mi>loc</a:mi></a:msup><a:mo>∪</a:mo><a:msup><a:mi>X</a:mi><a:mo>∘</a:mo></a:msup></a:math>, where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>X</c:mi><c:mi>loc</c:mi></c:msup></c:math> contains the singular locus and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>X</e:mi><e:mo>∘</e:mo></e:msup></e:math> is its complement. For the resulting supergravity theories, we construct a suitable symmetry topological field theory (SymTFT) associated with the boundary <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo>∂</g:mo><g:msup><g:mi>X</g:mi><g:mi>loc</g:mi></g:msup><g:mo>∐</g:mo><g:mo>∂</g:mo><g:msup><g:mi>X</g:mi><g:mo>∘</g:mo></g:msup></g:math>. 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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>d</i:mi><i:mo>≥</i:mo><i:mn>7</i:mn></i:math>, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="script">N</k:mi><k:mo>=</k:mo><k:mn>1</k:mn></k:math> 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. Published by the American Physical Society 2024