Cobordism Classes and the Swampland
Jacob McNamara, Cumrun Vafa
TL;DR
The paper proposes that cobordism groups of quantum-gravity configurations, Omega_k^QG, must vanish in a consistent theory of quantum gravity to preclude global symmetries, placing the Swampland constraint on viable theories. By mapping cobordism to domain walls between compactifications, the authors derive that known objects (D-branes, orientifolds, Hořava–Witten walls) and several novel defects must exist to kill nontrivial cobordism classes, and they explore how this structure is realized across p-form fields, spin and spin^c cobordism, M-theory on nonorientable manifolds, F-theory, and heterotic string theory. The work yields new predictions of non-supersymmetric defects required for consistency, and it argues that such defects must break all supersymmetry, with their charges behaving in a topologically gauged manner. Overall, the framework offers a topological, gravity-consistent lens on the string landscape and suggests concrete directions for discovering hitherto unknown defects and refining the mathematical structure of quantum gravity.
Abstract
We argue that any proposed quantum theory of gravity with non-trivial cobordism classes in the space of configurations belongs to the Swampland. The argument is based on the assumption that there are no global symmetries in a consistent theory of quantum gravity. The triviality of the cobordism classes requires the existence of certain stringy defects that trivialize the potential cobordism classes. We provide evidence for this conjecture by identifying those defects demanded by this argument that could preserve supersymmetry, and predict the existence of new non-supersymmetric defects in string theory.