For every consistent theory of quantum gravity with spacetime dimension d, tangential structure xi, and background data encoded by a space X, the associated bordism group is trivial: Omega_d^xi(X)=0. Equivalently, every admissible d-dimensional configuration at infinity represents the zero class in the relevant bordism group and therefore bounds a (d+1)-dimensional configuration.

Cobordism conjecture

The Cobordism Conjecture, introduced by Miguel Montero and Cumrun Vafa in 2020, proposes that the bordism group associated with a consistent theory of quantum gravity is trivial. The problem remains whether this principle can be established in general.

Known results

Montero and Vafa, 2020: related anomalies and string defects to the conjecture, predicting additional objects.
Andriot, Carqueville, and Cribiori, 2022: developed a structural framework involving geometric structures, magnetic defects, and dualities.
Recent work computed bordism groups for U-duality bundles in supergravity dimensions from three through eleven, producing predicted defects without resolving the conjecture.

Recent brane-network extension

Markus Dierigl and Ignacio Ruiz analyze networks and junctions using bordism and homology, with examples in four-dimensional supergravity models from string and M theory. This is a claimed theoretical extension of the conjecture's applications, not a complete resolution.

Current status (as of September 2026): The conjecture remains unresolved; the brane-network extension is a claimed advance, not a proof or disproof.

Cobordism conjecture

Primary source

Additional references

Progress summary

Solutions 0