Cobordism conjecture for quantum-gravity structures

Fix a standalone target-space string/M-theory quantum-gravity setting in integer dimension $d\ge 3$ (the full system, rather than a localized subsystem requiring a higher-dimensional gravitational host), an integer compactification dimension $0\le k<d$, and a specified set $B_k$ of its allowed compact closed $k$-dimensional backgrounds, geometric or non-geometric. Define the pointed termination relation $M⇝\mathrm{\varnothing }$ to mean that there exists an allowed noncompact $(k+1)$-dimensional quantum-gravity background $W$ with boundary $\mathrm{\partial }W=M$, whose Lorentzian interpretation gives an allowed finite-energy $(d-k-1)$-dimensional domain wall/defect across which the $D=d-k$-dimensional spacetime ends; the allowed $W$ and defect class are part of the input. The McNamara–Vafa cobordism conjecture proposes $M⇝\mathrm{\varnothing }$ for every $M\in B_k$, with the motivating hypothesis that the full target-space theory has no exact ungauged conserved charges, including charges carried by extended defects. This hypothesis is adopted in the proposal, not proved by this item. For $k\ge 3$ this is the conservative range identified in their black-hole argument; retaining $k<3$ is explicitly the stronger conjectural extension under that same no-global-symmetry hypothesis. It is not an assertion about arbitrary low-dimensional quantum-gravity theories such as string-worldsheet gravity. The paper packages this proposal as vanishing quantum-gravity cobordism groups, but this item records only the pointed termination criterion and asserts no quotient-group structure. For comparison, ordinary smooth $G$-bordism to the empty manifold is the separate compact-witness condition defined below. A noncompact termination witness does not by itself establish that condition.

The boundary convention is outward-normal-first. “Allowed” means admitted by the specified quantum-gravity theory, including its structure, flux/charge data and boundary conditions. The finite-energy condition is evaluated in the Lorentzian defect interpretation in $D=d-k$ dimensions. For the smooth geometric comparison, specify groups $G_j$ with homomorphisms $G_j\to O(j)$ and stabilization maps compatible with adding a trivial normal line. A $G$-structure on a smooth $j$-manifold is a principal $G_j$-bundle together with an equivariant map to the orthonormal tangent-frame bundle inducing an isomorphism after extension of structure group to $O(j)$; fix the boundary restriction through the outward normal. Ordinary $G$-bordism to $\mathrm{\varnothing }$ means a compact smooth $(k+1)$-manifold $\bar{W}$ with this structure and a boundary identification with $M$ preserving the induced structure. To deduce this compact condition from a geometric termination witness, additionally require a compactification $\bar{W}$ with no extra boundary components and with the $G$-structure extending over it. No group operation or quotient relation is inferred from the pointed relation for the general non-geometric backgrounds.

Scalar, gauge and theory-family data for swampland conjectures

McNamara–Vafa, pp. 1–3 and 7–9

Status review dated 2025-09-05: the bounded record is McNamara–Vafa, PDF pp. 1–3 and 7–9; Agmon–Bedroya–Kang–Vafa, PDF pp. 162–177; and Lehnert, §II.B, PDF pp. 11–13. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8). McNamara–Vafa propose the termination/defect relation and express a version through quantum-gravity cobordism groups; their examples do not prove the criterion for every allowed background or theory. The present formulation retains its pointed relation and separate compact smooth $G$-bordism comparison without asserting a quotient-group structure. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search.

Step-5 source qualification: McNamara–Vafa, §1, printed pp. 2–3 (PDF pp. 3–4), explicitly state the no-global-symmetry motivation, a conservative $k\ge 3$ restriction, and their decision to adopt the stronger target-space string-landscape hypothesis for the remaining discussion. Section 2.1, printed pp. 7–8 (PDF pp. 8–9), gives the noncompact termination interpretation. This qualification preserves the small-$k$ research question with its hypothesis; it does not deduce it for worldsheet quantum gravity.