Anti-de Sitter distance conjecture

Fix $d\ge 3$. Here a $d$-dimensional AdS spacetime is a Lorentzian manifold whose Riemann tensor has constant sectional curvature $-L_{\mathrm{A}\mathrm{d}\mathrm{S}}^{-2}<0$, with convention $R_{\mu \nu \rho \sigma }=-L_{\mathrm{A}\mathrm{d}\mathrm{S}}^{-2}(g_{\mu \rho }{g}_{\nu \sigma }-g_{\mu \sigma }{g}_{\nu \rho })$; its global quotient/topology and boundary conditions are additional family data. In particular, $R_{\mu \nu }=-(d-1)L_{\mathrm{A}\mathrm{d}\mathrm{S}}^{-2}{g}_{\mu \nu }$. In the convention $R_{\mu \nu }=2\mathrm{\Lambda }{g}_{\mu \nu }\mathrm{/}(d-2)$ this gives $\mathrm{\Lambda }=-(d-1)(d-2)\mathrm{/}(2L_{\mathrm{A}\mathrm{d}\mathrm{S}}^2)$. For a specified sequence or family of such $d$-dimensional AdS quantum-gravity vacua with $\mathrm{\Lambda }\to 0^{-}$, an infinite tower is conjectured to have characteristic mass $m_{\mathrm{t}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}\mathrm{/}{M}_{P,d}\sim (\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }\mathrm{/}{M}_{P,d}^2{)}^{\alpha }$ for fixed positive order-one $\alpha$; the strong version proposes $\alpha =1\mathrm{/}2$ for supersymmetric AdS vacua.

ADC abbreviates the AdS distance conjecture, and $M_{P,d}$ is the reduced $d$-dimensional Planck mass of the supplied swampland-claim datum. The parameter $\mathrm{\Lambda }$ has mass dimension two; hence both displayed ratios are dimensionless. The notation $\sim$ refers to asymptotic scaling along the specified family, and the tower species and characteristic gap must be identified. The limit is $\mathrm{\Lambda }\mathrm{/}{M}_{P,d}^2\to 0^{-}$, not a claim about each individual AdS vacuum. The strong $\alpha =1\mathrm{/}2$ version is restricted to the specified supersymmetric class. The separate no-scale-separation question concerns $M_{\mathrm{K}\mathrm{K}}{L}_{\mathrm{A}\mathrm{d}\mathrm{S}}$ and is not equivalent to this tower-mass statement.

Scalar, gauge and theory-family data for swampland conjectures

Lüst, Palti, and Vafa, pp. 1–5

Status review dated 2025-09-05: the bounded record is Lüst–Palti–Vafa, PDF pp. 1–5; Agmon–Bedroya–Kang–Vafa, PDF pp. 198–203; and Lehnert, §II.A, PDF pp. 6–8. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8). These sources present the tower scaling as conjectural, distinguish the strong supersymmetric exponent, and discuss examples and qualifications without proving the universal family statement. The separate no-scale-separation proposal is not used as proof. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search.