Charge-completeness conjecture

Fix a compact topological internal gauge group $G$, including its global form. A finite-dimensional representation is a pair $(V,\rho )$ where $V$ is a finite-dimensional complex vector space and $\rho :G\to \mathrm{G}\mathrm{L}(V)$ is a continuous group homomorphism; it is irreducible if $V$ has no nonzero proper $G$-invariant complex subspace. For this state-spectrum version, use a charge sector with fixed asymptotic boundary conditions in which $G$ acts by a strongly continuous unitary representation $U:G\to \mathcal{U}({\mathcal{H}}_{\mathrm{d}\mathrm{y}\mathrm{n}})$ on the physical dynamical Hilbert space, with all required gauge dressing included. Supply the nonnegative self-adjoint energy operator $H$ relative to the stated vacuum; its spectral projections commute with $U$. Count the full spectrum, including multi-particle configurations, bound states, black holes and wrapped extended objects of finite total energy. Exclude nondynamical external probe charges and isolated unwrapped objects of infinite total energy. A representation is realized precisely when there is an injective linear map $j:V\to \mathcal{D}(H^{1\mathrm{/}2})\subseteq {\mathcal{H}}_{\mathrm{d}\mathrm{y}\mathrm{n}}$ satisfying $U(g)j=j\rho (g)$ for every $g\in G$; $\mathcal{D}(H^{1\mathrm{/}2})$ is the finite-energy quadratic-form domain, so $\mathrm{\parallel }{H}^{1\mathrm{/}2}j(v){\mathrm{\parallel }}^2<\mathrm{\infty }$. The charge-completeness conjecture says that every finite-dimensional irreducible representation of $G$ is realized. For compact $U(1)$ with character lattice $\mathbb{Z}$, this includes every integer-charge representation. The universal conjecture is stronger than the Harlow–Ooguri result for a specified anti-de Sitter/conformal-field-theory holographic pair under its boundary and duality assumptions.

Occurrence in a multi-particle spectrum counts in this formulation. The conjecture concerns every ultraviolet-complete quantum-gravity theory admitting the specified asymptotic charge-sector data; a confined phase without that action requires a different formulation. Completeness does not say the state is light, stable, elementary, or a single-particle asymptotic state. Here AdS/CFT denotes a specified anti-de Sitter quantum-gravity theory and conformal field theory related by the stated holographic duality, with its boundary conditions and duality hypotheses.

Scalar, gauge and theory-family data for swampland conjectures

Polchinski, pp. 1–5

Harlow–Ooguri, pp. 1–6

Status review dated 2025-09-05: the bounded record is Polchinski, PDF pp. 1–5; Harlow–Ooguri, PDF pp. 1–6; and Lehnert, §II.C, PDF pp. 13–14. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8). Polchinski presents completeness as a principle/conjecture. Harlow–Ooguri's conditional argument assumes their asymptotically AdS/CFT setting, a compact internal gauge group with long-range gauge symmetry, and their stated charged-object and holographic hypotheses; their argument then uses the faithful asymptotic action, a Wilson line through the two-sided AdS-Schwarzschild geometry, and representation theory to obtain the irreducible representations. It does not establish the universal assertion for nonholographic backgrounds or every extended-object convention. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search; it is not a logical supplier.