Charge-Lattice and Gauge-Completeness Conjectures and Tests

Charge completeness is not one statement. One must distinguish the lattice allowed by the gauge group’s global form, genuine line charges, dynamical particle charges, a finite-index populated sublattice, and charges sufficient to discharge black holes. Their equality is a conjectural property to test, not a definition.

Required background. Quantum-Gravity Consistency Claims and Comparison Contract fixes the quantifiers; Genuine Lines, Screening, and Charge Lattices supplies line and screening data.

Helpful background. Abelian Electric–Magnetic Dualization, Charge Lattices, and Global Form supplies duality conventions; Charge Lattices, Duality Frames, and Local Systems supplies moduli dependence.

Evidence cutoff: 25 July 2026.

Three lattices and a spectrum

For $U(1)^r$, electric and magnetic line charges lie in dual lattices

$$\Lambda_{\rm e}\times\Lambda_{\rm m},\qquad \langle q,p\rangle=q_Ip^I\in\mathbb Z .$$

The global form and chosen genuine lines determine these lattices. Dynamical particles populate a subset $\Gamma_{\rm dyn}\subseteq\Lambda_{\rm e}$; screening identifies line sectors modulo the subgroup they generate. “Full completeness” asserts $\Gamma_{\rm dyn}=\Lambda_{\rm e}$ in an appropriate sense, whereas sublattice completeness requires a finite-index $\Lambda_{\rm sub}\subseteq\Gamma_{\rm dyn}$. Black-hole discharge asks instead whether each sufficiently large charged black hole has an allowed decay chain.

First application: circle compactification

Compactify a five-dimensional theory on a circle of radius $R$. The Kaluza–Klein photon couples to momentum,

$$m_n=\frac{\lvert n\rvert}{R},\qquad q_n=n,\qquad n\in\mathbb Z .$$

The complete KK tower populates the full electric lattice generated by unit momentum. A KK monopole supplies the magnetic sector in the appropriate global description. The test has four separate outputs:

1. allowed line lattice $\mathbb Z$;
2. dynamical electric spectrum containing every $n$;
3. minimal charge $1$;
4. kinematic discharge channels, which additionally depend on masses and extremality.

This example supports completeness in a controlled construction but does not derive it for all compactifications. Banks and Seiberg relate charge completeness and absence of global symmetries in quantum gravity Banks and Seiberg 2011.

Adversarial control: change the global form

Replace a theory described locally by $SU(2)$ with global form $SO(3)=SU(2)/\mathbb Z_2$. The Lie algebra is unchanged, but allowed Wilson lines and electric charge lattices differ. Alternatively, include a charge-$k$ dynamical field: it screens lines only modulo $k$ and does not populate charges coprime to $k$. A claim inferred from the Lie algebra alone fails both tests.

Completeness must therefore report the global gauge group, genuine lines, electric–magnetic pairing, dynamical objects, masses, and EFT range. Existence of some charged state is weaker than full lattice completeness, and completeness alone does not imply a weak-gravity mass inequality.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Banks, Tom, and Nathan Seiberg. “Symmetries and Strings in Field Theory and Gravity.” Physical Review D 83, 084019 (2011). DOI. Open PDF.