Weak-Gravity Conjectures and Their Variants

The weak-gravity conjecture (WGC) is a family of inequivalent statements. The electric, magnetic, convex-hull, sublattice, tower, scalar, and higher-form versions have different hypotheses and consequences. Every test must fix the gauge kinetic metric, charge lattice, long-range forces, extremal benchmark, and EFT domain.

Required background. Charge-Lattice and Gauge-Completeness Conjectures and Tests fixes allowed charges; Applying EFT Power Counting to Gravity fixes the cutoff.

Helpful background. Massless Exchange and Infrared Subtractions supplies force comparisons; D3-Branes and AdS5/CFT4: Parameter-Controlled Regimes and Evidence supplies a controlled top-down example.

Evidence cutoff: 25 July 2026.

Electric and convex-hull statements

With four-dimensional action

$$S=\int d^4x\sqrt{-g}\left[ \frac{M_{\rm Pl}^2}{2}R-\frac14(f^{-1})_{ab}F^a_{\mu\nu}F^{b\,\mu\nu} \right],$$

define each state’s charge-to-mass vector, in the chosen extremality normalization, by

$$\mathbf z_i=\frac{\sqrt2\,M_{\rm Pl}}{m_i}\,f^{1/2}\mathbf q_i .$$

For one $U(1)$ without massless scalar forces, the electric WGC asks for a state with $\lvert z\rvert\ge1$, so an extremal Reissner–Nordström black hole can decay. For several fields, the convex hull of $\{\pm\mathbf z_i\}$ must contain the unit ball. Scalar forces deform the extremality surface and must be included rather than hidden in an “order-one” factor.

First application: two U(1) fields

Take $f_{ab}=g^2\delta_{ab}$ and only two charged species,

$$\mathbf q_1=(1,0),\qquad \mathbf q_2=(0,1),\qquad m_1=m_2=m.$$

Their vectors have common length $z=\sqrt2\,gM_{\rm Pl}/m$. The convex hull is a diamond $\lvert x\rvert+\lvert y\rvert\le z$. Its nearest edge to the origin is at distance $z/\sqrt2$; it contains the unit disk only if

$$z\ge\sqrt2.$$

Thus satisfying the single-field inequality $z\ge1$ for each axis does not satisfy the two-charge convex-hull condition. Cheung and Remmen formulated this multi-field discharge criterion Cheung and Remmen 2014.

Other variants

The magnetic WGC estimates a cutoff $\Lambda\lesssim gM_{\rm Pl}$ by requiring monopoles not to be black holes. Sublattice and tower variants populate infinitely many charges, avoiding failures after circle reduction. Higher-form versions replace particles by charged branes and compare tension with charge. These strengthen or generalize the original proposal; none follows solely from existence of one superextremal particle Arkani-Hamed et al. 2007.

Adversarial control: move in moduli space

Vary a modulus so $f_{ab}$, scalar forces, and masses change. Integrate out a candidate state only if its mass exceeds the working cutoff, then retest the same variant in the remaining EFT. A rescaling of $A_\mu$ changes $g$ and integer charge oppositely; a conclusion that changes under that convention is invalid.

Current evidence is broad across string compactifications and black-hole arguments but does not make every WGC variant a theorem. A counterexample must target a fixed variant with its full domain; a model satisfying a weaker version does not establish a stronger one.

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

• Arkani-Hamed, Nima, Luboš Motl, Alberto Nicolis, and Cumrun Vafa. “The String Landscape, Black Holes and Gravity as the Weakest Force.” Journal of High Energy Physics 2007, 6 (2007): 060. DOI. Open PDF.
• Cheung, Clifford, and Grant N. Remmen. “Infrared Consistency and the Weak Gravity Conjecture.” Journal of High Energy Physics 2014, 12 (2014): 087. DOI. Open PDF.