Single-axion weak-gravity bound for instantons

Use natural units $\mathrm{\hslash }=c=1$ and the four-dimensional Einstein-frame convention $M_P^{-2}=8\pi G_N$. A single axion is a smooth field $\theta :\mathcal{X}\to \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ with kinetic term $-f^2{g}^{\mu \nu }{\mathrm{\partial }}_{\mu }\theta {\mathrm{\partial }}_{\nu }\theta \mathrm{/}2$, where $f>0$ is constant on the declared background patch and $a=f\theta$ is a local canonical field. Fix a proposed ultraviolet-complete quantum-gravity theory, its Euclidean boundary conditions, and the full set $\mathcal{I}$ of its charged instanton sectors: localized nonperturbative Euclidean sources with nonzero amplitudes (possibly with operator insertions), finite real vacuum-subtracted actions $S_i\ge 0$, and angular dependence $e^{i{n}_i\theta }$ with $n_i\in \mathbb{Z}\setminus \{0\}$. The charge, boundary conditions, action prescription, and fluctuation/zero-mode treatment are supplied for each $i\in \mathcal{I}$. An arbitrary selected subset of known saddles is not the full set $\mathcal{I}$.

The weak single-axion conjecture asks whether there is at least one $i\in \mathcal{I}$ satisfying $f{S}_i\mathrm{/}\mathrm{\mid }{n}_i\mathrm{\mid }\le c_0{M}_P$. Fix here the macroscopic-semiwormhole reference normalization $c_0:=\sqrt{6}\pi \mathrm{/}4$ for the four-dimensional axion-gravity system with no varying dilaton; this is a definite proposed version, rather than an adjustable constant fitted to a chosen instanton. A dilaton-dependent bound, other reference normalizations, and multi-axion lattice or convex-hull versions are separate questions.

Here $f$ and $M_P$ have mass dimension one; $S_i,n_i,c_0$ are dimensionless. There is no requirement that the satisfying sector dominate the potential or have unit charge. If a sector has no unsaturated zero modes and produces a scalar-potential harmonic, its term can be written ${\mathrm{\Lambda }}_i^4{e}^{-S_i}[1-\cos (n_i\theta +{\delta }_i)]$ with supplied ${\mathrm{\Lambda }}_i\ge 0$ of mass dimension one and dimensionless ${\delta }_i$; a charged instanton need not generate such a bosonic term. A semiclassical evaluation of $S_i$ requires its own controlled saddle expansion and corrections. In particular, $S_i\gg 1$ is not an extra requirement on the conjectured sector, and no instanton or ultraviolet completion follows from periodicity alone.

Periodic axion EFT data and sources

Scalar, gauge and theory-family data for swampland conjectures

Andriolo et al., Eq. (1.1), printed p. 2

Arkani-Hamed et al., pp. 11–12

Marsh, §2.4 and §§4.1–4.3

Status review dated 2025-09-05: the bounded record is Arkani-Hamed–Motl–Nicolis–Vafa, PDF pp. 11–12; Palti, PDF pp. 28–31; Lehnert, §II.S, PDF pp. 95–96 and its axion-compatibility note at PDF p. 15, note 54; and Marsh, §2.4 and §§4.1–4.3. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8). These sources give conventions, conjectural bounds and model-dependent examples, not a proof of the universal single- or multi-axion statement. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search. The referenced axion definition supplies the fixed-$f$ convention used here; this open problem supplies no proof premise.

Selective status supplement dated 2026-11-05: Di Ubaldo–Iliesiu–Lin–Yan, arXiv:2605.05305v1, §§II–V, give a conditional argument from path-integral positivity, contributing perturbatively stable axion wormholes, analytic continuation of boundary axion values, suppression of higher-boundary connected contributions, and an instanton-induced instability. Their §IV also identifies exclusion of those wormholes as a logical alternative. Section V says higher-boundary suppression is difficult to check and does not prove that instanton proliferation restores positivity. Their background-dependent constants do not establish the universal fixed-normalization conjecture above for every ultraviolet completion. This selective supplement assesses that argument and is not an exhaustive later-literature review.