Positivity of the gravitational path integral implies the axionic weak gravity conjecture

TL;DR

This paper shows that the positivity of the gravitational path integral imposes a sharp constraint on axion theories, leading to a version of the weak gravity conjecture with specific numerical bounds.

Contribution

It derives a new, precise form of the axion weak gravity conjecture from positivity constraints of the gravitational path integral, linking it to wormhole stability.

Findings

Positivity constraints are violated by certain wormholes unless shift symmetry is broken.

Stable wormholes imply non-perturbative shift symmetry breaking in low-energy theories.

The derived bound relates to extensions of the swampland conjectures, especially the distance conjecture.

Abstract

The gravitational path integral can compute inner products between different states of open and closed universes. To have a well-defined Hilbert space, these inner products should be positive semi-definite, which is not manifest in the low-energy effective theory. In this letter, we analyze the constraints that the positivity of inner products imposes on gravitational theories coupled to axions. If the axion has an exact shift symmetry, we show that, under mild assumptions, a combined positivity constraint on closed and open universes is violated when one includes certain wormholes. In low-energy effective theories where these wormholes are perturbatively stable, positivity requires that the wormholes have a non-perturbative instability that breaks the shift symmetry. This leads to a sharp version of the axion weak gravity conjecture, including precise numerical constants. We relate the bound to possible extensions of other swampland conjectures, arguing for an imaginary continuation of the distance conjecture. We comment on how the bound applies to axions in string theory and discuss phenomenological implications.