# Gauge and gravitational instantons: From 3-forms and fermions to Weak Gravity and flat axion potentials

## TL;DR

This paper explores how gauge and gravitational instantons influence global symmetry breaking and axion potentials within quantum gravity, proposing bounds and analyzing the role of fermions and spacetime topology.

## Contribution

It provides a detailed description of instanton effects via 3-form gauge theories and conjectures a cutoff-dependent lower bound on axion potentials due to quantum gravity constraints.

## Key findings

- Decoupling of instantons explained through 3-form gauge theory.
- Fermionic operators induced by K3 instantons may be phenomenologically relevant.
- A conjectured lower bound on axion potential from quantum gravity considerations.

## Abstract

We investigate the role of gauge and gravitational instantons in the context of the Swampland program. Our focus is on the global symmetry breaking they induce, especially in the presence of fermions. We first recall and make more precise the description of the dilute instanton gas through a 3-form gauge theory. In this language, the familiar suppression of instanton effects by light fermions can be understood as the decoupling of the 3-form. Even if all fermions remain massive, such decoupling may occur on the basis of an explicitly unbroken but anomalous global symmetry in the fermionic sector. This should be forbidden by quantum gravity, which leads us to conjecture a related, cutoff-dependent lower bound on the induced axion potential. Finally, we note that the gravitational counterpart of the above are K3 instantons. These are small fluctuations of Euclidean spacetime with K3 topology, which induce fermionic operators analogous to the 't Hooft vertex in gauge theories. Although Planck-suppressed, they may be phenomenologically relevant if accompanied by other higher-dimension fermion operators or if the K3 carries appropriate gauge fluxes.