# Infinitely Many M2-instanton Corrections to M-theory on $G_2$-manifolds

## TL;DR

This paper demonstrates that M-theory compactified on certain $G_2$-manifolds receives infinitely many M2-instanton contributions to its superpotential, linked through dualities to known infinite instanton effects in other string theories.

## Contribution

It reveals the existence of infinitely many M2-instanton corrections in M-theory on twisted connected sum $G_2$-manifolds, a novel result in understanding non-perturbative effects.

## Key findings

- Infinite M2-instanton contributions from three-spheres.
- Duality chain connects these contributions to known F-theory instantons.
- Conjecture that these cycles are supersymmetric (associative).

## Abstract

We consider the non-perturbative superpotential for a class of four-dimensional $\mathcal N=1$ vacua obtained from M-theory on seven-manifolds with holonomy $G_2$. The class of $G_2$-holonomy manifolds we consider are so-called twisted connected sum (TCS) constructions, which have the topology of a K3-fibration over $S^3$. We show that the non-perturbative superpotential of M-theory on a class of TCS geometries receives infinitely many inequivalent M2-instanton contributions from infinitely many three-spheres, which we conjecture are supersymmetric (and thus associative) cycles. The rationale for our construction is provided by the duality chain of arXiv:1708.07215, which relates M-theory on TCS $G_2$-manifolds to $E_8\times E_8$ heterotic backgrounds on the Schoen Calabi-Yau threefold, as well as to F-theory on a K3-fibered Calabi-Yau fourfold. The latter are known to have an infinite number of instanton corrections to the superpotential and it is these contributions that we trace through the duality chain back to the $G_2$-compactification.