# Current progress on $G_2$--instantons over twisted connected sums

## TL;DR

This paper reviews a method for constructing $
m{G}_2$--instantons on compact $
m{G}_2$--manifolds formed via twisted connected sums of Calabi-Yau 3-folds, using holomorphic bundles and explicit examples from semi-Fano 3-folds.

## Contribution

It introduces a gluing construction for $
m{G}_2$--instantons on twisted connected sum manifolds, with explicit examples derived from semi-Fano 3-folds using an algorithmic approach.

## Key findings

- Construction of $
m{G}_2$--instantons via gluing methods.
- Explicit examples from semi-Fano 3-folds.
- Algorithmic procedure based on Hartshorne-Serre correspondence.

## Abstract

We review a method to construct $\rm{G}_2$--instantons over compact $\rm{G}_2$--manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau $3$-folds with cylindrical end, based on the series of articles [SE15, SEW15, JMPSE17, MNSE17] by the author and others. The construction is based on gluing $\rm{G}_2$--instantons obtained from holomorphic bundles over such building blocks, subject to natural compatibility and transversality conditions. Explicit examples are obtained from matching pairs of semi-Fano $3$-folds by an algorithmic procedure based on the Hartshorne-Serre correspondence.