# Triply Periodic Monopoles and Difference Modules on Elliptic Curves

## TL;DR

This paper explores the deep mathematical correspondences between twisted monopoles, mini-holomorphic bundles, and difference modules on elliptic curves, revealing their equivalences on a 3-torus.

## Contribution

It establishes new equivalences between twisted monopoles, holomorphic bundles, and difference modules on elliptic curves and tori.

## Key findings

- Twisted monopoles correspond to polystable twisted mini-holomorphic bundles.
- These bundles are equivalent to polystable parabolic twisted difference modules.
- The work clarifies the relationships among geometric and algebraic structures on elliptic curves.

## Abstract

We explain the correspondences between twisted monopoles with Dirac type singularity and polystable twisted mini-holomorphic bundles with Dirac type singularity on a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic twisted difference modules on elliptic curves.