# Gauge modules for the Lie algebras of vector fields on affine varieties

## TL;DR

This paper investigates gauge modules over the Lie algebra of vector fields on affine varieties, establishing conditions for their irreducibility and linking them to de Rham complexes.

## Contribution

It introduces a class of gauge modules with compatible actions and characterizes their irreducibility based on simple ngle9 modules, connecting to de Rham complexes.

## Key findings

- Gauge modules are irreducible unless associated with the de Rham complex.
- A correspondence between simple e9gle9 modules and irreducibility is established.
- The study advances understanding of module structures over Lie algebras of vector fields.

## Abstract

For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge module corresponding to a simple $\mathfrak{gl}_N$-module is irreducible as a module over the Lie algebra of vector fields unless it appears in the de Rham complex.