# Structure of Gauge-Invariant Lagrangians

## TL;DR

This paper proves the existence of a finite generating set of gauge-invariant Lagrangians for gauge theories, enabling any such Lagrangian to be expressed as a smooth function of these generators.

## Contribution

It establishes the finite generation of gauge-invariant Lagrangians and provides explicit examples, advancing the mathematical understanding of gauge field theories.

## Key findings

- Existence of finite generating set of gauge-invariant Lagrangians
- Any gauge-invariant Lagrangian can be expressed as a smooth function of generators
- Explicit examples illustrating the theory

## Abstract

The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below we tackle one of these problems: The existence of a finite system of generators of gauge-invariant Lagrangians and how to compute them. More precisely, if $p\colon C\to M$ is the bundle of connections on a principal $G$-bundle $\pi\colon P\to M$, then a finite number $L_1,\dotsc,L_{N^\prime }$ of gauge-invariant Lagrangians defined on $J^1C$ is proved to exist such that for any other gauge-invariant Lagrangian $L\in C^\infty (J^1C)$ there exists a function $F\in C^\infty (\mathbb{R}^{N^\prime })$ such that $L=F(L_1,\dotsc,L_{N^\prime})$. Several examples are dealt with explicitly.